id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-18000 | Metric formed by Arc Length on Circle is Lipschitz Equivalent to Euclidean Metric | Let $A \subseteq \R^2$ be the set defined as:
:$A = \set {\tuple {x_1, x_2}: x_1^2 + y_2^2 = 1}$
Thus from Equation of Unit Circle, $A$ is the unit circle embedded in the Cartesian plane.
Let $d: A^2 \to \R$ be the metric defined as:
:$\forall \tuple {x, y} \in A^2: \map d {x, y} = \begin {cases} 0 & : x = y \\ \pi & :... | That $d$ forms a metric is demonstrated in Arc Length on Circle forms Metric.
Let $p_1 = \tuple {x_1, y_1}$ and $p_2 = \tuple {x_2, y_2}$ be points in $A$.
We have that $\map {d_2} {p_1, p_2}$ is the length of a line segment between $p_1$ and $p_2$.
This can never be longer than the length of the arc between $p_1$ and ... | Let $A \subseteq \R^2$ be the [[Definition:Set|set]] defined as:
:$A = \set {\tuple {x_1, x_2}: x_1^2 + y_2^2 = 1}$
Thus from [[Equation of Unit Circle]], $A$ is the [[Definition:Unit Circle|unit circle]] embedded in the [[Definition:Cartesian Plane|Cartesian plane]].
Let $d: A^2 \to \R$ be the [[Definition:Metric|met... | That $d$ forms a [[Definition:Metric|metric]] is demonstrated in [[Arc Length on Circle forms Metric]].
Let $p_1 = \tuple {x_1, y_1}$ and $p_2 = \tuple {x_2, y_2}$ be [[Definition:Point|points]] in $A$.
We have that $\map {d_2} {p_1, p_2}$ is the [[Definition:Length of Line|length]] of a [[Definition:Line Segment|lin... | Metric formed by Arc Length on Circle is Lipschitz Equivalent to Euclidean Metric | https://proofwiki.org/wiki/Metric_formed_by_Arc_Length_on_Circle_is_Lipschitz_Equivalent_to_Euclidean_Metric | https://proofwiki.org/wiki/Metric_formed_by_Arc_Length_on_Circle_is_Lipschitz_Equivalent_to_Euclidean_Metric | [
"Examples of Metric Spaces",
"Lipschitz Equivalence"
] | [
"Definition:Set",
"Equation of Unit Circle",
"Definition:Unit Circle",
"Definition:Cartesian Plane",
"Definition:Metric Space/Metric",
"Definition:Arc Length",
"Definition:Circle/Arc/Minor",
"Definition:Lipschitz Equivalence/Metrics",
"Definition:Euclidean Metric/Real Number Plane"
] | [
"Definition:Metric Space/Metric",
"Arc Length on Circle forms Metric",
"Definition:Point",
"Definition:Linear Measure/Length",
"Definition:Line/Segment",
"Definition:Arc Length",
"Definition:Circle/Arc",
"Definition:Circle/Diameter",
"Definition:Lipschitz Equivalence/Metrics"
] |
proofwiki-18001 | Standard Bounded Metric is Metric/Topological Equivalence | $\bar d$ is topologically equivalent to $d$. | That $\bar d$ forms a metric on $M$ is demonstrated in Standard Bounded Metric is Metric.
We have that:
:$\forall x, y \in A^2: \map {\bar d} {x, y} \le \map d {x, y}$
Hence:
:$\map {B_\epsilon} {x; d} \subseteq \map {B_\epsilon} {x; \bar d}$
where $\map {B_\epsilon} {x; d}$ denotes the open $\epsilon$-ball of $x$ in $... | $\bar d$ is [[Definition:Topologically Equivalent Metrics|topologically equivalent]] to $d$. | That $\bar d$ forms a [[Definition:Metric|metric]] on $M$ is demonstrated in [[Standard Bounded Metric is Metric]].
We have that:
:$\forall x, y \in A^2: \map {\bar d} {x, y} \le \map d {x, y}$
Hence:
:$\map {B_\epsilon} {x; d} \subseteq \map {B_\epsilon} {x; \bar d}$
where $\map {B_\epsilon} {x; d}$ denotes the [[... | Standard Bounded Metric is Metric/Topological Equivalence | https://proofwiki.org/wiki/Standard_Bounded_Metric_is_Metric/Topological_Equivalence | https://proofwiki.org/wiki/Standard_Bounded_Metric_is_Metric/Topological_Equivalence | [
"Standard Bounded Metric is Metric"
] | [
"Definition:Topologically Equivalent Metrics"
] | [
"Definition:Metric Space/Metric",
"Standard Bounded Metric is Metric",
"Definition:Open Ball",
"Definition:Open Set/Metric Space",
"Definition:Open Set/Metric Space",
"Definition:Subset",
"Definition:Open Set/Metric Space",
"Definition:Open Set/Metric Space"
] |
proofwiki-18002 | Metric over 1 plus Metric forms Metric/Topological Equivalence | $d_3$ is topologically equivalent to $d$. | That $d_3$ forms a metric on $M$ is demonstrated in Metric over 1 plus Metric forms Metric.
We have that:
:$\forall x, y \in A^2: \map {d_3} {x, y} \le \map d {x, y}$
Hence:
:$\map {B_\epsilon} {x; d} \subseteq \map {B_\epsilon} {x; d_3}$
where $\map {B_\epsilon} {x; d}$ denotes the open $\epsilon$-ball of $x$ in $\str... | $d_3$ is [[Definition:Topologically Equivalent Metrics|topologically equivalent]] to $d$. | That $d_3$ forms a [[Definition:Metric|metric]] on $M$ is demonstrated in [[Metric over 1 plus Metric forms Metric]].
We have that:
:$\forall x, y \in A^2: \map {d_3} {x, y} \le \map d {x, y}$
Hence:
:$\map {B_\epsilon} {x; d} \subseteq \map {B_\epsilon} {x; d_3}$
where $\map {B_\epsilon} {x; d}$ denotes the [[Defi... | Metric over 1 plus Metric forms Metric/Topological Equivalence | https://proofwiki.org/wiki/Metric_over_1_plus_Metric_forms_Metric/Topological_Equivalence | https://proofwiki.org/wiki/Metric_over_1_plus_Metric_forms_Metric/Topological_Equivalence | [
"Examples of Metric Spaces"
] | [
"Definition:Topologically Equivalent Metrics"
] | [
"Definition:Metric Space/Metric",
"Metric over 1 plus Metric forms Metric",
"Definition:Open Ball",
"Definition:Open Set/Metric Space",
"Definition:Open Set/Metric Space",
"Definition:Subset",
"Definition:Open Set/Metric Space",
"Definition:Open Set/Metric Space",
"Definition:Strictly Increasing/Rea... |
proofwiki-18003 | Euclidean Metric on Real Number Space is Translation Invariant | Let $\tau_{\mathbf t}: \R^n \to \R^n$ denote the translation of the real Euclidean space of $n$ dimensions by the vector $\mathbf t = \tuple {t_1, t_2, \ldots, t_n}$.
Let $d_2$ denote the Euclidean metric on $\R^n$.
Then $d_2$ is unchanged by application of $\tau$:
:$\forall \mathbf x, \mathbf y \in \R^n: \map {d_2} {\... | Let $\mathbf x = \tuple {x_1, x_2, \ldots, x_n}$ and $\mathbf y = \tuple {y_1, y_2, \ldots, y_n}$ be arbitrary points in $\R^n$.
Then:
{{begin-eqn}}
{{eqn | l = \map {d_2} {\map \tau {\mathbf x}, \map \tau {\mathbf y} }
| r = \map {d_2} {\mathbf x - \mathbf t, \mathbf y - \mathbf t}
| c = {{Defof|Translatio... | Let $\tau_{\mathbf t}: \R^n \to \R^n$ denote the [[Definition:Translation in Euclidean Space|translation]] of the [[Definition:Real Euclidean Space|real Euclidean space]] of $n$ [[Definition:Dimension of Vector Space|dimensions]] by the [[Definition:Vector|vector]] $\mathbf t = \tuple {t_1, t_2, \ldots, t_n}$.
Let $d_... | Let $\mathbf x = \tuple {x_1, x_2, \ldots, x_n}$ and $\mathbf y = \tuple {y_1, y_2, \ldots, y_n}$ be arbitrary [[Definition:Point|points]] in $\R^n$.
Then:
{{begin-eqn}}
{{eqn | l = \map {d_2} {\map \tau {\mathbf x}, \map \tau {\mathbf y} }
| r = \map {d_2} {\mathbf x - \mathbf t, \mathbf y - \mathbf t}
|... | Euclidean Metric on Real Number Space is Translation Invariant | https://proofwiki.org/wiki/Euclidean_Metric_on_Real_Number_Space_is_Translation_Invariant | https://proofwiki.org/wiki/Euclidean_Metric_on_Real_Number_Space_is_Translation_Invariant | [
"Translation Mappings",
"Euclidean Metric"
] | [
"Definition:Translation Mapping/Euclidean Space",
"Definition:Euclidean Space/Real",
"Definition:Dimension of Vector Space",
"Definition:Vector",
"Definition:Euclidean Metric/Real Vector Space"
] | [
"Definition:Point"
] |
proofwiki-18004 | Euclidean Metric on Real Number Plane is Rotation Invariant | Let $r_\alpha: \R^2 \to \R^2$ denote the rotation of the Euclidean plane about the origin through an angle of $\alpha$.
Let $d_2$ denote the Euclidean metric on $\R^2$.
Then $d_2$ is unchanged by application of $r_\alpha$:
:$\forall x, y \in \R^2: \map {d_2} {\map {r_\alpha} x, \map {r_\alpha} y} = \map {d_2} {x, y}$ | Let $x = \tuple {x_1, x_2}$ and $y = \tuple {y_1, y_2}$ be arbitrary points in $\R^2$.
Note that $\paren {\map {d_2} {x, y} }^2$ can be expressed as:
:$\paren {\map {d_2} {x, y} }^2 = \paren {\mathbf x - \mathbf y}^\intercal \paren {\mathbf x - \mathbf y}$
where:
:$x$ and $y$ are expressed in vector form: $\mathbf x = ... | Let $r_\alpha: \R^2 \to \R^2$ denote the [[Definition:Plane Rotation|rotation]] of the [[Definition:Euclidean Plane|Euclidean plane]] about the [[Definition:Origin|origin]] through an [[Definition:Angle|angle]] of $\alpha$.
Let $d_2$ denote the [[Definition:Euclidean Metric on Real Number Plane|Euclidean metric]] on $... | Let $x = \tuple {x_1, x_2}$ and $y = \tuple {y_1, y_2}$ be arbitrary [[Definition:Point|points]] in $\R^2$.
Note that $\paren {\map {d_2} {x, y} }^2$ can be expressed as:
:$\paren {\map {d_2} {x, y} }^2 = \paren {\mathbf x - \mathbf y}^\intercal \paren {\mathbf x - \mathbf y}$
where:
:$x$ and $y$ are expressed in [[De... | Euclidean Metric on Real Number Plane is Rotation Invariant | https://proofwiki.org/wiki/Euclidean_Metric_on_Real_Number_Plane_is_Rotation_Invariant | https://proofwiki.org/wiki/Euclidean_Metric_on_Real_Number_Plane_is_Rotation_Invariant | [
"Geometric Rotations",
"Euclidean Metric"
] | [
"Definition:Rotation (Geometry)/Plane",
"Definition:Euclidean Plane",
"Definition:Coordinate System/Origin",
"Definition:Angle",
"Definition:Euclidean Metric/Real Number Plane"
] | [
"Definition:Point",
"Definition:Vector",
"Definition:Transpose of Matrix",
"Matrix Form of Plane Rotation",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-18005 | Taxicab Metric on Real Number Plane is Translation Invariant | Let $\tau_{\mathbf t}: \R^2 \to \R^2$ denote the translation of the Euclidean plane by the vector $\mathbf t = \begin {pmatrix} a \\ b \end {pmatrix}$.
Let $d_1$ denote the taxicab metric on $\R^2$.
Then $d_1$ is unchanged by application of $\tau$:
:$\forall x, y \in \R^2: \map {d_1} {\map \tau x, \map \tau y} = \map {... | Let $x = \tuple {x_1, x_2}$ and $y = \tuple {y_1, y_2}$ be arbitrary points in $\R^2$.
Then:
{{begin-eqn}}
{{eqn | l = \map {d_1} {\map \tau x, \map \tau y}
| r = \map {d_1} {x - \mathbf t, y - \mathbf t}
| c = {{Defof|Translation in Euclidean Space}}
}}
{{eqn | r = \size {\paren {x_1 - a} - \paren {y_1 - a... | Let $\tau_{\mathbf t}: \R^2 \to \R^2$ denote the [[Definition:Translation in Euclidean Space|translation]] of the [[Definition:Euclidean Plane|Euclidean plane]] by the [[Definition:Vector|vector]] $\mathbf t = \begin {pmatrix} a \\ b \end {pmatrix}$.
Let $d_1$ denote the [[Definition:Taxicab Metric on Real Number Plan... | Let $x = \tuple {x_1, x_2}$ and $y = \tuple {y_1, y_2}$ be arbitrary [[Definition:Point|points]] in $\R^2$.
Then:
{{begin-eqn}}
{{eqn | l = \map {d_1} {\map \tau x, \map \tau y}
| r = \map {d_1} {x - \mathbf t, y - \mathbf t}
| c = {{Defof|Translation in Euclidean Space}}
}}
{{eqn | r = \size {\paren {x_1... | Taxicab Metric on Real Number Plane is Translation Invariant | https://proofwiki.org/wiki/Taxicab_Metric_on_Real_Number_Plane_is_Translation_Invariant | https://proofwiki.org/wiki/Taxicab_Metric_on_Real_Number_Plane_is_Translation_Invariant | [
"Translation Mappings",
"Taxicab Metric"
] | [
"Definition:Translation Mapping/Euclidean Space",
"Definition:Euclidean Plane",
"Definition:Vector",
"Definition:Taxicab Metric/Real Number Plane"
] | [
"Definition:Point"
] |
proofwiki-18006 | Chebyshev Distance on Real Number Plane is Translation Invariant | Let $\tau_{\mathbf t}: \R^2 \to \R^2$ denote the translation of the Euclidean plane by the vector $\mathbf t = \begin {pmatrix} a \\ b \end {pmatrix}$.
Let $d_\infty$ denote the Chebyshev distance on $\R^2$.
Then $d_1$ is unchanged by application of $\tau$:
:$\forall x, y \in \R^2: \map {d_\infty} {\map \tau x, \map \t... | Let $x = \tuple {x_1, x_2}$ and $y = \tuple {y_1, y_2}$ be arbitrary points in $\R^2$.
Then:
{{begin-eqn}}
{{eqn | l = \map {d_\infty} {\map \tau x, \map \tau y}
| r = \map {d_\infty} {x - \mathbf t, y - \mathbf t}
| c = {{Defof|Translation in Euclidean Space}}
}}
{{eqn | r = \max \set {\size {\paren {x_1 -... | Let $\tau_{\mathbf t}: \R^2 \to \R^2$ denote the [[Definition:Translation in Euclidean Space|translation]] of the [[Definition:Euclidean Plane|Euclidean plane]] by the [[Definition:Vector|vector]] $\mathbf t = \begin {pmatrix} a \\ b \end {pmatrix}$.
Let $d_\infty$ denote the [[Definition:Chebyshev Distance on Real Nu... | Let $x = \tuple {x_1, x_2}$ and $y = \tuple {y_1, y_2}$ be arbitrary [[Definition:Point|points]] in $\R^2$.
Then:
{{begin-eqn}}
{{eqn | l = \map {d_\infty} {\map \tau x, \map \tau y}
| r = \map {d_\infty} {x - \mathbf t, y - \mathbf t}
| c = {{Defof|Translation in Euclidean Space}}
}}
{{eqn | r = \max \se... | Chebyshev Distance on Real Number Plane is Translation Invariant | https://proofwiki.org/wiki/Chebyshev_Distance_on_Real_Number_Plane_is_Translation_Invariant | https://proofwiki.org/wiki/Chebyshev_Distance_on_Real_Number_Plane_is_Translation_Invariant | [
"Translation Mappings",
"Chebyshev Distance"
] | [
"Definition:Translation Mapping/Euclidean Space",
"Definition:Euclidean Plane",
"Definition:Vector",
"Definition:Chebyshev Distance/Real Number Plane"
] | [
"Definition:Point"
] |
proofwiki-18007 | Taxicab Metric on Real Number Plane is not Rotation Invariant | Let $r_\alpha: \R^2 \to \R^2$ denote the rotation of the Euclidean plane about the origin through an angle of $\alpha$.
Let $d_1$ denote the taxicab metric on $\R^2$.
Then it is not necessarily the case that:
:$\forall x, y \in \R^2: \map {d_1} {\map {r_\alpha} x, \map {r_\alpha} y} = \map {d_1} {x, y}$ | ;Proof by Counterexample:
Let $x = \tuple {0, 0}$ and $y = \tuple {0, 1}$ be arbitrary points in $\R^2$.
Then:
{{begin-eqn}}
{{eqn | l = \map {d_1} {x, y}
| r = \map {d_1} {\tuple {0, 0}, \tuple {0, 1} }
| c = Definition of $x$ and $y$
}}
{{eqn | r = \size {0 - 0} + \size {0 - 1}
| c = {{Defof|Taxicab... | Let $r_\alpha: \R^2 \to \R^2$ denote the [[Definition:Plane Rotation|rotation]] of the [[Definition:Euclidean Plane|Euclidean plane]] about the [[Definition:Origin|origin]] through an [[Definition:Angle|angle]] of $\alpha$.
Let $d_1$ denote the [[Definition:Taxicab Metric on Real Number Plane|taxicab metric]] on $\R^2... | ;[[Proof by Counterexample]]:
Let $x = \tuple {0, 0}$ and $y = \tuple {0, 1}$ be arbitrary [[Definition:Point|points]] in $\R^2$.
Then:
{{begin-eqn}}
{{eqn | l = \map {d_1} {x, y}
| r = \map {d_1} {\tuple {0, 0}, \tuple {0, 1} }
| c = Definition of $x$ and $y$
}}
{{eqn | r = \size {0 - 0} + \size {0 - 1}... | Taxicab Metric on Real Number Plane is not Rotation Invariant | https://proofwiki.org/wiki/Taxicab_Metric_on_Real_Number_Plane_is_not_Rotation_Invariant | https://proofwiki.org/wiki/Taxicab_Metric_on_Real_Number_Plane_is_not_Rotation_Invariant | [
"Geometric Rotations",
"Taxicab Metric"
] | [
"Definition:Rotation (Geometry)/Plane",
"Definition:Euclidean Plane",
"Definition:Coordinate System/Origin",
"Definition:Angle",
"Definition:Taxicab Metric/Real Number Plane"
] | [
"Proof by Counterexample",
"Definition:Point"
] |
proofwiki-18008 | Chebyshev Distance on Real Number Plane is not Rotation Invariant | Let $r_\alpha: \R^2 \to \R^2$ denote the rotation of the Euclidean plane about the origin through an angle of $\alpha$.
Let $d_\infty$ denote the Chebyshev distance on $\R^2$.
Then it is not necessarily the case that:
:$\forall x, y \in \R^2: \map {d_\infty} {\map {r_\alpha} x, \map {r_\alpha} y} = \map {d_\infty} {x, ... | ;Proof by Counterexample:
Let $x = \tuple {0, 0}$ and $y = \tuple {1, 1}$ be arbitrary points in $\R^2$.
Then:
{{begin-eqn}}
{{eqn | l = \map {d_\infty} {x, y}
| r = \map {d_\infty} {\tuple {0, 0}, \tuple {1, 1} }
| c = Definition of $x$ and $y$
}}
{{eqn | r = \max \set {\size {0 - 1}, \size {0 - 1} }
... | Let $r_\alpha: \R^2 \to \R^2$ denote the [[Definition:Plane Rotation|rotation]] of the [[Definition:Euclidean Plane|Euclidean plane]] about the [[Definition:Origin|origin]] through an [[Definition:Angle|angle]] of $\alpha$.
Let $d_\infty$ denote the [[Definition:Chebyshev Distance on Real Number Plane|Chebyshev distan... | ;[[Proof by Counterexample]]:
Let $x = \tuple {0, 0}$ and $y = \tuple {1, 1}$ be arbitrary [[Definition:Point|points]] in $\R^2$.
Then:
{{begin-eqn}}
{{eqn | l = \map {d_\infty} {x, y}
| r = \map {d_\infty} {\tuple {0, 0}, \tuple {1, 1} }
| c = Definition of $x$ and $y$
}}
{{eqn | r = \max \set {\size {0... | Chebyshev Distance on Real Number Plane is not Rotation Invariant | https://proofwiki.org/wiki/Chebyshev_Distance_on_Real_Number_Plane_is_not_Rotation_Invariant | https://proofwiki.org/wiki/Chebyshev_Distance_on_Real_Number_Plane_is_not_Rotation_Invariant | [
"Chebyshev Distance",
"Geometric Rotations"
] | [
"Definition:Rotation (Geometry)/Plane",
"Definition:Euclidean Plane",
"Definition:Coordinate System/Origin",
"Definition:Angle",
"Definition:Chebyshev Distance/Real Number Plane"
] | [
"Proof by Counterexample",
"Definition:Point"
] |
proofwiki-18009 | Open Balls of Supremum Metric on Continuous Real Functions on Closed Interval | Let $\mathscr C \closedint a b$ be the set of all continuous functions $f: \closedint a b \to \R$.
Let $d: \mathscr C^2 \to \R$ be the supremum metric on $\mathscr C \closedint a b$ defined as:
:$\ds \forall f, g \in \mathscr C \closedint a b: \map d {f, g} := \sup_{x \mathop \in \closedint a b} \size {\map f x - \map ... | Recall the definition of open ball:
The '''open $\epsilon$-ball of $a$ in $M = \struct {A, d}$''' is defined as:
:$\map {B_\epsilon} a := \set {x \in A: \map d {x, a} < \epsilon}$
In this context, the open $\epsilon$-ball of $\phi$ in $\mathscr C \closedint a b$ is defined as:
:$\ds \map {B_\epsilon} \phi := \set {\rho... | Let $\mathscr C \closedint a b$ be the [[Definition:Set|set]] of all [[Definition:Continuous Real Function on Closed Interval|continuous functions]] $f: \closedint a b \to \R$.
Let $d: \mathscr C^2 \to \R$ be the [[Definition:Supremum Metric on Bounded Real Functions on Closed Interval|supremum metric]] on $\mathscr C... | Recall the definition of [[Definition:Open Ball of Metric Space|open ball]]:
The '''[[Definition:Open Ball of Metric Space|open $\epsilon$-ball of $a$ in $M = \struct {A, d}$]]''' is defined as:
:$\map {B_\epsilon} a := \set {x \in A: \map d {x, a} < \epsilon}$
In this context, the [[Definition:Open Ball of Metric ... | Open Balls of Supremum Metric on Continuous Real Functions on Closed Interval | https://proofwiki.org/wiki/Open_Balls_of_Supremum_Metric_on_Continuous_Real_Functions_on_Closed_Interval | https://proofwiki.org/wiki/Open_Balls_of_Supremum_Metric_on_Continuous_Real_Functions_on_Closed_Interval | [
"Supremum Metric",
"Open Balls"
] | [
"Definition:Set",
"Definition:Continuous Real Function/Closed Interval",
"Definition:Supremum Metric/Bounded Real Functions on Interval",
"Definition:Supremum of Mapping/Real-Valued Function",
"Definition:Set",
"Definition:Open Ball",
"Definition:Constant"
] | [
"Definition:Open Ball",
"Definition:Open Ball",
"Definition:Open Ball",
"Definition:Open Ball",
"Definition:Open Ball",
"Definition:Open Ball"
] |
proofwiki-18010 | Equality of Open Balls does not imply Equality of Centers | Let $M = \struct {A, d}$ be a metric space.
Let $x, y \in A$ and $r, s \in \R$ such that:
:$\map {B_r} x = \map {B_s} y$
Then it is not necessarily the case that their centers $x$ and $y$ are equal. | Let $A$ be arbitrary.
Let $d$ be the (standard) discrete metric on $A$.
Let $r \ge 1$ and $s \ge 1$.
Then from Open Ball in Standard Discrete Metric Space:
:$\forall x, y \in A: \map {B_r} x = \map {B_s} y = A$
whether $x = y$ or not.
{{qed}} | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $x, y \in A$ and $r, s \in \R$ such that:
:$\map {B_r} x = \map {B_s} y$
Then it is not necessarily the case that their [[Definition:Center of Open Ball|centers]] $x$ and $y$ are equal. | Let $A$ be arbitrary.
Let $d$ be the [[Definition:Standard Discrete Metric|(standard) discrete metric]] on $A$.
Let $r \ge 1$ and $s \ge 1$.
Then from [[Open Ball in Standard Discrete Metric Space]]:
:$\forall x, y \in A: \map {B_r} x = \map {B_s} y = A$
whether $x = y$ or not.
{{qed}} | Equality of Open Balls does not imply Equality of Centers | https://proofwiki.org/wiki/Equality_of_Open_Balls_does_not_imply_Equality_of_Centers | https://proofwiki.org/wiki/Equality_of_Open_Balls_does_not_imply_Equality_of_Centers | [
"Open Balls"
] | [
"Definition:Metric Space",
"Definition:Open Ball/Center"
] | [
"Definition:Standard Discrete Metric",
"Open Ball in Standard Discrete Metric Space"
] |
proofwiki-18011 | Equality of Open Balls does not imply Equality of Radii | Let $M = \struct {A, d}$ be a metric space.
Let $x, y \in A$ and $r, s \in \R$ such that:
:$\map {B_r} x = \map {B_s} y$
Then it is not necessarily the case that their radii $r$ and $s$ are equal. | Let $A$ be arbitrary.
Let $d$ be the (standard) discrete metric on $A$.
Let $r \ge 1$ and $s \ge 1$ such that $r \ne s$.
Then from Open Ball in Standard Discrete Metric Space:
:$\forall x, y \in A: \map {B_r} x = \map {B_s} y = A$
{{qed}} | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $x, y \in A$ and $r, s \in \R$ such that:
:$\map {B_r} x = \map {B_s} y$
Then it is not necessarily the case that their [[Definition:Radius of Open Ball|radii]] $r$ and $s$ are equal. | Let $A$ be arbitrary.
Let $d$ be the [[Definition:Standard Discrete Metric|(standard) discrete metric]] on $A$.
Let $r \ge 1$ and $s \ge 1$ such that $r \ne s$.
Then from [[Open Ball in Standard Discrete Metric Space]]:
:$\forall x, y \in A: \map {B_r} x = \map {B_s} y = A$
{{qed}} | Equality of Open Balls does not imply Equality of Radii | https://proofwiki.org/wiki/Equality_of_Open_Balls_does_not_imply_Equality_of_Radii | https://proofwiki.org/wiki/Equality_of_Open_Balls_does_not_imply_Equality_of_Radii | [
"Open Balls"
] | [
"Definition:Metric Space",
"Definition:Open Ball/Radius"
] | [
"Definition:Standard Discrete Metric",
"Open Ball in Standard Discrete Metric Space"
] |
proofwiki-18012 | Mapping between Subspaces of Homeomorphic Spaces is Homeomorphism | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.
Let $f: T_1 \to T_2$ be a homeomorphism.
Let $H \subseteq S_1$.
Let $T_H = \struct {H, \tau_H}$ be the topological subspace of $T_1$ under the subspace topology $\tau_H$ induced by $\tau_1$.
Let $K = f \sqbrk H$ be the image of $... | Let $U \in \tau_K$ be open in $K$.
Then either:
:$U \in \tau_2$
or:
:$U = K \cap V$
where $V \in \tau_2$.
Suppose $U \in \tau_2$.
Because $f$ is continuous:
:$f^{-1} \sqbrk U \in \tau_1$
{{begin-eqn}}
{{eqn | l = f^{-1} \sqbrk U
| o = \in
| r = \tau_1
| c = {{Defof|Continuous Mapping}}
}}
{{eqn | ll= ... | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $f: T_1 \to T_2$ be a [[Definition:Homeomorphism (Topological Spaces)|homeomorphism]].
Let $H \subseteq S_1$.
Let $T_H = \struct {H, \tau_H}$ be the [[Definition:Topological Subspace|topol... | Let $U \in \tau_K$ be [[Definition:Open Set (Topology)|open]] in $K$.
Then either:
:$U \in \tau_2$
or:
:$U = K \cap V$
where $V \in \tau_2$.
Suppose $U \in \tau_2$.
Because $f$ is [[Definition:Continuous Mapping|continuous]]:
:$f^{-1} \sqbrk U \in \tau_1$
{{begin-eqn}}
{{eqn | l = f^{-1} \sqbrk U
| o = \in
... | Mapping between Subspaces of Homeomorphic Spaces is Homeomorphism | https://proofwiki.org/wiki/Mapping_between_Subspaces_of_Homeomorphic_Spaces_is_Homeomorphism | https://proofwiki.org/wiki/Mapping_between_Subspaces_of_Homeomorphic_Spaces_is_Homeomorphism | [
"Mapping between Subspaces of Homeomorphic Spaces is Homeomorphism",
"Homeomorphisms (Topological Spaces)",
"Topological Subspaces"
] | [
"Definition:Topological Space",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Topological Subspace",
"Definition:Topological Subspace",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Topological Subspace",
"Definition:Topological Subspace",
"Definition:Restriction/Mapping",
... | [
"Definition:Open Set/Topology",
"Definition:Continuous Mapping",
"Definition:Topological Subspace",
"Definition:Open Set/Topology",
"Preimage of Intersection under Mapping",
"Definition:Continuous Mapping"
] |
proofwiki-18013 | Mapping between Subspaces of Homeomorphic Spaces is Homeomorphism/Corollary | Let $\overline H$ denote the complement $S_1 \setminus H$ of $H$ relative to $S_1$.
Let $T_{\overline H} = \struct {\overline H, \tau_{\overline H} }$ be the topological subspace of $T_1$ under the subspace topology $\tau_{\overline H}$ induced by $\tau_1$.
Let $\overline K = f \sqbrk {\overline H}$ be the image of $\o... | By definition of relative complement, $\overline H \subseteq S_1$.
As $f$ is a homeomorphism, it is a fortiori a bijection.
Hence from Image of Relative Complement under Bijection is Relative Complement of Image:
:$f \sqbrk {\overline H} = \overline K$
Hence Mapping between Subspaces of Homeomorphic Spaces is Homeomorp... | Let $\overline H$ denote the [[Definition:Relative Complement|complement $S_1 \setminus H$ of $H$ relative to $S_1$]].
Let $T_{\overline H} = \struct {\overline H, \tau_{\overline H} }$ be the [[Definition:Topological Subspace|topological subspace]] of $T_1$ under the [[Definition:Subspace Topology|subspace topology $... | By definition of [[Definition:Relative Complement|relative complement]], $\overline H \subseteq S_1$.
As $f$ is a [[Definition:Homeomorphism (Topological Spaces)|homeomorphism]], it is [[Definition:A Fortiori|a fortiori]] a [[Definition:Bijection|bijection]].
Hence from [[Image of Relative Complement under Bijection ... | Mapping between Subspaces of Homeomorphic Spaces is Homeomorphism/Corollary | https://proofwiki.org/wiki/Mapping_between_Subspaces_of_Homeomorphic_Spaces_is_Homeomorphism/Corollary | https://proofwiki.org/wiki/Mapping_between_Subspaces_of_Homeomorphic_Spaces_is_Homeomorphism/Corollary | [
"Mapping between Subspaces of Homeomorphic Spaces is Homeomorphism"
] | [
"Definition:Relative Complement",
"Definition:Topological Subspace",
"Definition:Topological Subspace",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Topological Subspace",
"Definition:Topological Subspace",
"Definition:Restriction/Mapping",
"Definition:Homeomorphism/Topological Spaces"
... | [
"Definition:Relative Complement",
"Definition:Homeomorphism/Topological Spaces",
"Definition:A Fortiori",
"Definition:Bijection",
"Image of Relative Complement under Bijection is Relative Complement of Image",
"Mapping between Subspaces of Homeomorphic Spaces is Homeomorphism"
] |
proofwiki-18014 | Image of Relative Complement under Bijection is Relative Complement of Image | Let $S$ and $T$ be sets.
Let $f: S \to T$ be a bijection.
Let $H \subseteq S$.
Let $f \sqbrk H = K$ be the image of $H$ under $f$.
Let $\relcomp S H$ denote the relative complement of $H$ in $S$.
Then:
:$f \sqbrk {\relcomp S H} = \relcomp T K$ | From Set with Relative Complement forms Partition, $\set {H \mid \relcomp S H}$ forms a partition of $S$.
The result follows from Bijection Preserves Set Partition.
{{qed}}
Category:Relative Complement
Category:Bijections
ok4glh22j4a3wuzk94eewx7lqh50ya9 | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $f: S \to T$ be a [[Definition:Bijection|bijection]].
Let $H \subseteq S$.
Let $f \sqbrk H = K$ be the [[Definition:Image of Subset under Mapping|image]] of $H$ under $f$.
Let $\relcomp S H$ denote the [[Definition:Relative Complement|relative complement]] of $H$ in $... | From [[Set with Relative Complement forms Partition]], $\set {H \mid \relcomp S H}$ forms a [[Definition:Set Partition|partition]] of $S$.
The result follows from [[Bijection Preserves Set Partition]].
{{qed}}
[[Category:Relative Complement]]
[[Category:Bijections]]
ok4glh22j4a3wuzk94eewx7lqh50ya9 | Image of Relative Complement under Bijection is Relative Complement of Image | https://proofwiki.org/wiki/Image_of_Relative_Complement_under_Bijection_is_Relative_Complement_of_Image | https://proofwiki.org/wiki/Image_of_Relative_Complement_under_Bijection_is_Relative_Complement_of_Image | [
"Relative Complement",
"Bijections"
] | [
"Definition:Set",
"Definition:Bijection",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Relative Complement"
] | [
"Set Difference and Intersection form Partition/Corollary 2",
"Definition:Set Partition",
"Bijection Preserves Set Partition",
"Category:Relative Complement",
"Category:Bijections"
] |
proofwiki-18015 | Bijection Preserves Set Partition | Let $S$ and $T$ be sets.
Let $f: S \to T$ be a bijection
Let $I$ be an indexing set.
Let $\family {S_i}_{i \mathop \in I}$ be a partitioning of $S$ indexed by $I$.
Hence, let $\set {S_i: i \in I}$ be the resulting partition of $S$.
Then the image of $f$ is a partition of $T$ indexed by $I$ such that:
:$T = \set {f \sqb... | By definition of partitioning:
:$(1): \quad \forall i \in I: S_i \ne \O$, that is, none of $S_i$ is empty
:$(2): \quad \ds S = \bigcup_{i \mathop \in I} S_i$, that is, $S$ is the union of $\family {S_i}_{i \mathop \in I}$
:$(3): \quad \forall i, j \in I: i \ne j \implies S_i \cap S_j = \O$, that is, the elements of $\f... | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $f: S \to T$ be a [[Definition:Bijection|bijection]]
Let $I$ be an [[Definition:Indexing Set|indexing set]].
Let $\family {S_i}_{i \mathop \in I}$ be a [[Definition:Partitioning|partitioning]] of $S$ [[Definition:Indexed Set|indexed]] by $I$.
Hence, let $\set {S_i: i... | By definition of [[Definition:Partitioning|partitioning]]:
:$(1): \quad \forall i \in I: S_i \ne \O$, that is, none of $S_i$ is [[Definition:Empty Set|empty]]
:$(2): \quad \ds S = \bigcup_{i \mathop \in I} S_i$, that is, $S$ is the [[Definition:Union of Family|union]] of $\family {S_i}_{i \mathop \in I}$
:$(3): \quad ... | Bijection Preserves Set Partition | https://proofwiki.org/wiki/Bijection_Preserves_Set_Partition | https://proofwiki.org/wiki/Bijection_Preserves_Set_Partition | [
"Set Partitions",
"Bijections"
] | [
"Definition:Set",
"Definition:Bijection",
"Definition:Indexing Set",
"Definition:Partitioning",
"Definition:Indexing Set/Indexed Set",
"Definition:Set Partition",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Set Partition",
"Definition:Indexing Set/Indexed Set"
] | [
"Definition:Partitioning",
"Definition:Empty Set",
"Definition:Set Union/Family of Sets",
"Definition:Pairwise Disjoint",
"Definition:Empty Set",
"Definition:Set Union/Family of Sets",
"Definition:Pairwise Disjoint",
"Definition:Bijection",
"Definition:A Fortiori",
"Definition:Surjection",
"Defi... |
proofwiki-18016 | Product Space is Homeomorphic to Product Space with Factors Commuted | Let $T_1$ and $T_2$ be topological spaces.
Let $T_1 \times T_2$ denote the product space of $T_1$ and $T_2$.
Let $t: T_1 \times T_2 \to T_2 \times T_1$ be the mapping defined as:
:$\forall \tuple {x, y} \in T_1 \times T_2: \map t {x, y} = \tuple {y, x}$
Then $t$ is a homeomorphism. | $t$ is trivially a bijection.
Let $U$ be open in $T_2 \times T_1$.
Then by definition of product space:
:$U = U_2 \times U_1$
where:
:$U_2$ is open in $T_2$
:$U_1$ is open in $T_1$.
Hence by definition of product space:
:$t^{-1} \sqbrk {U_2 \times U_1} = U_1 \times U_2$ is open in $T_1 \times T_2$.
Hence it has been sh... | Let $T_1$ and $T_2$ be [[Definition:Topological Space|topological spaces]].
Let $T_1 \times T_2$ denote the [[Definition:Product Space (Topology) of Two Factor Spaces|product space]] of $T_1$ and $T_2$.
Let $t: T_1 \times T_2 \to T_2 \times T_1$ be the [[Definition:Mapping|mapping]] defined as:
:$\forall \tuple {x, y... | $t$ is trivially a [[Definition:Bijection|bijection]].
Let $U$ be [[Definition:Open Set (Topology)|open]] in $T_2 \times T_1$.
Then by definition of [[Definition:Product Space|product space]]:
:$U = U_2 \times U_1$
where:
:$U_2$ is [[Definition:Open Set (Topology)|open]] in $T_2$
:$U_1$ is [[Definition:Open Set (Topo... | Product Space is Homeomorphic to Product Space with Factors Commuted | https://proofwiki.org/wiki/Product_Space_is_Homeomorphic_to_Product_Space_with_Factors_Commuted | https://proofwiki.org/wiki/Product_Space_is_Homeomorphic_to_Product_Space_with_Factors_Commuted | [
"Product Topology",
"Homeomorphisms (Topological Spaces)"
] | [
"Definition:Topological Space",
"Definition:Product Space (Topology)/Two Factor Spaces",
"Definition:Mapping",
"Definition:Homeomorphism/Topological Spaces"
] | [
"Definition:Bijection",
"Definition:Open Set/Topology",
"Definition:Product Space",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Product Space",
"Definition:Open Set/Topology",
"Definition:Continuous",
"Definition:Open Set/Topology",
"Definition:Product Space",
"De... |
proofwiki-18017 | Cartesian Product of Homeomorphisms is Homeomorphism | Let $S_1, S_2, T_1, T_2$ be topological spaces.
Let $f_1: S_1 \to T_1$ and $f_2: S_2 \to T_2$ be mappings.
Let:
:$f_1 \times f_2: S_1 \times S_2 \to T_1 \times T_2$
be defined as:
:$\forall \tuple {x, y} \in S_1 \times S_2: \map {\paren {f_1 \times f_2} } {x, y} = \tuple {\map {f_1} x, \map {f_2} y}$
where $S_1 \times ... | From Cartesian Product of Mappings is Continuous iff Factor Mappings are Continuous:
:$f_1 \times f_2$ is continuous.
From Cartesian Product of Bijections is Bijection:
:$f_1 \times f_2$ is a bijection.
From Cartesian Product of Mappings is Continuous iff Factor Mappings are Continuous:
:$\paren {f_1 \times f_2}^{-1} =... | Let $S_1, S_2, T_1, T_2$ be [[Definition:Topological Space|topological spaces]].
Let $f_1: S_1 \to T_1$ and $f_2: S_2 \to T_2$ be [[Definition:Mapping|mappings]].
Let:
:$f_1 \times f_2: S_1 \times S_2 \to T_1 \times T_2$
be defined as:
:$\forall \tuple {x, y} \in S_1 \times S_2: \map {\paren {f_1 \times f_2} } {x, y}... | From [[Cartesian Product of Mappings is Continuous iff Factor Mappings are Continuous]]:
:$f_1 \times f_2$ is [[Definition:Continuous Mapping (Topology)|continuous]].
From [[Cartesian Product of Bijections is Bijection]]:
:$f_1 \times f_2$ is a [[Definition:Bijection|bijection]].
From [[Cartesian Product of Mappings ... | Cartesian Product of Homeomorphisms is Homeomorphism | https://proofwiki.org/wiki/Cartesian_Product_of_Homeomorphisms_is_Homeomorphism | https://proofwiki.org/wiki/Cartesian_Product_of_Homeomorphisms_is_Homeomorphism | [
"Cartesian Product",
"Homeomorphisms (Topological Spaces)"
] | [
"Definition:Topological Space",
"Definition:Mapping",
"Definition:Product Space (Topology)/Two Factor Spaces",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Homeomorphism/Topological Spaces"
] | [
"Cartesian Product of Mappings is Continuous iff Factor Mappings are Continuous",
"Definition:Continuous Mapping (Topology)",
"Cartesian Product of Bijections is Bijection",
"Definition:Bijection",
"Cartesian Product of Mappings is Continuous iff Factor Mappings are Continuous",
"Definition:Continuous Map... |
proofwiki-18018 | Product of Closed and Half-Open Unit Intervals is Homeomorphic to Product of Half-Open Unit Intervals | Let $\closedint 0 1$ denote the closed unit interval $\set {x \in \R: 0 \le x \le 1}$.
Let $\hointr 0 1$ denote the half-open unit interval $\set {x \in \R: 0 \le x < 1}$.
Let both $\closedint 0 1$ and $\hointr 0 1$ have the Euclidean topology.
Then the product space:
:$\closedint 0 1 \times \hointr 0 1$
is homeomorphi... | First we take the square $\Box ABCD$ embedded in the Cartesian plane such that $AD$ corresponds to $\closedint 0 1$ and $AB$ corresponds to $\hointr 0 1$:
:300px
This corresponds to the set $\closedint 0 1 \times \hointr 0 1$.
It is noted that the line segment $BC$ which corresponds to $\closedint 0 1 \times \set 1$ is... | Let $\closedint 0 1$ denote the [[Definition:Closed Unit Interval|closed unit interval]] $\set {x \in \R: 0 \le x \le 1}$.
Let $\hointr 0 1$ denote the [[Definition:Unit Interval|half-open unit interval]] $\set {x \in \R: 0 \le x < 1}$.
Let both $\closedint 0 1$ and $\hointr 0 1$ have the [[Definition:Real Number Lin... | First we take the [[Definition:Square (Geometry)|square]] $\Box ABCD$ embedded in the [[Definition:Cartesian Plane|Cartesian plane]] such that $AD$ corresponds to $\closedint 0 1$ and $AB$ corresponds to $\hointr 0 1$:
:[[File:Closed-0-1-by-halfopen-0-1.png|300px]]
This corresponds to the [[Definition:Set|set]] $\clo... | Product of Closed and Half-Open Unit Intervals is Homeomorphic to Product of Half-Open Unit Intervals | https://proofwiki.org/wiki/Product_of_Closed_and_Half-Open_Unit_Intervals_is_Homeomorphic_to_Product_of_Half-Open_Unit_Intervals | https://proofwiki.org/wiki/Product_of_Closed_and_Half-Open_Unit_Intervals_is_Homeomorphic_to_Product_of_Half-Open_Unit_Intervals | [
"Product Spaces",
"Homeomorphisms (Topological Spaces)"
] | [
"Definition:Real Interval/Unit Interval/Closed",
"Definition:Real Interval/Unit Interval",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Product Space",
"Definition:Homeomorphism/Topological Spaces"
] | [
"Definition:Quadrilateral/Square",
"Definition:Cartesian Plane",
"File:Closed-0-1-by-halfopen-0-1.png",
"Definition:Set",
"Definition:Line/Segment",
"Definition:Set",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Perimeter",
"Definition:Circle",
"Definition:Circle/Center",
"Definiti... |
proofwiki-18019 | Closed Image of Closure of Set under Continuous Mapping equals Closure of Image | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.
Let $H \subseteq S_1$ be a subset of $S_1$.
Let $\map \cl H$ denote the closure of $H$.
Let $f: T_1 \to T_2$ be a continuous mapping.
Let $f \sqbrk {\map \cl H}$ be closed in $T_2$.
Then:
:$f \sqbrk {\map \cl H} = \map \cl {f \sq... | By Continuity Defined by Closure:
:$f \sqbrk {\map \cl H} \subseteq \map \cl {f \sqbrk H}$
{{qed|lemma}}
{{begin-eqn}}
{{eqn | l = H
| o = \subseteq
| r = \map \cl H
| c = Set is Subset of its Topological Closure
}}
{{eqn | ll= \leadsto
| l = f \sqbrk H
| o = \subseteq
| r = f \sqbrk... | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $H \subseteq S_1$ be a [[Definition:Subset|subset]] of $S_1$.
Let $\map \cl H$ denote the [[Definition:Closure (Topology)|closure]] of $H$.
Let $f: T_1 \to T_2$ be a [[Definition:Continuous... | By [[Continuity Defined by Closure]]:
:$f \sqbrk {\map \cl H} \subseteq \map \cl {f \sqbrk H}$
{{qed|lemma}}
{{begin-eqn}}
{{eqn | l = H
| o = \subseteq
| r = \map \cl H
| c = [[Set is Subset of its Topological Closure]]
}}
{{eqn | ll= \leadsto
| l = f \sqbrk H
| o = \subseteq
| r ... | Closed Image of Closure of Set under Continuous Mapping equals Closure of Image | https://proofwiki.org/wiki/Closed_Image_of_Closure_of_Set_under_Continuous_Mapping_equals_Closure_of_Image | https://proofwiki.org/wiki/Closed_Image_of_Closure_of_Set_under_Continuous_Mapping_equals_Closure_of_Image | [
"Set Closures",
"Continuous Mappings"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Closure (Topology)",
"Definition:Continuous Mapping",
"Definition:Closed Set/Topology"
] | [
"Continuity Defined by Closure",
"Set is Subset of its Topological Closure",
"Image of Subset under Relation is Subset of Image",
"Definition:Closed Set/Topology",
"Closure of Subset of Closed Set of Topological Space is Subset",
"Definition:Set Equality"
] |
proofwiki-18020 | Closure of Image under Continuous Mapping is not necessarily Image of Closure | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.
Let $H \subseteq S_1$ be a subset of $S_1$.
Let $\map \cl H$ denote the closure of $H$.
Let $f: T_1 \to T_2$ be a continuous mapping.
Then it is not necessarily the case that:
:$f \sqbrk {\map \cl H} = \map \cl {f \sqbrk H}$ | Proof by Counterexample:
Let $\R$ be the real numbers under the usual (Euclidean) topology.
Let $f: \R \to \R$ be the (real) hyperbolic tangent function:
:$\forall x \in \R: \map f x = \tanh x$
It is accepted that $f$ is continuous.
Let $H \subseteq \R$ be the subset of $\R$ defined as:
{{begin-eqn}}
{{eqn | l = H
... | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $H \subseteq S_1$ be a [[Definition:Subset|subset]] of $S_1$.
Let $\map \cl H$ denote the [[Definition:Closure (Topology)|closure]] of $H$.
Let $f: T_1 \to T_2$ be a [[Definition:Continuous... | [[Proof by Counterexample]]:
Let $\R$ be the [[Definition:Real Number Line with Euclidean Topology|real numbers under the usual (Euclidean) topology]].
Let $f: \R \to \R$ be the [[Definition:Real Hyperbolic Tangent|(real) hyperbolic tangent]] function:
:$\forall x \in \R: \map f x = \tanh x$
It is accepted that $f$... | Closure of Image under Continuous Mapping is not necessarily Image of Closure | https://proofwiki.org/wiki/Closure_of_Image_under_Continuous_Mapping_is_not_necessarily_Image_of_Closure | https://proofwiki.org/wiki/Closure_of_Image_under_Continuous_Mapping_is_not_necessarily_Image_of_Closure | [
"Set Closures",
"Continuous Mappings"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Closure (Topology)",
"Definition:Continuous Mapping"
] | [
"Proof by Counterexample",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Hyperbolic Tangent/Real",
"Definition:Continuous Mapping",
"Definition:Subset",
"Definition:Open Set/Topology"
] |
proofwiki-18021 | Closure of Preimage under Continuous Mapping is not necessarily Preimage of Closure | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.
Let $H \subseteq S_1$ be a subset of $S_1$.
Let $\map \cl H$ denote the closure of $H$.
Let $f: T_1 \to T_2$ be a continuous mapping.
Then it is not necessarily the case that:
:$f^{-1} \sqbrk {\map \cl H} = \map \cl {f^{-1} \sqbr... | Proof by Counterexample:
Let $\R$ be the real numbers under the usual (Euclidean) topology.
Let $f: \R \to \R$ be the real function:
:$\forall x \in \R: \map f x = \begin {cases} -1 & : x \le -1 \\ x & : -1 \le x \le 1 \\ 1 & : x \ge 1 \end {cases}$
It is accepted that $f$ is a continuous mapping.
Let $H \subseteq \R$ ... | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $H \subseteq S_1$ be a [[Definition:Subset|subset]] of $S_1$.
Let $\map \cl H$ denote the [[Definition:Closure (Topology)|closure]] of $H$.
Let $f: T_1 \to T_2$ be a [[Definition:Continuous... | [[Proof by Counterexample]]:
Let $\R$ be the [[Definition:Real Number Line with Euclidean Topology|real numbers under the usual (Euclidean) topology]].
Let $f: \R \to \R$ be the [[Definition:Real Function|real function]]:
:$\forall x \in \R: \map f x = \begin {cases} -1 & : x \le -1 \\ x & : -1 \le x \le 1 \\ 1 & : ... | Closure of Preimage under Continuous Mapping is not necessarily Preimage of Closure | https://proofwiki.org/wiki/Closure_of_Preimage_under_Continuous_Mapping_is_not_necessarily_Preimage_of_Closure | https://proofwiki.org/wiki/Closure_of_Preimage_under_Continuous_Mapping_is_not_necessarily_Preimage_of_Closure | [
"Set Closures",
"Continuous Mappings"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Closure (Topology)",
"Definition:Continuous Mapping"
] | [
"Proof by Counterexample",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Real Function",
"Definition:Continuous Mapping",
"Definition:Subset",
"Definition:Open Set/Topology"
] |
proofwiki-18022 | Normed Vector Space of Rational Numbers is not Banach Space | Let $\struct {\Q, \size {\, \cdot \,}}$ be the normed vector space of rational numbers.
Then $\struct {\Q, \size {\, \cdot \,}}$ is not a Banach space. | Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $\Q$ defined recursively in the following way:
:$\ds x_0 = \frac 3 2$
:$\ds \forall n \in \N_{> 0} : x_{n \mathop + 1} = \frac {4 + 3 x_n} {3 + 2 x_n}$
We have that:
:$\forall n \in \N : x_n \ge 0$
Note that:
{{begin-eqn}}
{{eqn | l = x_{n \mathop + 1}^2 - 2
... | Let $\struct {\Q, \size {\, \cdot \,}}$ be the [[Rational Numbers with Absolute Norm form Normed Vector Space|normed vector space of rational numbers]].
Then $\struct {\Q, \size {\, \cdot \,}}$ is not a [[Definition:Banach Space|Banach space]]. | Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $\Q$ defined [[Definition:Recursive Sequence|recursively]] in the following way:
:$\ds x_0 = \frac 3 2$
:$\ds \forall n \in \N_{> 0} : x_{n \mathop + 1} = \frac {4 + 3 x_n} {3 + 2 x_n}$
We have that:
:$\forall n \in \N : x_n \ge 0$
N... | Normed Vector Space of Rational Numbers is not Banach Space | https://proofwiki.org/wiki/Normed_Vector_Space_of_Rational_Numbers_is_not_Banach_Space | https://proofwiki.org/wiki/Normed_Vector_Space_of_Rational_Numbers_is_not_Banach_Space | [
"Rational Number Space"
] | [
"Rational Numbers with Absolute Norm form Normed Vector Space",
"Definition:Banach Space"
] | [
"Definition:Sequence",
"Definition:Recursive Sequence",
"Monotone Convergence Theorem (Real Analysis)/Decreasing Sequence",
"Definition:Convergent Sequence/Normed Vector Space",
"Convergent Sequence is Cauchy Sequence/Normed Vector Space",
"Definition:Cauchy Sequence/Normed Vector Space",
"Definition:Ca... |
proofwiki-18023 | Intersection of Open Set with Closure of Set is Subset of Closure of Intersection | Let $T = \struct {S, \tau}$ be a topological space.
Let $H \in \tau$ be an open set of $T$.
Let $K \subseteq S$ be an arbitrary subset of $S$.
Then:
:$H \cap \map \cl K \subseteq \map \cl {H \cap K}$
where $\cl$ denotes set closure. | Let $x \in H \cap \map \cl K$.
Then:
:$x \in H$
and:
:$x \in \map \cl K$
Let $N_1$ be an arbitrary neighborbood of $x$.
Because $x \in H$, there exists a neighborbood $N_2$ of $x$ entirely within $U$.
Let $N_3 = N_1 \cap N_2$.
By Intersection of Neighborhoods in Topological Space is Neighborhood, $N_3$ is also a neighb... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $H \in \tau$ be an [[Definition:Open Set (Topology)|open set]] of $T$.
Let $K \subseteq S$ be an arbitrary [[Definition:Subset|subset]] of $S$.
Then:
:$H \cap \map \cl K \subseteq \map \cl {H \cap K}$
where $\cl$ denotes [[Defi... | Let $x \in H \cap \map \cl K$.
Then:
:$x \in H$
and:
:$x \in \map \cl K$
Let $N_1$ be an arbitrary [[Definition:Neighborhood of Point|neighborbood of $x$]].
Because $x \in H$, there exists a [[Definition:Neighborhood of Point|neighborbood $N_2$ of $x$]] entirely within $U$.
Let $N_3 = N_1 \cap N_2$.
By [[Intersect... | Intersection of Open Set with Closure of Set is Subset of Closure of Intersection | https://proofwiki.org/wiki/Intersection_of_Open_Set_with_Closure_of_Set_is_Subset_of_Closure_of_Intersection | https://proofwiki.org/wiki/Intersection_of_Open_Set_with_Closure_of_Set_is_Subset_of_Closure_of_Intersection | [
"Set Closures",
"Set Intersection"
] | [
"Definition:Topological Space",
"Definition:Open Set/Topology",
"Definition:Subset",
"Definition:Closure (Topology)"
] | [
"Definition:Neighborhood (Topology)/Point",
"Definition:Neighborhood (Topology)/Point",
"Intersection of Neighborhoods in Topological Space is Neighborhood",
"Definition:Neighborhood (Topology)/Point",
"Definition:Closure (Topology)",
"Definition:Neighborhood (Topology)/Point",
"Definition:Subset"
] |
proofwiki-18024 | Closure of Non-Empty Bounded Subset of Metric Space is Bounded | Let $M = \struct {A, d}$ be a metric space.
Let $S \subseteq A$ be bounded in $M$.
Then:
:$\map \cl S$ is also bounded in $M$.
where $\map \cl S$ denotes the closure of $S$ in $M$. | By definition of bounded:
:$S$ is '''bounded''' {{iff}}:
:$\exists K \in \R: \forall x, y \in M': \map {d_B} {x, y} \le K$
That is, such that $S$ has a diameter $K$.
From Diameter of Closure of Subset is Diameter of Subset, if $S$ has a diameter $K$, then so does $\map \cl S$.
That is, $\map \cl S$ is also bounded.
{{q... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $S \subseteq A$ be [[Definition:Bounded Metric Space|bounded]] in $M$.
Then:
:$\map \cl S$ is also [[Definition:Bounded Metric Space|bounded]] in $M$.
where $\map \cl S$ denotes the [[Definition:Closure (Metric Space)|closure]] of $S$ in $M... | By definition of [[Definition:Bounded Metric Space|bounded]]:
:$S$ is '''bounded''' {{iff}}:
:$\exists K \in \R: \forall x, y \in M': \map {d_B} {x, y} \le K$
That is, such that $S$ has a [[Definition:Diameter of Subset of Metric Space|diameter]] $K$.
From [[Diameter of Closure of Subset is Diameter of Subset]], if ... | Closure of Non-Empty Bounded Subset of Metric Space is Bounded | https://proofwiki.org/wiki/Closure_of_Non-Empty_Bounded_Subset_of_Metric_Space_is_Bounded | https://proofwiki.org/wiki/Closure_of_Non-Empty_Bounded_Subset_of_Metric_Space_is_Bounded | [
"Metric Spaces",
"Set Closures",
"Boundedness"
] | [
"Definition:Metric Space",
"Definition:Bounded Metric Space",
"Definition:Bounded Metric Space",
"Definition:Closure (Topology)/Metric Space"
] | [
"Definition:Bounded Metric Space",
"Definition:Diameter of Subset of Metric Space",
"Diameter of Closure of Subset is Diameter of Subset",
"Definition:Diameter of Subset of Metric Space",
"Definition:Bounded Metric Space"
] |
proofwiki-18025 | Image under Projection from Closed Set of Product Topology is not necessarily Closed | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.
Let $T = \struct {T_1 \times T_2, \tau}$ be the product space of $T_1$ and $T_2$, where $\tau$ is the product topology on $S$.
Let $\pr_1: T \to T_1$ and $\pr_2: T \to T_2$ be the first and second projections from $T$ onto its fa... | Proof by Counterexample:
Let $K = \set {\tuple {x, y} \in \R^2: x \ge 0, y \ge \dfrac 1 x}$
Then:
:$\pr_1 \sqbrk K = \openint 0 \to$
$K$ is closed in $T$, as follows:
Consider the mapping $f: \R^2 \to \R$ defined as:
:$\map f {x, y} = x y$
which is continuous on $\R^2$ (see Preimages of $\map f {x, y} = \tuple {x^2 + y... | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $T = \struct {T_1 \times T_2, \tau}$ be the [[Definition:Product Space (Topology) of Two Factor Spaces|product space]] of $T_1$ and $T_2$, where $\tau$ is the [[Definition:Product Topology on... | [[Proof by Counterexample]]:
Let $K = \set {\tuple {x, y} \in \R^2: x \ge 0, y \ge \dfrac 1 x}$
Then:
:$\pr_1 \sqbrk K = \openint 0 \to$
$K$ is [[Definition:Closed Set (Topology)|closed]] in $T$, as follows:
Consider the [[Definition:Mapping|mapping]] $f: \R^2 \to \R$ defined as:
:$\map f {x, y} = x y$
which is [[D... | Image under Projection from Closed Set of Product Topology is not necessarily Closed | https://proofwiki.org/wiki/Image_under_Projection_from_Closed_Set_of_Product_Topology_is_not_necessarily_Closed | https://proofwiki.org/wiki/Image_under_Projection_from_Closed_Set_of_Product_Topology_is_not_necessarily_Closed | [
"Product Topology",
"Open Mappings",
"Projections",
"Projection from Product Topology is Open and Continuous"
] | [
"Definition:Topological Space",
"Definition:Product Space (Topology)/Two Factor Spaces",
"Definition:Product Topology/Two Factor Spaces",
"Definition:Projection (Mapping Theory)",
"Definition:Product Topology/Factor Space",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology"
] | [
"Proof by Counterexample",
"Definition:Closed Set/Topology",
"Definition:Mapping",
"Definition:Continuous Mapping",
"Preimage of Subset under Mapping/Examples/Preimages of f(x, y) = (x^2 + y^2, x y)/Continuity"
] |
proofwiki-18026 | Interior of Cartesian Product is Product of Interiors | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.
Let $T = \struct {T_1 \times T_2, \tau}$ be the product space of $T_1$ and $T_2$, where $\tau$ is the product topology on $S$.
Let $H \subseteq T_1$ and $K \subseteq T_2$.
Then:
:$\Int {H \times K} = \Int H \times \Int K$
where $... | By definition of interior, both $\Int H$ and $\Int K$ are open in $T_1$ and $T_2$ respectively.
By Projection from Product Topology is Continuous, it follows that $\Int {H \times K}$ is an open set of $T$.
It remains to be shown that $\Int {H \times K}$ is the largest open subset of $H \times K$.
Let $H' \times K'$ be ... | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $T = \struct {T_1 \times T_2, \tau}$ be the [[Definition:Product Space (Topology) of Two Factor Spaces|product space]] of $T_1$ and $T_2$, where $\tau$ is the [[Definition:Product Topology on... | By definition of [[Definition:Interior (Topology)|interior]], both $\Int H$ and $\Int K$ are [[Definition:Open Set (Topology)|open]] in $T_1$ and $T_2$ respectively.
By [[Projection from Product Topology is Continuous]], it follows that $\Int {H \times K}$ is an [[Definition:Open Set (Topology)|open set]] of $T$.
It... | Interior of Cartesian Product is Product of Interiors | https://proofwiki.org/wiki/Interior_of_Cartesian_Product_is_Product_of_Interiors | https://proofwiki.org/wiki/Interior_of_Cartesian_Product_is_Product_of_Interiors | [
"Set Interiors",
"Product Topology"
] | [
"Definition:Topological Space",
"Definition:Product Space (Topology)/Two Factor Spaces",
"Definition:Product Topology/Two Factor Spaces",
"Definition:Interior (Topology)"
] | [
"Definition:Interior (Topology)",
"Definition:Open Set/Topology",
"Projection from Product Topology is Continuous",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Subset",
"Definition:Open Set/Topology",
"Projection from Product Topology is Open",
"Definition:Open Set/To... |
proofwiki-18027 | Closure of Cartesian Product is Product of Closures | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.
Let $T = \struct {T_1 \times T_2, \tau}$ be the product space of $T_1$ and $T_2$, where $\tau$ is the product topology on $S$.
Let $H \subseteq T_1$ and $K \subseteq T_2$.
Then:
:$\map \cl {H \times K} = \map \cl H \times \map \c... | Consider the relative complements of $H$ and $K$ in $T_1$ and $T_2$ respectively:
:$\overline H = \relcomp {S_1} H$
:$\overline K = \relcomp {S_2} K$
Then from Interior of Cartesian Product is Product of Interiors:
:$\Int {\overline H \times \overline K} = \Int {\overline H} \times \Int {\overline K}$
From Complement o... | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $T = \struct {T_1 \times T_2, \tau}$ be the [[Definition:Product Space (Topology) of Two Factor Spaces|product space]] of $T_1$ and $T_2$, where $\tau$ is the [[Definition:Product Topology on... | Consider the [[Definition:Relative Complement|relative complements]] of $H$ and $K$ in $T_1$ and $T_2$ respectively:
:$\overline H = \relcomp {S_1} H$
:$\overline K = \relcomp {S_2} K$
Then from [[Interior of Cartesian Product is Product of Interiors]]:
:$\Int {\overline H \times \overline K} = \Int {\overline H} \t... | Closure of Cartesian Product is Product of Closures | https://proofwiki.org/wiki/Closure_of_Cartesian_Product_is_Product_of_Closures | https://proofwiki.org/wiki/Closure_of_Cartesian_Product_is_Product_of_Closures | [
"Set Closures",
"Product Topology"
] | [
"Definition:Topological Space",
"Definition:Product Space (Topology)/Two Factor Spaces",
"Definition:Product Topology/Two Factor Spaces",
"Definition:Closure (Topology)"
] | [
"Definition:Relative Complement",
"Interior of Cartesian Product is Product of Interiors",
"Complement of Interior equals Closure of Complement"
] |
proofwiki-18028 | Boundary of Cartesian Product of Subsets | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.
Let $T = \struct {T_1 \times T_2, \tau}$ be the product space of $T_1$ and $T_2$, where $\tau$ is the product topology on $S$.
Let $H \subseteq T_1$ and $K \subseteq T_2$.
Then:
:$\map \partial {H \times K} = \paren {\map \partia... | {{begin-eqn}}
{{eqn | l = \map \partial {H \times K}
| r = \map \cl {H \times K} \setminus \Int {H \times K}
| c = {{Defof|Boundary (Topology)}}
}}
{{eqn | r = \paren {\map \cl H \times \map \cl K} \setminus \Int {H \times K}
| c = Closure of Cartesian Product is Product of Closures
}}
{{eqn | r = \pa... | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $T = \struct {T_1 \times T_2, \tau}$ be the [[Definition:Product Space (Topology) of Two Factor Spaces|product space]] of $T_1$ and $T_2$, where $\tau$ is the [[Definition:Product Topology on... | {{begin-eqn}}
{{eqn | l = \map \partial {H \times K}
| r = \map \cl {H \times K} \setminus \Int {H \times K}
| c = {{Defof|Boundary (Topology)}}
}}
{{eqn | r = \paren {\map \cl H \times \map \cl K} \setminus \Int {H \times K}
| c = [[Closure of Cartesian Product is Product of Closures]]
}}
{{eqn | r =... | Boundary of Cartesian Product of Subsets | https://proofwiki.org/wiki/Boundary_of_Cartesian_Product_of_Subsets | https://proofwiki.org/wiki/Boundary_of_Cartesian_Product_of_Subsets | [
"Set Boundaries",
"Product Topology"
] | [
"Definition:Topological Space",
"Definition:Product Space (Topology)/Two Factor Spaces",
"Definition:Product Topology/Two Factor Spaces",
"Definition:Closure (Topology)",
"Definition:Boundary (Topology)"
] | [
"Closure of Cartesian Product is Product of Closures",
"Interior of Cartesian Product is Product of Interiors",
"Set Difference of Cartesian Products"
] |
proofwiki-18029 | Composition of Identification Mappings is Identification Mapping | Let $T_1 = \struct {S_1, \tau_1}$ be a topological space.
Let $S_2$ and $S_3$ be sets.
Let $f_1: S_1 \to S_2$ and $f_2: S_2 \to S_3$ be mappings.
Let $\tau_2$ be the identification topology on $S_2$ with respect to $f_1$ and $\tau_1$.
Let $\tau_3$ be the identification topology on $S_3$ with respect to $f_2$ and $\tau_... | Suppose $V \subseteq S_3$ is an arbitrary subset.
We have the following chain of equivalences:
{{begin-eqn}}
{{eqn | l = V
| o = \in
| r = \tau_3
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = f_2^{-1} \sqbrk V
| o = \in
| r = \tau_2
| c = $f_2$ is an identification mapping
}}
{{eq... | Let $T_1 = \struct {S_1, \tau_1}$ be a [[Definition:Topological Space|topological space]].
Let $S_2$ and $S_3$ be [[Definition:Set|sets]].
Let $f_1: S_1 \to S_2$ and $f_2: S_2 \to S_3$ be [[Definition:Mapping|mappings]].
Let $\tau_2$ be the [[Definition:Identification Topology|identification topology on $S_2$ with r... | Suppose $V \subseteq S_3$ is an arbitrary [[Definition:Subset|subset]].
We have the following chain of equivalences:
{{begin-eqn}}
{{eqn | l = V
| o = \in
| r = \tau_3
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = f_2^{-1} \sqbrk V
| o = \in
| r = \tau_2
| c = $f_2$ is an [[Defi... | Composition of Identification Mappings is Identification Mapping | https://proofwiki.org/wiki/Composition_of_Identification_Mappings_is_Identification_Mapping | https://proofwiki.org/wiki/Composition_of_Identification_Mappings_is_Identification_Mapping | [
"Identification Topology",
"Composite Mappings"
] | [
"Definition:Topological Space",
"Definition:Set",
"Definition:Mapping",
"Definition:Identification Topology",
"Definition:Identification Topology",
"Definition:Composition of Mappings",
"Definition:Identification Topology/Identification Mapping"
] | [
"Definition:Subset",
"Definition:Identification Topology/Identification Mapping",
"Definition:Identification Topology/Identification Mapping",
"Preimage of Subset under Composite Mapping",
"Definition:Identification Topology/Identification Mapping"
] |
proofwiki-18030 | Graph of Continuous Mapping to Hausdorff Space is Closed in Product | Let $T_A = \struct {A, \tau_A}$ and $T_B = \struct {B, \tau_B}$ be topological spaces.
Let $T_B$ be a $T_2$ (Hausdorff) space.
Let $f: T_A \to T_B$ be a continuous mapping.
Then the graph of $f$ is a closed subset of $T_A \times T_B$ under the product topology. | Let $G_f$ be the graph of $f$:
:$G_f = \set {\tuple {x, y} \in A \times B: \map f x = y}$
Let $I_B: T_B \to T_B$ be the identity mapping on $B$:
:$\forall y \in B: \map {I_B} y = y$
From Identity Mapping is Continuous, $I_B$ is continuous on $T_B$.
Let $f \times I_B: T_A \times T_B \to T_B \times T_B$ be the product ma... | Let $T_A = \struct {A, \tau_A}$ and $T_B = \struct {B, \tau_B}$ be [[Definition:Topological Space|topological spaces]].
Let $T_B$ be a [[Definition:T2 Space|$T_2$ (Hausdorff) space]].
Let $f: T_A \to T_B$ be a [[Definition:Continuous Mapping|continuous mapping]].
Then the [[Definition:Graph of Mapping|graph]] of $f... | Let $G_f$ be the [[Definition:Graph of Mapping|graph]] of $f$:
:$G_f = \set {\tuple {x, y} \in A \times B: \map f x = y}$
Let $I_B: T_B \to T_B$ be the [[Definition:Identity Mapping|identity mapping]] on $B$:
:$\forall y \in B: \map {I_B} y = y$
From [[Identity Mapping is Continuous]], $I_B$ is [[Definition:Everywhe... | Graph of Continuous Mapping to Hausdorff Space is Closed in Product/Proof 1 | https://proofwiki.org/wiki/Graph_of_Continuous_Mapping_to_Hausdorff_Space_is_Closed_in_Product | https://proofwiki.org/wiki/Graph_of_Continuous_Mapping_to_Hausdorff_Space_is_Closed_in_Product/Proof_1 | [
"Graph of Continuous Mapping to Hausdorff Space is Closed in Product",
"Hausdorff Spaces",
"Continuous Mappings",
"Product Spaces"
] | [
"Definition:Topological Space",
"Definition:T2 Space",
"Definition:Continuous Mapping",
"Definition:Graph of Mapping",
"Definition:Closed Set/Topology",
"Definition:Subset",
"Definition:Product Topology"
] | [
"Definition:Graph of Mapping",
"Definition:Identity Mapping",
"Identity Mapping is Continuous",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Continuous Mapping to Product Space",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Diagonal Relation",
"Hausdorff Space iff Diag... |
proofwiki-18031 | Intersection of Open Sets of Hausdorff Space containing Point is Singleton | Let $T = \struct {S, \tau}$ be a Hausdorff space.
Let $x \in S$ be arbitrary.
Then the intersection of all open sets containing $x$ is $\set x$:
:$\ds \bigcap_{\substack {H \mathop \in \tau \\ x \mathop \in H} } = \set x$ | Let $x \in S$.
Let $K = \ds \bigcap_{\substack {H \mathop \in \tau \\ x \mathop \in H} }$, that is, the intersection of all open sets containing $x$.
{{AimForCont}} there exists $y \in S$ such that $y \in K$ but $y \ne x$.
By definition of Hausdorff space, there exist disjoint open sets $U, V \in \tau$ containing $x$ a... | Let $T = \struct {S, \tau}$ be a [[Definition:Hausdorff Space|Hausdorff space]].
Let $x \in S$ be arbitrary.
Then the [[Definition:Set Intersection|intersection]] of all [[Definition:Open Set (Topology)|open sets]] containing $x$ is $\set x$:
:$\ds \bigcap_{\substack {H \mathop \in \tau \\ x \mathop \in H} } = \set... | Let $x \in S$.
Let $K = \ds \bigcap_{\substack {H \mathop \in \tau \\ x \mathop \in H} }$, that is, the [[Definition:Set Intersection|intersection]] of all [[Definition:Open Set (Topology)|open sets]] containing $x$.
{{AimForCont}} there exists $y \in S$ such that $y \in K$ but $y \ne x$.
By definition of [[Definiti... | Intersection of Open Sets of Hausdorff Space containing Point is Singleton | https://proofwiki.org/wiki/Intersection_of_Open_Sets_of_Hausdorff_Space_containing_Point_is_Singleton | https://proofwiki.org/wiki/Intersection_of_Open_Sets_of_Hausdorff_Space_containing_Point_is_Singleton | [
"Hausdorff Spaces"
] | [
"Definition:T2 Space",
"Definition:Set Intersection",
"Definition:Open Set/Topology"
] | [
"Definition:Set Intersection",
"Definition:Open Set/Topology",
"Definition:T2 Space",
"Definition:Disjoint Sets",
"Definition:Open Set/Topology",
"Intersection is Subset",
"Definition:Contradiction",
"Definition:Disjoint Sets",
"Definition:Contradiction",
"Definition:T2 Space",
"Proof by Contrad... |
proofwiki-18032 | Space such that Intersection of Open Sets containing Point is Singleton may not be Hausdorff | Let $T = \struct {S, \tau}$ be a topological space.
Let $x \in S$ be arbitrary.
Let $T$ be such that the intersection of all open sets containing $x$ is $\set x$:
:$\ds \bigcap_{\substack {H \mathop \in \tau \\ x \mathop \in H} } = \set x$
Then it is not necessarily the case that $T$ is a $T_2$ (Hausdorff) space. | Let $T$ be the finite complement topology on the real numbers $\R$, for example.
The open sets of $T$ are subsets of $\R$ of the form $U$ such that $\R \setminus U$ is finite, together with $\O$.
Let $x \in \R$ be arbitrary.
Let $K = \ds \bigcap_{\substack {H \mathop \in \tau \\ x \mathop \in H} }$, that is, the inters... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x \in S$ be arbitrary.
Let $T$ be such that the [[Definition:Set Intersection|intersection]] of all [[Definition:Open Set (Topology)|open sets]] containing $x$ is $\set x$:
:$\ds \bigcap_{\substack {H \mathop \in \tau \\ x \mat... | Let $T$ be the [[Definition:Uncountable Finite Complement Topology|finite complement topology]] on the [[Definition:Real Number|real numbers]] $\R$, for example.
The [[Definition:Open Set|open sets]] of $T$ are [[Definition:Subset|subsets]] of $\R$ of the form $U$ such that $\R \setminus U$ is [[Definition:Finite Set|... | Space such that Intersection of Open Sets containing Point is Singleton may not be Hausdorff | https://proofwiki.org/wiki/Space_such_that_Intersection_of_Open_Sets_containing_Point_is_Singleton_may_not_be_Hausdorff | https://proofwiki.org/wiki/Space_such_that_Intersection_of_Open_Sets_containing_Point_is_Singleton_may_not_be_Hausdorff | [
"Hausdorff Spaces"
] | [
"Definition:Topological Space",
"Definition:Set Intersection",
"Definition:Open Set/Topology",
"Definition:T2 Space"
] | [
"Definition:Finite Complement Topology/Uncountable",
"Definition:Real Number",
"Definition:Open Set",
"Definition:Subset",
"Definition:Finite Set",
"Definition:Set Intersection",
"Definition:Open Set/Topology",
"Definition:Set",
"Definition:Finite Set",
"Definition:Open Set/Topology",
"Definitio... |
proofwiki-18033 | Point at Zero Distance from Subset of Metric Space is Limit Point or Element | Let $M = \struct {A, d}$ be a metric space.
Let $H \subseteq A$ be an arbitrary subset of $A$.
Let $x \in A$ be arbitrary.
Let $\map d {x, H}$ denote the distance between $x$ and $H$:
:$\ds \map d {x, H} = \inf_{y \mathop \in H} \paren {\map d {x, y} }$
Then:
:$\map d {x, H} = 0$
{{iff}}:
:either $x \in H$ or $x$ is a ... | === Necessary Condition ===
Let $x$ be such that either $x \in H$ or $x$ is a limit point of $H$.
If $x \in H$ then:
:$\map d {x, H} = 0$
from Distance from Subset to Element.
Otherwise $x$ is a limit point of $H$.
Then from Limit Point of Subset of Metric Space is at Zero Distance:
:$\map d {x, H} = 0$
{{qed|lemma}} | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $H \subseteq A$ be an arbitrary [[Definition:Subset|subset]] of $A$.
Let $x \in A$ be arbitrary.
Let $\map d {x, H}$ denote the [[Definition:Distance between Element and Subset of Metric Space|distance between $x$ and $H$]]:
:$\ds \map d {x,... | === Necessary Condition ===
Let $x$ be such that either $x \in H$ or $x$ is a [[Definition:Limit Point (Metric Space)|limit point]] of $H$.
If $x \in H$ then:
:$\map d {x, H} = 0$
from [[Distance from Subset to Element]].
Otherwise $x$ is a [[Definition:Limit Point (Metric Space)|limit point]] of $H$.
Then from [[L... | Point at Zero Distance from Subset of Metric Space is Limit Point or Element | https://proofwiki.org/wiki/Point_at_Zero_Distance_from_Subset_of_Metric_Space_is_Limit_Point_or_Element | https://proofwiki.org/wiki/Point_at_Zero_Distance_from_Subset_of_Metric_Space_is_Limit_Point_or_Element | [
"Limit Points",
"Distance Function"
] | [
"Definition:Metric Space",
"Definition:Subset",
"Definition:Distance/Sets/Metric Spaces",
"Definition:Limit Point/Metric Space"
] | [
"Definition:Limit Point/Metric Space",
"Distance from Subset to Element",
"Definition:Limit Point/Metric Space",
"Limit Point of Subset of Metric Space is at Zero Distance",
"Definition:Limit Point/Metric Space",
"Definition:Limit Point/Metric Space",
"Definition:Limit Point/Metric Space",
"Definition... |
proofwiki-18034 | Limit Point of Subset of Metric Space is at Zero Distance | Let $M = \struct {A, d}$ be a metric space.
Let $H \subseteq A$ be an arbitrary subset of $A$.
Let $x \in A$ be a limit point of $H$.
Let $\map d {x, H}$ denote the distance between $x$ and $H$:
:$\ds \map d {x, H} = \inf_{y \mathop \in H} \paren {\map d {x, y} }$
Then:
:$\map d {x, H} = 0$ | Let $x$ be a limit point of $H$.
{{AimForCont}} $\map d {x, H} \ne 0$.
By definition of metric, that means:
:$\map d {x, H} > 0$
Then:
:$\exists \epsilon \in \R_{>0}: \forall y \in H: \map d {x, y} > \epsilon$
That is:
:$\forall y \in H: y \notin \map {B_\epsilon} x \setminus \set x$
where $\map {B_\epsilon} x \setminu... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $H \subseteq A$ be an arbitrary [[Definition:Subset|subset]] of $A$.
Let $x \in A$ be a [[Definition:Limit Point (Metric Space)|limit point]] of $H$.
Let $\map d {x, H}$ denote the [[Definition:Distance between Element and Subset of Metric S... | Let $x$ be a [[Definition:Limit Point (Metric Space)|limit point]] of $H$.
{{AimForCont}} $\map d {x, H} \ne 0$.
By definition of [[Definition:Metric|metric]], that means:
:$\map d {x, H} > 0$
Then:
:$\exists \epsilon \in \R_{>0}: \forall y \in H: \map d {x, y} > \epsilon$
That is:
:$\forall y \in H: y \notin \map ... | Limit Point of Subset of Metric Space is at Zero Distance | https://proofwiki.org/wiki/Limit_Point_of_Subset_of_Metric_Space_is_at_Zero_Distance | https://proofwiki.org/wiki/Limit_Point_of_Subset_of_Metric_Space_is_at_Zero_Distance | [
"Limit Points",
"Distance Function"
] | [
"Definition:Metric Space",
"Definition:Subset",
"Definition:Limit Point/Metric Space",
"Definition:Distance/Sets/Metric Spaces"
] | [
"Definition:Limit Point/Metric Space",
"Definition:Metric Space/Metric",
"Definition:Deleted Neighborhood/Metric Space",
"Definition:Limit Point/Metric Space",
"Definition:Contradiction"
] |
proofwiki-18035 | Point not in Subset of Metric Space iff Distance from Elements is Greater than Zero | Let $M = \struct {A, d}$ be a metric space.
Let $H \subseteq A$ be an arbitrary subset of $A$.
Let $x \in A$ be arbitrary.
Then:
:$x \notin H$
{{iff}}:
:$\forall y \in H: \map d {x, y} > 0$ | {{begin-eqn}}
{{eqn | q = \forall y \in H
| l = \map d {x, y}
| o = >
| r = 0
| c =
}}
{{eqn | ll= \leadstoandfrom
| q = \forall y \in H
| l = \map d {x, y}
| o = \ne
| r = \map d {x, x}
| c = {{Metric-space-axiom|1}}
}}
{{eqn | ll= \leadstoandfrom
| q = \for... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $H \subseteq A$ be an arbitrary [[Definition:Subset|subset]] of $A$.
Let $x \in A$ be arbitrary.
Then:
:$x \notin H$
{{iff}}:
:$\forall y \in H: \map d {x, y} > 0$ | {{begin-eqn}}
{{eqn | q = \forall y \in H
| l = \map d {x, y}
| o = >
| r = 0
| c =
}}
{{eqn | ll= \leadstoandfrom
| q = \forall y \in H
| l = \map d {x, y}
| o = \ne
| r = \map d {x, x}
| c = {{Metric-space-axiom|1}}
}}
{{eqn | ll= \leadstoandfrom
| q = \for... | Point not in Subset of Metric Space iff Distance from Elements is Greater than Zero | https://proofwiki.org/wiki/Point_not_in_Subset_of_Metric_Space_iff_Distance_from_Elements_is_Greater_than_Zero | https://proofwiki.org/wiki/Point_not_in_Subset_of_Metric_Space_iff_Distance_from_Elements_is_Greater_than_Zero | [
"Distance Function"
] | [
"Definition:Metric Space",
"Definition:Subset"
] | [
"Category:Distance Function"
] |
proofwiki-18036 | Point at Distance Zero from Closed Set is Element | Let $M = \struct {A, d}$ be a metric space.
Let $H \subseteq A$ be an arbitrary subset of $A$.
Let $x \in A$ be arbitrary.
Let $\map d {x, H}$ denote the distance between $x$ and $H$:
:$\ds \map d {x, H} = \inf_{y \mathop \in H} \paren {\map d {x, y} }$
Let $H$ be closed in $M$.
Then:
:$\map d {x, H} = 0$ {{iff}} $x \i... | === Necessary Condition ===
Let $x \in H$.
Then from Distance from Subset to Element:
:$\map d {x, H} = 0$
whether $H$ is closed or not.
{{qed|lemma}} | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $H \subseteq A$ be an arbitrary [[Definition:Subset|subset]] of $A$.
Let $x \in A$ be arbitrary.
Let $\map d {x, H}$ denote the [[Definition:Distance between Element and Subset of Metric Space|distance between $x$ and $H$]]:
:$\ds \map d {x,... | === Necessary Condition ===
Let $x \in H$.
Then from [[Distance from Subset to Element]]:
:$\map d {x, H} = 0$
whether $H$ is [[Definition:Closed Set (Metric Space)|closed]] or not.
{{qed|lemma}} | Point at Distance Zero from Closed Set is Element | https://proofwiki.org/wiki/Point_at_Distance_Zero_from_Closed_Set_is_Element | https://proofwiki.org/wiki/Point_at_Distance_Zero_from_Closed_Set_is_Element | [
"Metric Spaces",
"Closed Sets"
] | [
"Definition:Metric Space",
"Definition:Subset",
"Definition:Distance/Sets/Metric Spaces",
"Definition:Closed Set/Metric Space"
] | [
"Distance from Subset to Element",
"Definition:Closed Set/Metric Space",
"Definition:Closed Set/Metric Space"
] |
proofwiki-18037 | Square Root is not Lipschitz Continuous | Let $\sqrt {\size x} : \R \to \R_{\ge 0}$ be a real function.
$\sqrt {\size x}$ is not Lipschitz continuous. | {{AimForCont}} $\sqrt {\size x}$ is Lipschitz continuous.
Then:
:$\exists L \in \R_{> 0} : \forall x, y \in \R : \size {\sqrt {\size x} - \sqrt {\size y} } \le L \size {x - y}$
Suppose $x = \dfrac 1 {n^2}$ with $n \in \N$ and $y = 0$.
Then:
:$\dfrac 1 n \le \dfrac L {n^2}$
In other words:
:$\forall n \in \N : n \le L$
... | Let $\sqrt {\size x} : \R \to \R_{\ge 0}$ be a [[Definition:Real Function|real function]].
$\sqrt {\size x}$ is not [[Definition:Lipschitz Continuous Real Function|Lipschitz continuous]]. | {{AimForCont}} $\sqrt {\size x}$ is [[Definition:Lipschitz Continuous Real Function|Lipschitz continuous]].
Then:
:$\exists L \in \R_{> 0} : \forall x, y \in \R : \size {\sqrt {\size x} - \sqrt {\size y} } \le L \size {x - y}$
Suppose $x = \dfrac 1 {n^2}$ with $n \in \N$ and $y = 0$.
Then:
:$\dfrac 1 n \le \dfrac ... | Square Root is not Lipschitz Continuous | https://proofwiki.org/wiki/Square_Root_is_not_Lipschitz_Continuous | https://proofwiki.org/wiki/Square_Root_is_not_Lipschitz_Continuous | [
"Lipschitz Continuous Functions"
] | [
"Definition:Real Function",
"Definition:Lipschitz Continuity/Real Function"
] | [
"Definition:Lipschitz Continuity/Real Function",
"Definition:Finite",
"Definition:Contradiction"
] |
proofwiki-18038 | Triangle Inequality on Distance from Point to Subset | Let $M = \struct {A, d}$ be a metric space.
Let $H \subseteq A$.
Then:
:$\forall x, y \in A: \map d {x, H} \le \map d {x, y} + \map d {y, H}$
where $\map d {x, H}$ denotes the distance between $x$ and $H$. | {{begin-eqn}}
{{eqn | q = \forall z \in H
| l = \map d {y, z}
| o = \ge
| r = \map d {x, z} - \map d {x, y}
| c = {{Metric-space-axiom|2}}
}}
{{eqn | ll= \leadsto
| q = \forall z \in H
| l = \map d {y, z}
| o = \ge
| r = \map d {x, H} - \map d {x, y}
| c = {{Defof|D... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $H \subseteq A$.
Then:
:$\forall x, y \in A: \map d {x, H} \le \map d {x, y} + \map d {y, H}$
where $\map d {x, H}$ denotes the [[Definition:Distance between Element and Subset of Metric Space|distance between $x$ and $H$]]. | {{begin-eqn}}
{{eqn | q = \forall z \in H
| l = \map d {y, z}
| o = \ge
| r = \map d {x, z} - \map d {x, y}
| c = {{Metric-space-axiom|2}}
}}
{{eqn | ll= \leadsto
| q = \forall z \in H
| l = \map d {y, z}
| o = \ge
| r = \map d {x, H} - \map d {x, y}
| c = {{Defof|D... | Triangle Inequality on Distance from Point to Subset | https://proofwiki.org/wiki/Triangle_Inequality_on_Distance_from_Point_to_Subset | https://proofwiki.org/wiki/Triangle_Inequality_on_Distance_from_Point_to_Subset | [
"Distance Function",
"Triangle Inequality"
] | [
"Definition:Metric Space",
"Definition:Distance/Sets/Metric Spaces"
] | [] |
proofwiki-18039 | Equal Images of Mappings to Hausdorff Space form Closed Set | Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be topological spaces.
Let $T_B$ be a Hausdorff space.
Let $f, g: T_A \to T_B$ be continuous mappings.
Let $W$ be the set defined as:
:$W = \set {x \in T_A: \map f x = \map g x}$
Then $W$ is closed in $T_A$. | Consider the set $V = S_A \setminus W$.
Hence:
:$V = \set {x \in T_A: \map f x \ne \map g x}$
Let $x \in V$.
Then:
:$\map f x \ne \map g x$
and as $T_B$ is Hausdorff:
:$\exists U_1, U_2 \in \tau_B: \map f x \in U_1, \map g x \in U_2, U_1 \cap U_2 = \O$
As $f$ and $g$ are continuous mappings:
:$f^{-1} \sqbrk {U_1}$ and ... | Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be [[Definition:Topological Space|topological spaces]].
Let $T_B$ be a [[Definition:Hausdorff Space|Hausdorff space]].
Let $f, g: T_A \to T_B$ be [[Definition:Continuous Mapping|continuous mappings]].
Let $W$ be the [[Definition:Set|set]] defined as... | Consider the set $V = S_A \setminus W$.
Hence:
:$V = \set {x \in T_A: \map f x \ne \map g x}$
Let $x \in V$.
Then:
:$\map f x \ne \map g x$
and as $T_B$ is [[Definition:Hausdorff Space|Hausdorff]]:
:$\exists U_1, U_2 \in \tau_B: \map f x \in U_1, \map g x \in U_2, U_1 \cap U_2 = \O$
As $f$ and $g$ are [[Definitio... | Equal Images of Mappings to Hausdorff Space form Closed Set | https://proofwiki.org/wiki/Equal_Images_of_Mappings_to_Hausdorff_Space_form_Closed_Set | https://proofwiki.org/wiki/Equal_Images_of_Mappings_to_Hausdorff_Space_form_Closed_Set | [
"Hausdorff Spaces"
] | [
"Definition:Topological Space",
"Definition:T2 Space",
"Definition:Continuous Mapping",
"Definition:Set",
"Definition:Closed Set/Topology"
] | [
"Definition:T2 Space",
"Definition:Continuous Mapping",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology"
] |
proofwiki-18040 | Fixed Point Set of Continuous Self-Map on Hausdorff Space is Closed | Let $T = \struct {S, \tau}$ be a Hausdorff space.
Let $f: T \to T$ be a continuous mapping on $T$.
Let $W$ be the set defined as:
:$W = \set {x \in T: \map f x = x}$
Then $W$ is closed in $T$. | Let $g: T \to T$ be the identity mapping on $T$:
:$\forall x \in T: \map g x = x$
From Identity Mapping is Continuous, $g$ is a continuous mapping on $T$.
From Equal Images of Mappings to Hausdorff Space form Closed Set:
:$\set {x \in T: \map f x = \map g x}$ is closed in $T$.
and the result follows.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Hausdorff Space|Hausdorff space]].
Let $f: T \to T$ be a [[Definition:Continuous Mapping|continuous mapping]] on $T$.
Let $W$ be the [[Definition:Set|set]] defined as:
:$W = \set {x \in T: \map f x = x}$
Then $W$ is [[Definition:Closed Set (Topology)|closed in $T$]]. | Let $g: T \to T$ be the [[Definition:Identity Mapping|identity mapping]] on $T$:
:$\forall x \in T: \map g x = x$
From [[Identity Mapping is Continuous]], $g$ is a [[Definition:Continuous Mapping|continuous mapping]] on $T$.
From [[Equal Images of Mappings to Hausdorff Space form Closed Set]]:
:$\set {x \in T: \map f... | Fixed Point Set of Continuous Self-Map on Hausdorff Space is Closed | https://proofwiki.org/wiki/Fixed_Point_Set_of_Continuous_Self-Map_on_Hausdorff_Space_is_Closed | https://proofwiki.org/wiki/Fixed_Point_Set_of_Continuous_Self-Map_on_Hausdorff_Space_is_Closed | [
"Hausdorff Spaces"
] | [
"Definition:T2 Space",
"Definition:Continuous Mapping",
"Definition:Set",
"Definition:Closed Set/Topology"
] | [
"Definition:Identity Mapping",
"Identity Mapping is Continuous",
"Definition:Continuous Mapping",
"Equal Images of Mappings to Hausdorff Space form Closed Set",
"Definition:Closed Set/Topology"
] |
proofwiki-18041 | Continuous Real Function is Bounded on Neighborhood of Argument | Let $A \subseteq \R$ be a subset of the real number line $\R$.
Let $f: A \to \R$ be a continuous real function on $A$.
Let $a \in A$.
Then there exists a bound:
:$K_a = 1 + \size {\map f a}$ for $\size {\map f x}$
for all $x$ in some neighborhood:
:$\openint {a - \map \delta a} {a + \map \delta a}$ of $a$
where $\map \... | Let $a \in A$.
By definition of continuous real function, there exists $\delta \in \R_{>0}$ such that:
:$\forall x \in A: 0 < \size {x - a} < \delta \implies \size {\map f x - \map f a} < \epsilon$
for all $\epsilon \in \R_{>0}$.
Putting $\epsilon = 1$, say, gives us:
:$\forall x \in A: 0 < \size {x - a} < \delta \impl... | Let $A \subseteq \R$ be a [[Definition:Subset|subset]] of the [[Definition:Real Number Line|real number line]] $\R$.
Let $f: A \to \R$ be a [[Definition:Continuous Real Function|continuous real function]] on $A$.
Let $a \in A$.
Then there exists a [[Definition:Bound of Real-Valued Function|bound]]:
:$K_a = 1 + \siz... | Let $a \in A$.
By definition of [[Definition:Continuous Real Function|continuous real function]], there exists $\delta \in \R_{>0}$ such that:
:$\forall x \in A: 0 < \size {x - a} < \delta \implies \size {\map f x - \map f a} < \epsilon$
for all $\epsilon \in \R_{>0}$.
Putting $\epsilon = 1$, say, gives us:
:$\for... | Continuous Real Function is Bounded on Neighborhood of Argument | https://proofwiki.org/wiki/Continuous_Real_Function_is_Bounded_on_Neighborhood_of_Argument | https://proofwiki.org/wiki/Continuous_Real_Function_is_Bounded_on_Neighborhood_of_Argument | [
"Continuous Real-Valued Functions",
"Bounded Real-Valued Functions"
] | [
"Definition:Subset",
"Definition:Real Number/Real Number Line",
"Definition:Continuous Real Function",
"Definition:Bound of Real-Valued Function",
"Definition:Neighborhood (Real Analysis)/Epsilon",
"Definition:Positive/Real Number",
"Definition:Constant"
] | [
"Definition:Continuous Real Function"
] |
proofwiki-18042 | Continuous Real Function Bounded on Finite Subdivision | Let $A = \closedint a b$ be a closed real interval of the set $\R$ of real numbers.
Let $f: A \to \R$ be a continuous real function on $A$.
Let $P = \set {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ be a finite subdivision of $A$ such that:
:each $\closedint {x_j} {x_{j + 1} }$ is a neighborhood of some $a_j$ such that $f$... | Follows directly from:
:Continuous Real Function is Bounded on Neighborhood of Argument
and:
:Mapping is Bounded on Union iff Bounded on Each Component/Real-Valued Function.
{{qed}} | Let $A = \closedint a b$ be a [[Definition:Closed Real Interval|closed real interval]] of the set $\R$ of [[Definition:Real Number|real numbers]].
Let $f: A \to \R$ be a [[Definition:Continuous Real Function|continuous real function]] on $A$.
Let $P = \set {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ be a [[Definition:Fi... | Follows directly from:
:[[Continuous Real Function is Bounded on Neighborhood of Argument]]
and:
:[[Mapping is Bounded on Union iff Bounded on Each Component/Real-Valued Function]].
{{qed}} | Continuous Real Function Bounded on Finite Subdivision | https://proofwiki.org/wiki/Continuous_Real_Function_Bounded_on_Finite_Subdivision | https://proofwiki.org/wiki/Continuous_Real_Function_Bounded_on_Finite_Subdivision | [
"Continuous Real-Valued Functions",
"Boundedness"
] | [
"Definition:Real Interval/Closed",
"Definition:Real Number",
"Definition:Continuous Real Function",
"Definition:Subdivision of Interval/Finite",
"Definition:Neighborhood (Real Analysis)/Epsilon",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Bounded Mapping/Real-Valued"
] | [
"Continuous Real Function is Bounded on Neighborhood of Argument",
"Mapping is Bounded on Union iff Bounded on Each Component/Real-Valued Function"
] |
proofwiki-18043 | Linear Transformation from Finite-Dimensional Vector Space is Injective iff Surjective | Let $K$ be a field.
Let $V$ be a finite dimensional vector space over $K$.
Let $f: V \to V$ be a linear transformation on $V$.
Then $f$ is an injection {{iff}} $f$ is a surjection. | Let $n = \dim V$.
From Vector Space has Basis, there exists a basis:
:$\BB = \set {e_1, \ldots, e_n}$
for $V$.
Then from Image of Generating Set of Vector Space under Linear Transformation is Generating Set of Image, $f \sqbrk \BB$ is a generating set for $f \sqbrk V$. | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $V$ be a [[Definition:Finite Dimensional Vector Space|finite dimensional vector space]] over $K$.
Let $f: V \to V$ be a [[Definition:Linear Transformation|linear transformation]] on $V$.
Then $f$ is an [[Definition:Injection|injection]] {{iff}} $f$ is... | Let $n = \dim V$.
From [[Vector Space has Basis]], there exists a [[Definition:Basis of Vector Space|basis]]:
:$\BB = \set {e_1, \ldots, e_n}$
for $V$.
Then from [[Image of Generating Set of Vector Space under Linear Transformation is Generating Set of Image]], $f \sqbrk \BB$ is a [[Definition:Generator of Modu... | Linear Transformation from Finite-Dimensional Vector Space is Injective iff Surjective | https://proofwiki.org/wiki/Linear_Transformation_from_Finite-Dimensional_Vector_Space_is_Injective_iff_Surjective | https://proofwiki.org/wiki/Linear_Transformation_from_Finite-Dimensional_Vector_Space_is_Injective_iff_Surjective | [
"Linear Transformations"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Dimension of Vector Space/Finite",
"Definition:Linear Transformation",
"Definition:Injection",
"Definition:Surjection"
] | [
"Vector Space has Basis",
"Definition:Basis of Vector Space",
"Image of Generating Set of Vector Space under Linear Transformation is Generating Set of Image",
"Definition:Generator of Module",
"Definition:Generator of Module",
"Definition:Basis of Vector Space"
] |
proofwiki-18044 | Closed and Bounded Subspace is not necessarily Compact | Let $M = \struct {A, d}$ be a metric space.
Let $H \subseteq A$ be a subset of $A$.
Let $M_H = \struct {H, d_H}$ be the subspace of $M$ induced by $d$.
Let $H$ be closed and bounded.
Then it is not necessarily the case that $M_H$ is compact. | Proof by Counterexample:
Let $A = \openint 0 1$ be the open unit interval.
From Open Real Interval is not Compact, $\openint 0 1$ is not a compact space.
Let $H = \openint 0 1$.
Then $H \subseteq A$.
From Underlying Set of Topological Space is Clopen, $\openint 0 1$ is both closed and open in $\openint 0 1$.
Thus $H$ i... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $H \subseteq A$ be a [[Definition:Subset|subset]] of $A$.
Let $M_H = \struct {H, d_H}$ be the [[Definition:Metric Subspace|subspace]] of $M$ induced by $d$.
Let $H$ be [[Definition:Closed Set (Metric Space)|closed]] and [[Definition:Bounded ... | [[Proof by Counterexample]]:
Let $A = \openint 0 1$ be the [[Definition:Open Unit Interval|open unit interval]].
From [[Open Real Interval is not Compact]], $\openint 0 1$ is not a [[Definition:Compact Metric Space|compact space]].
Let $H = \openint 0 1$.
Then $H \subseteq A$.
From [[Underlying Set of Topological ... | Closed and Bounded Subspace is not necessarily Compact | https://proofwiki.org/wiki/Closed_and_Bounded_Subspace_is_not_necessarily_Compact | https://proofwiki.org/wiki/Closed_and_Bounded_Subspace_is_not_necessarily_Compact | [
"Compact Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Subset",
"Definition:Metric Subspace",
"Definition:Closed Set/Metric Space",
"Definition:Bounded Metric Space",
"Definition:Compact Space/Metric Space"
] | [
"Proof by Counterexample",
"Definition:Real Interval/Unit Interval/Open",
"Open Real Interval is not Compact",
"Definition:Compact Space/Metric Space",
"Underlying Set of Topological Space is Clopen",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Definition:Closed Set/Metric Space... |
proofwiki-18045 | Union of Two Compact Sets is Compact | Let $T = \struct {S, \tau}$ be a topological spaces.
Let $H$ and $K$ be compact subsets of $T$.
Then $H \cup K$ is compact in $T$. | Let $\CC$ be an open cover of $H \cup K$.
Then $\CC$ is an open cover of both $H$ and $K$.
As $H$ and $K$ are both compact in $T$:
:$H$ has a finite subcover $C_H$ of $\CC$
:$K$ has a finite subcover $C_K$ of $\CC$.
Their union $C_H \cup C_K$ is a finite subcover of $\CC$ for $H \cup K$.
From Union of Finite Sets is Fi... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological spaces]].
Let $H$ and $K$ be [[Definition:Compact Topological Subspace|compact subsets]] of $T$.
Then $H \cup K$ is [[Definition:Compact Topological Subspace|compact]] in $T$. | Let $\CC$ be an [[Definition:Open Cover|open cover]] of $H \cup K$.
Then $\CC$ is an [[Definition:Open Cover|open cover]] of both $H$ and $K$.
As $H$ and $K$ are both [[Definition:Compact Topological Subspace|compact]] in $T$:
:$H$ has a [[Definition:Finite Subcover|finite subcover]] $C_H$ of $\CC$
:$K$ has a [[Defin... | Union of Two Compact Sets is Compact | https://proofwiki.org/wiki/Union_of_Two_Compact_Sets_is_Compact | https://proofwiki.org/wiki/Union_of_Two_Compact_Sets_is_Compact | [
"Compact Topological Spaces",
"Set Union"
] | [
"Definition:Topological Space",
"Definition:Compact Topological Space/Subspace",
"Definition:Compact Topological Space/Subspace"
] | [
"Definition:Open Cover",
"Definition:Open Cover",
"Definition:Compact Topological Space/Subspace",
"Definition:Subcover/Finite",
"Definition:Subcover/Finite",
"Definition:Set Union",
"Definition:Subcover/Finite",
"Union of Finite Sets is Finite",
"Definition:Subcover/Finite",
"Definition:Compact T... |
proofwiki-18046 | Coarser Topology than Compact Space is Compact | Let $S$ be a set.
Let $\tau_1$ and $\tau_2$ be topologies on $S$ such that $\tau_1$ is coarser than $\tau_2$:
:$\tau_1 \subseteq \tau_2$
Let $\struct {S, \tau_2}$ be a compact space.
Then $\struct {S, \tau_1}$ is also compact. | Let $\struct {S, \tau_2}$ be a compact space as asserted.
Let $I_S: \struct {S, \tau_2} \to \struct {S, \tau_1}$ denote the identity mapping on $S$:
:$\forall x \in S: \map {I_S} x = x$
From Identity Mapping to Coarser Topology is Continuous, $I_S$ is continuous.
We also have the result Identity Mapping is Surjection.
... | Let $S$ be a [[Definition:Set|set]].
Let $\tau_1$ and $\tau_2$ be [[Definition:Topology|topologies]] on $S$ such that $\tau_1$ is [[Definition:Coarser Topology|coarser]] than $\tau_2$:
:$\tau_1 \subseteq \tau_2$
Let $\struct {S, \tau_2}$ be a [[Definition:Compact Topological Space|compact space]].
Then $\struct {S,... | Let $\struct {S, \tau_2}$ be a [[Definition:Compact Topological Space|compact space]] as asserted.
Let $I_S: \struct {S, \tau_2} \to \struct {S, \tau_1}$ denote the [[Definition:Identity Mapping|identity mapping]] on $S$:
:$\forall x \in S: \map {I_S} x = x$
From [[Identity Mapping to Coarser Topology is Continuous]]... | Coarser Topology than Compact Space is Compact | https://proofwiki.org/wiki/Coarser_Topology_than_Compact_Space_is_Compact | https://proofwiki.org/wiki/Coarser_Topology_than_Compact_Space_is_Compact | [
"Coarser Topology",
"Compact Topological Spaces"
] | [
"Definition:Set",
"Definition:Topology",
"Definition:Coarser Topology",
"Definition:Compact Topological Space",
"Definition:Compact Topological Space"
] | [
"Definition:Compact Topological Space",
"Definition:Identity Mapping",
"Identity Mapping to Coarser Topology is Continuous",
"Definition:Continuous Mapping",
"Identity Mapping is Surjection",
"Compactness is Preserved under Continuous Surjection"
] |
proofwiki-18047 | Space of Almost-Zero Sequences is not Closed in 2-Sequence Space | Let $\struct {\ell^2, \norm {\, \cdot \,}_2}$ be the normed 2-sequence vector space.
Let $\struct {c_{00}, \norm {\, \cdot \,}_2}$ be the normed vector space of almost-zero sequences.
Then $\struct {c_{00}, \norm {\, \cdot \,}_2}$ is not closed in $\struct {\ell^2, \norm {\, \cdot \,}_2}$. | Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $c_{00}$ such that:
:$\ds x_n := \tuple {1, \frac 1 2, \ldots \frac 1 n, 0, \ldots}$
Let $\ds x := \tuple {1, \frac 1 2, \ldots, \frac 1 n, \ldots}$ with $n \in \N_{>0}$.
We have that $x \in \ell^2 \setminus c_{00}$ where $\setminus$ denotes set difference.
Then... | Let $\struct {\ell^2, \norm {\, \cdot \,}_2}$ be the [[P-Sequence Space with P-Norm forms Normed Vector Space|normed 2-sequence vector space]].
Let $\struct {c_{00}, \norm {\, \cdot \,}_2}$ be the [[Space of Almost-Zero Sequences with P-Norm forms Normed Vector Space (or something similar)|normed vector space of almos... | Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $c_{00}$ such that:
:$\ds x_n := \tuple {1, \frac 1 2, \ldots \frac 1 n, 0, \ldots}$
Let $\ds x := \tuple {1, \frac 1 2, \ldots, \frac 1 n, \ldots}$ with $n \in \N_{>0}$.
We have that $x \in \ell^2 \setminus c_{00}$ where $\setminus$ d... | Space of Almost-Zero Sequences is not Closed in 2-Sequence Space | https://proofwiki.org/wiki/Space_of_Almost-Zero_Sequences_is_not_Closed_in_2-Sequence_Space | https://proofwiki.org/wiki/Space_of_Almost-Zero_Sequences_is_not_Closed_in_2-Sequence_Space | [
"Closed Sets"
] | [
"P-Sequence Space with P-Norm forms Normed Vector Space",
"Space of Almost-Zero Sequences with P-Norm forms Normed Vector Space (or something similar)",
"Definition:Closed Set/Normed Vector Space/Definition 2"
] | [
"Definition:Sequence",
"Definition:Set Difference",
"Definition:Limit of Sequence/Normed Vector Space",
"Definition:Subset",
"Definition:Limit Point/Normed Vector Space/Sequence",
"Definition:Closed Set/Normed Vector Space/Definition 2"
] |
proofwiki-18048 | Equivalence of Definitions of Preimage of Subset under Mapping | Let $f: S \to T$ be a mapping from a set $S$ to a set $T$.
Let $Y \subseteq T$ be a subset of $T$.
{{TFAE|def = Preimage of Subset under Mapping}} | The difference in definitions is no more than a difference in notations.
Let $X$ be a preimage by definition $2$.
Then by definition:
:$X := \map {f^\gets} Y$
By definition of inverse image mapping of mapping:
:$\forall Y \in \powerset T: \map {f^\gets} Y = \set {s \in S: \exists t \in Y: \map f s = t}$
Thus $X$ is a p... | Let $f: S \to T$ be a [[Definition:Mapping|mapping]] from a [[Definition:Set|set]] $S$ to a [[Definition:Set|set]] $T$.
Let $Y \subseteq T$ be a [[Definition:Subset|subset]] of $T$.
{{TFAE|def = Preimage of Subset under Mapping}} | The difference in definitions is no more than a difference in notations.
Let $X$ be a [[Definition:Preimage of Subset under Mapping/Definition 2|preimage by definition $2$]].
Then by definition:
:$X := \map {f^\gets} Y$
By definition of [[Definition:Inverse Image Mapping of Mapping|inverse image mapping of mapping]... | Equivalence of Definitions of Preimage of Subset under Mapping | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Preimage_of_Subset_under_Mapping | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Preimage_of_Subset_under_Mapping | [
"Preimage of Subset under Mapping"
] | [
"Definition:Mapping",
"Definition:Set",
"Definition:Set",
"Definition:Subset"
] | [
"Definition:Preimage of Subset under Mapping/Definition 2",
"Definition:Inverse Image Mapping/Mapping",
"Definition:Preimage of Subset under Mapping/Definition 1",
"Category:Preimage of Subset under Mapping"
] |
proofwiki-18049 | 1-Sequence Space is Proper Subset of 2-Sequence Space | Let $\ell^1$ and $\ell^2$ be the $1$-sequence space and $2$-sequence space respectively.
Then $\ell^1$ is a proper subset of $\ell^2$. | === $\ell^1$ is a subset of $\ell^2$ ===
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $\ell^1$.
By definition of $1$-sequence space:
:$\ds \sum_{n \mathop = 0}^\infty \size {x_n} < \infty$
By Terms in Convergent Series Converge to Zero:
:$\ds \lim_{n \mathop \to \infty} \size {x_n} = 0$
By definition of co... | Let $\ell^1$ and $\ell^2$ be the [[Definition:P-Sequence Space|$1$-sequence space]] and [[Definition:P-Sequence Space|$2$-sequence space]] respectively.
Then $\ell^1$ is a [[Definition:Proper Subset|proper subset]] of $\ell^2$. | === $\ell^1$ is a [[Definition:Subset|subset]] of $\ell^2$ ===
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $\ell^1$.
By definition of [[Definition:P-Sequence Space|$1$-sequence space]]:
:$\ds \sum_{n \mathop = 0}^\infty \size {x_n} < \infty$
By [[Terms in Convergent Series Conv... | 1-Sequence Space is Proper Subset of 2-Sequence Space | https://proofwiki.org/wiki/1-Sequence_Space_is_Proper_Subset_of_2-Sequence_Space | https://proofwiki.org/wiki/1-Sequence_Space_is_Proper_Subset_of_2-Sequence_Space | [
"P-Sequence Spaces",
"Proper Subsets",
"P-Sequence Spaces"
] | [
"Definition:P-Sequence Space",
"Definition:P-Sequence Space",
"Definition:Proper Subset"
] | [
"Definition:Subset",
"Definition:Sequence",
"Definition:P-Sequence Space",
"Terms in Convergent Series Converge to Zero",
"Definition:Convergent Sequence/Normed Vector Space",
"Comparison Test",
"Definition:Subset"
] |
proofwiki-18050 | Cardinality of Set of Characteristic Functions on Finite Set | Let $I$ be a finite set.
The number of characteristic functions on $I$ is:
:$2^{\card I}$
where $\card I$ denotes the cardinality of $I$. | Let $A = \set {0, 1}$.
A characteristic function of $I$ is a mapping from $I$ to $A$.
Hence the set of characteristic functions on $I$ is the indexed Cartesian space $A_I$:
:$A^I = \ds \prod_{i \mathop \in I} A := \set {f: \paren {f: I \to A} \land \paren {\forall i \in I: \paren {\map f i \in A} } }$
Hence from Cardin... | Let $I$ be a [[Definition:Finite Set|finite set]].
The number of [[Definition:Characteristic Function of Set|characteristic functions]] on $I$ is:
:$2^{\card I}$
where $\card I$ denotes the [[Definition:Cardinality|cardinality]] of $I$. | Let $A = \set {0, 1}$.
A [[Definition:Characteristic Function of Set|characteristic function]] of $I$ is a [[Definition:Mapping|mapping]] from $I$ to $A$.
Hence the [[Definition:Set|set]] of [[Definition:Characteristic Function of Set|characteristic functions]] on $I$ is the [[Definition:Indexed Cartesian Space|index... | Cardinality of Set of Characteristic Functions on Finite Set | https://proofwiki.org/wiki/Cardinality_of_Set_of_Characteristic_Functions_on_Finite_Set | https://proofwiki.org/wiki/Cardinality_of_Set_of_Characteristic_Functions_on_Finite_Set | [
"Characteristic Functions",
"Indexed Families"
] | [
"Definition:Finite Set",
"Definition:Characteristic Function (Set Theory)/Set",
"Definition:Cardinality"
] | [
"Definition:Characteristic Function (Set Theory)/Set",
"Definition:Mapping",
"Definition:Set",
"Definition:Characteristic Function (Set Theory)/Set",
"Definition:Cartesian Product/Cartesian Space/Family of Sets",
"Cardinality of Set of All Mappings/Finite Sets"
] |
proofwiki-18051 | Cartesian Product of Projections is Projection on Cartesian Product of Mappings | Let $I$ be an indexing set.
Let $\family {S_\alpha}_{\alpha \mathop \in I}$ and $\family {T_\alpha}_{\alpha \mathop \in I}$ be families of sets both indexed by $I$.
For each $\alpha \in I$, let $f_\alpha: S_\alpha \to T_\alpha$ be a mapping.
There exists a unique mapping:
:$\ds f: \prod_{\alpha \mathop \in I} S_\alpha ... | === Proof of Existence ===
Let $\mathbf x \in \ds \prod_{\alpha \mathop \in I} S_\alpha$ be arbitrary:
:$\mathbf x = \family {x_\alpha \in S_\alpha}_{\alpha \mathop \in I}$
Let $\ds f: \prod_{\alpha \mathop \in I} S_\alpha \to \prod_{\alpha \mathop \in I} T_\alpha$ be defined as:
:$\forall \mathbf x \in \ds \prod_{\alp... | Let $I$ be an [[Definition:Indexing Set|indexing set]].
Let $\family {S_\alpha}_{\alpha \mathop \in I}$ and $\family {T_\alpha}_{\alpha \mathop \in I}$ be [[Definition:Indexed Family of Sets|families of sets both indexed by $I$]].
For each $\alpha \in I$, let $f_\alpha: S_\alpha \to T_\alpha$ be a [[Definition:Mappin... | === Proof of Existence ===
Let $\mathbf x \in \ds \prod_{\alpha \mathop \in I} S_\alpha$ be arbitrary:
:$\mathbf x = \family {x_\alpha \in S_\alpha}_{\alpha \mathop \in I}$
Let $\ds f: \prod_{\alpha \mathop \in I} S_\alpha \to \prod_{\alpha \mathop \in I} T_\alpha$ be defined as:
:$\forall \mathbf x \in \ds \prod_{\a... | Cartesian Product of Projections is Projection on Cartesian Product of Mappings | https://proofwiki.org/wiki/Cartesian_Product_of_Projections_is_Projection_on_Cartesian_Product_of_Mappings | https://proofwiki.org/wiki/Cartesian_Product_of_Projections_is_Projection_on_Cartesian_Product_of_Mappings | [
"Projections"
] | [
"Definition:Indexing Set",
"Definition:Indexing Set/Family of Sets",
"Definition:Mapping",
"Definition:Unique",
"Definition:Mapping",
"Definition:Composition of Mappings",
"Definition:Projection (Mapping Theory)/Family of Sets"
] | [] |
proofwiki-18052 | Space of Zero-Limit Sequences with Supremum Norm forms Banach Space | Let $c_0$ be the space of zero-limit sequences.
Let $\norm {\, \cdot \,}_\infty$ be the supremum norm.
Then $\struct {c_0, \norm {\, \cdot \,}_\infty}$ is a Banach space. | Let $\sequence {a_n}_{n \mathop \in \N}$ be a Cauchy sequence in $\struct {c_0, \norm {\, \cdot \,}_\infty}$.
Let $\struct {\ell^\infty, \norm {\, \cdot \,}_\infty}$ be the normed vector space of bounded sequences.
By Space of Zero-Limit Sequences with Supremum Norm forms Normed Vector Space, $\struct {c_0, \norm {\, \... | Let $c_0$ be the [[Definition:Space of Zero-Limit Sequences|space of zero-limit sequences]].
Let $\norm {\, \cdot \,}_\infty$ be the [[Definition:Supremum Norm|supremum norm]].
Then $\struct {c_0, \norm {\, \cdot \,}_\infty}$ is a [[Definition:Banach Space|Banach space]]. | Let $\sequence {a_n}_{n \mathop \in \N}$ be a [[Definition:Cauchy Sequence|Cauchy sequence]] in $\struct {c_0, \norm {\, \cdot \,}_\infty}$.
Let $\struct {\ell^\infty, \norm {\, \cdot \,}_\infty}$ be the [[Space of Bounded Sequences with Supremum Norm forms Normed Vector Space|normed vector space of bounded sequences]... | Space of Zero-Limit Sequences with Supremum Norm forms Banach Space | https://proofwiki.org/wiki/Space_of_Zero-Limit_Sequences_with_Supremum_Norm_forms_Banach_Space | https://proofwiki.org/wiki/Space_of_Zero-Limit_Sequences_with_Supremum_Norm_forms_Banach_Space | [
"Banach Spaces"
] | [
"Definition:Space of Zero-Limit Sequences",
"Definition:Supremum Norm",
"Definition:Banach Space"
] | [
"Definition:Cauchy Sequence",
"Space of Bounded Sequences with Supremum Norm forms Normed Vector Space",
"Space of Zero-Limit Sequences with Supremum Norm forms Normed Vector Space",
"Definition:Vector Subspace",
"Definition:Cauchy Sequence",
"Space of Bounded Sequences with Supremum Norm forms Banach Spa... |
proofwiki-18053 | Composition of Cartesian Products of Mappings | Let $I$ be an indexing set.
Let $\family {S_\alpha}_{\alpha \mathop \in I}$, $\family {T_\alpha}_{\alpha \mathop \in I}$ and $\family {U_\alpha}_{\alpha \mathop \in I}$ be families of sets all indexed by $I$.
For each $\alpha \in I$, let:
:$f_\alpha: S_\alpha \to T_\alpha$ be a mapping
:$g_\alpha: T_\alpha \to U_\alpha... | First note that for all $\alpha \in I$:
:$\Dom {g_\alpha} = \Cdm {f_\alpha} = T_\alpha$
where $\Dom {g_\alpha}$ denotes the domain of $g_\alpha$ and $\Cdm {f_\alpha}$ denotes the codomain of $f_\alpha$.
So $g_\alpha \circ f_\alpha$ is defined for all $\alpha \in I$.
Similarly:
:$\Cdm f = \Dom g = T$
and so $g \circ f$ ... | Let $I$ be an [[Definition:Indexing Set|indexing set]].
Let $\family {S_\alpha}_{\alpha \mathop \in I}$, $\family {T_\alpha}_{\alpha \mathop \in I}$ and $\family {U_\alpha}_{\alpha \mathop \in I}$ be [[Definition:Indexed Family of Sets|families of sets all indexed by $I$]].
For each $\alpha \in I$, let:
:$f_\alpha: ... | First note that for all $\alpha \in I$:
:$\Dom {g_\alpha} = \Cdm {f_\alpha} = T_\alpha$
where $\Dom {g_\alpha}$ denotes the [[Definition:Domain of Mapping|domain]] of $g_\alpha$ and $\Cdm {f_\alpha}$ denotes the [[Definition:Codomain of Mapping|codomain]] of $f_\alpha$.
So $g_\alpha \circ f_\alpha$ is defined for al... | Composition of Cartesian Products of Mappings | https://proofwiki.org/wiki/Composition_of_Cartesian_Products_of_Mappings | https://proofwiki.org/wiki/Composition_of_Cartesian_Products_of_Mappings | [
"Cartesian Product",
"Composite Mappings"
] | [
"Definition:Indexing Set",
"Definition:Indexing Set/Family of Sets",
"Definition:Mapping",
"Definition:Mapping",
"Definition:Composition of Mappings"
] | [
"Definition:Domain (Set Theory)/Mapping",
"Definition:Codomain (Set Theory)/Mapping"
] |
proofwiki-18054 | Partition of Indexing Set induces Bijection on Family of Sets | Let $I$ be an indexing set.
Let $\family {S_\alpha}_{\alpha \mathop \in I}$ be a family of sets indexed by $I$.
Let $\family {I_\gamma}_{\gamma \mathop \in J}$ be a partitioning of $I$.
Then there exists a bijection:
:$\ds \phi: \prod_{\gamma \mathop \in J} \paren {\prod_{\alpha \mathop \in I_\gamma} S_\alpha} \to \pro... | First a lemma:
{{:Partition of Indexing Set induces Bijection on Family of Sets/Lemma}}{{qed|lemma}}
We can define a projection $\pr_\gamma$:
:$\ds \pr_\gamma: \prod_{\gamma \mathop \in J} \paren {\prod_{\alpha \mathop \in I_\gamma} S_\alpha} \to \prod_{\alpha \mathop \in I_\gamma} S_\alpha$
so that for $\ds X \in \pro... | Let $I$ be an [[Definition:Indexing Set|indexing set]].
Let $\family {S_\alpha}_{\alpha \mathop \in I}$ be a [[Definition:Indexed Family of Sets|family of sets indexed by $I$]].
Let $\family {I_\gamma}_{\gamma \mathop \in J}$ be a [[Definition:Partitioning|partitioning]] of $I$.
Then there exists a [[Definition:Bij... | First a [[Partition of Indexing Set induces Bijection on Family of Sets/Lemma|lemma]]:
{{:Partition of Indexing Set induces Bijection on Family of Sets/Lemma}}{{qed|lemma}}
We can define a [[Definition:Projection (Mapping Theory)|projection]] $\pr_\gamma$:
:$\ds \pr_\gamma: \prod_{\gamma \mathop \in J} \paren {\prod_... | Partition of Indexing Set induces Bijection on Family of Sets | https://proofwiki.org/wiki/Partition_of_Indexing_Set_induces_Bijection_on_Family_of_Sets | https://proofwiki.org/wiki/Partition_of_Indexing_Set_induces_Bijection_on_Family_of_Sets | [
"Indexed Families",
"Bijections",
"Set Partitions"
] | [
"Definition:Indexing Set",
"Definition:Indexing Set/Family of Sets",
"Definition:Partitioning",
"Definition:Bijection"
] | [
"Partition of Indexing Set induces Bijection on Family of Sets/Lemma",
"Definition:Projection (Mapping Theory)"
] |
proofwiki-18055 | Partition of Indexing Set induces Bijection on Family of Sets/Lemma | Let $I$ be an indexing set.
Let $\family {S_\alpha}_{\alpha \mathop \in I}$ be a family of sets indexed by $I$.
Let $I = I_1 \cup I_2$ such that $I_1 \cap I_2 = \O$.
Then there exists a bijection:
:$\ds \psi: \paren {\prod_{\alpha \mathop \in I_1} S_\alpha} \times \paren {\prod_{\alpha \mathop \in I_2} S_\alpha} \to \p... | Let us define the mapping:
:$\ds \psi: \paren {\prod_{\alpha \mathop \in I_1} S_\alpha} \times \paren {\prod_{\alpha \mathop \in I_2} S_\alpha} \to \prod_{\alpha \mathop \in I} S_\alpha$
$\psi$ can be injective {{iff}}:
:$\map \psi a = \map \psi {a'} \implies a = a'$
where $\ds a, a' \in \paren {\prod_{\alpha \mathop \... | Let $I$ be an [[Definition:Indexing Set|indexing set]].
Let $\family {S_\alpha}_{\alpha \mathop \in I}$ be a [[Definition:Indexed Family of Sets|family of sets indexed by $I$]].
Let $I = I_1 \cup I_2$ such that $I_1 \cap I_2 = \O$.
Then there exists a [[Definition:Bijection|bijection]]:
:$\ds \psi: \paren {\prod_{\... | Let us define the [[Definition:Mapping|mapping]]:
:$\ds \psi: \paren {\prod_{\alpha \mathop \in I_1} S_\alpha} \times \paren {\prod_{\alpha \mathop \in I_2} S_\alpha} \to \prod_{\alpha \mathop \in I} S_\alpha$
$\psi$ can be [[Definition:Injection|injective]] {{iff}}:
:$\map \psi a = \map \psi {a'} \implies a = a'$
w... | Partition of Indexing Set induces Bijection on Family of Sets/Lemma | https://proofwiki.org/wiki/Partition_of_Indexing_Set_induces_Bijection_on_Family_of_Sets/Lemma | https://proofwiki.org/wiki/Partition_of_Indexing_Set_induces_Bijection_on_Family_of_Sets/Lemma | [
"Indexed Families",
"Bijections",
"Set Partitions"
] | [
"Definition:Indexing Set",
"Definition:Indexing Set/Family of Sets",
"Definition:Bijection"
] | [
"Definition:Mapping",
"Definition:Injection",
"Definition:Projection (Mapping Theory)",
"Definition:Projection (Mapping Theory)",
"Definition:Surjection"
] |
proofwiki-18056 | Subsequence of Subsequence | Let $s$ be a set.
Let $\sequence {s_n}$ be a sequence in $S$.
Let $\sequence {s_m}$ be a subsequence of $\sequence {s_n}$.
Let $\sequence {s_k}$ be a subsequence of $\sequence {s_m}$.
Then $\sequence {s_k}$ is a subsequence of $\sequence {s_n}$. | By definition, there exists a strictly increasing sequence $\sequence {n_r}$ in $\N$ such that:
:$\forall m \in \N: s_m = s_{n_r}$
Similarly, there exists a strictly increasing sequence $\sequence {m_s}$ in $\N$ such that:
:$\forall k \in \N: s_k = s_{m_s}$
We have that:
:$\forall k \in \N: s_k \in \sequence {s_m}$
and... | Let $s$ be a [[Definition:Set|set]].
Let $\sequence {s_n}$ be a [[Definition:Sequence|sequence in $S$]].
Let $\sequence {s_m}$ be a [[Definition:Subsequence|subsequence]] of $\sequence {s_n}$.
Let $\sequence {s_k}$ be a [[Definition:Subsequence|subsequence]] of $\sequence {s_m}$.
Then $\sequence {s_k}$ is a [[Defi... | By definition, there exists a [[Definition:Strictly Increasing Sequence|strictly increasing sequence $\sequence {n_r}$ in $\N$]] such that:
:$\forall m \in \N: s_m = s_{n_r}$
Similarly, there exists a [[Definition:Strictly Increasing Sequence|strictly increasing sequence $\sequence {m_s}$ in $\N$]] such that:
:$\foral... | Subsequence of Subsequence | https://proofwiki.org/wiki/Subsequence_of_Subsequence | https://proofwiki.org/wiki/Subsequence_of_Subsequence | [
"Subsequences"
] | [
"Definition:Set",
"Definition:Sequence",
"Definition:Subsequence",
"Definition:Subsequence",
"Definition:Subsequence"
] | [
"Definition:Strictly Increasing/Sequence",
"Definition:Strictly Increasing/Sequence",
"Definition:Subsequence",
"Definition:Subsequence"
] |
proofwiki-18057 | Cardinality of Extensions of Function on Subset of Finite Set | Let $m, n \in \Z_{>0}$ be (strictly) positive integers.
Let $S$ be a set with $m$ elements.
Let $T$ be a set with $n$ elements.
Let $A$ be a subset of $S$ with $r$ elements where $0 \le r < m$.
Let $f: A \to T$ be a mapping.
Then there are $n^{m - r}$ distinct extensions of $f$ to $S$. | Let $N$ denote the number of distinct extensions of $f$ to $S$.
The question is equivalent to asking the number of distinct mappings from $S \setminus A$ to $T$.
There are $m - r$ elements in $S \setminus A$.
Hence from Cardinality of Set of All Mappings:
:$N = n^{m - r}$
{{qed}} | Let $m, n \in \Z_{>0}$ be [[Definition:Strictly Positive Integer|(strictly) positive integers]].
Let $S$ be a [[Definition:Set|set]] with $m$ [[Definition:Element|elements]].
Let $T$ be a [[Definition:Set|set]] with $n$ [[Definition:Element|elements]].
Let $A$ be a [[Definition:Subset|subset]] of $S$ with $r$ [[Defi... | Let $N$ denote the number of [[Definition:Distinct Elements|distinct]] [[Definition:Extension of Mapping|extensions]] of $f$ to $S$.
The question is equivalent to asking the number of [[Definition:Distinct Elements|distinct]] [[Definition:Mapping|mappings]] from $S \setminus A$ to $T$.
There are $m - r$ [[Definition:... | Cardinality of Extensions of Function on Subset of Finite Set | https://proofwiki.org/wiki/Cardinality_of_Extensions_of_Function_on_Subset_of_Finite_Set | https://proofwiki.org/wiki/Cardinality_of_Extensions_of_Function_on_Subset_of_Finite_Set | [
"Finite Sets",
"Restrictions",
"Mapping Theory",
"Cardinality"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Set",
"Definition:Element",
"Definition:Set",
"Definition:Element",
"Definition:Subset",
"Definition:Element",
"Definition:Mapping",
"Definition:Distinct/Plural",
"Definition:Extension of Mapping"
] | [
"Definition:Distinct/Plural",
"Definition:Extension of Mapping",
"Definition:Distinct/Plural",
"Definition:Mapping",
"Definition:Element",
"Cardinality of Set of All Mappings"
] |
proofwiki-18058 | L1 Metric is Topologically Equivalent to Supremum Metric on Bounded Continuous Real Functions | Let $\FF$ be the set of all real functions which are also bounded on the closed interval $\closedint a b$.
Let $d: \FF \times \FF \to \R$ be the $L^1$ metric on $\closedint a b$:
:$\ds \forall f, g \in \FF: \map d {f, g} := \int_a^b \size {\map f t - \map g t} \rd t$
Let $d': \FF \times \FF \to \R$ be the supremum metr... | Let $U$ be an upper bound of $\set {\size {\map f x - \map g x} }$.
Then:
:$\ds U \ge \sup_{x \mathop \in S} \size {\map f x - \map g x}$
Hence:
:$\ds \max_{x \mathop \in \closedint a b} \set {\size {\map f x - \map g x} } = \map {d'} {f, g}$
Then:
{{begin-eqn}}
{{eqn | l = \map d {f, g}
| r = \int_a^b \size {\ma... | Let $\FF$ be the [[Definition:Set|set]] of all [[Definition:Real Function|real functions]] which are also [[Definition:Bounded Real-Valued Function|bounded]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Let $d: \FF \times \FF \to \R$ be the [[Definition:L1 Metric on Closed Real Interval... | Let $U$ be an [[Definition:Upper Bound of Set|upper bound]] of $\set {\size {\map f x - \map g x} }$.
Then:
:$\ds U \ge \sup_{x \mathop \in S} \size {\map f x - \map g x}$
Hence:
:$\ds \max_{x \mathop \in \closedint a b} \set {\size {\map f x - \map g x} } = \map {d'} {f, g}$
Then:
{{begin-eqn}}
{{eqn | l = \map d {... | L1 Metric is Topologically Equivalent to Supremum Metric on Bounded Continuous Real Functions | https://proofwiki.org/wiki/L1_Metric_is_Topologically_Equivalent_to_Supremum_Metric_on_Bounded_Continuous_Real_Functions | https://proofwiki.org/wiki/L1_Metric_is_Topologically_Equivalent_to_Supremum_Metric_on_Bounded_Continuous_Real_Functions | [
"L1 Metric",
"Supremum Metric"
] | [
"Definition:Set",
"Definition:Real Function",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Real Interval/Closed",
"Definition:L1 Metric/Closed Real Interval",
"Definition:Supremum Metric",
"Definition:Topologically Equivalent Metrics"
] | [
"Definition:Upper Bound of Set"
] |
proofwiki-18059 | Mapping whose Graph is Closed in Chebyshev Product is not necessarily Continuous | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $\AA = A_1 \times A_2$ be the cartesian product of $A_1$ and $A_2$.
Let $d_\infty: \AA \times \AA \to \R$ be the Chebyshev distance on $\AA$:
:$\ds \map {d_\infty} {x, y} = \max \set {\map {d_1} {x_1, y_1}, \map {d_2} {x_2, y_2} }$
wher... | Consider the mapping $f: \R \to \R$ defined as:
:$\map f x = \begin {cases} \dfrac 1 x : & x > 0 \\ 0 : & x \le 0 \end {cases}$
It is seen that $\map f x$ is continuous everywhere except at $x = 0$.
Hence from Graph of Continuous Mapping between Metric Spaces is Closed in Chebyshev Product, $\Gamma_f$ contains all its ... | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be [[Definition:Metric Space|metric spaces]].
Let $\AA = A_1 \times A_2$ be the [[Definition:Cartesian Product|cartesian product]] of $A_1$ and $A_2$.
Let $d_\infty: \AA \times \AA \to \R$ be the [[Definition:Chebyshev Distance|Chebyshev distance]] on $\AA... | Consider the [[Definition:Mapping|mapping]] $f: \R \to \R$ defined as:
:$\map f x = \begin {cases} \dfrac 1 x : & x > 0 \\ 0 : & x \le 0 \end {cases}$
It is seen that $\map f x$ is [[Definition:Continuous Real Function|continuous]] everywhere except at $x = 0$.
Hence from [[Graph of Continuous Mapping between Metric... | Mapping whose Graph is Closed in Chebyshev Product is not necessarily Continuous | https://proofwiki.org/wiki/Mapping_whose_Graph_is_Closed_in_Chebyshev_Product_is_not_necessarily_Continuous | https://proofwiki.org/wiki/Mapping_whose_Graph_is_Closed_in_Chebyshev_Product_is_not_necessarily_Continuous | [
"Continuous Mappings",
"Chebyshev Distance"
] | [
"Definition:Metric Space",
"Definition:Cartesian Product",
"Definition:Chebyshev Distance",
"Definition:Graph of Mapping",
"Definition:Mapping",
"Definition:Closed Set/Metric Space",
"Definition:Continuous Mapping (Metric Space)"
] | [
"Definition:Mapping",
"Definition:Continuous Real Function",
"Graph of Continuous Mapping between Metric Spaces is Closed in Chebyshev Product",
"Definition:Limit Point/Metric Space",
"Definition:Bounded Mapping/Metric Space",
"Definition:Continuous Real Function",
"Definition:Limit Point/Metric Space",... |
proofwiki-18060 | Right Order Topology on Strictly Positive Integers is Topology | Let $\Z_{>0}$ be the set of strictly positive integers.
Let $\tau$ be the '''right order topology on $\Z_{>0}$'''.
Then $\tau$ forms a topology on $\Z_{>0}$.
That is:
:$T = \struct {\Z_{>0}, \tau}$ is a topological space. | Let $S := \Z_{>0}$ to ease notational clutter.
First we note that:
:$m \le n \implies O_n \subseteq O_m$
where $O_n := \set {x \in \Z_{>0}: x \ge n}$.
By definition we have that:
:$\O \in \tau$
Then each of the open set axioms is examined in turn: | Let $\Z_{>0}$ be the [[Definition:Set|set]] of [[Definition:Strictly Positive Integer|strictly positive integers]].
Let $\tau$ be the '''[[Definition:Right Order Topology on Strictly Positive Integers|right order topology on $\Z_{>0}$]]'''.
Then $\tau$ forms a [[Definition:Topology|topology]] on $\Z_{>0}$.
That is:... | Let $S := \Z_{>0}$ to ease notational clutter.
First we note that:
:$m \le n \implies O_n \subseteq O_m$
where $O_n := \set {x \in \Z_{>0}: x \ge n}$.
By definition we have that:
:$\O \in \tau$
Then each of the [[Axiom:Open Set Axioms|open set axioms]] is examined in turn: | Right Order Topology on Strictly Positive Integers is Topology | https://proofwiki.org/wiki/Right_Order_Topology_on_Strictly_Positive_Integers_is_Topology | https://proofwiki.org/wiki/Right_Order_Topology_on_Strictly_Positive_Integers_is_Topology | [
"Right Order Topologies"
] | [
"Definition:Set",
"Definition:Strictly Positive/Integer",
"Definition:Right Order Topology on Strictly Positive Integers",
"Definition:Topology",
"Definition:Topological Space"
] | [
"Axiom:Open Set Axioms",
"Axiom:Open Set Axioms"
] |
proofwiki-18061 | Right Order Topology on Strictly Positive Integers is not Metrizable | Let $\Z_{>0}$ be the set of strictly positive integers.
Let $T = \struct {\Z_{>0}, \tau}$ denote the '''right order space on $\Z_{>0}$'''.
Then $T = \struct {\Z_{>0}, \tau}$ is not a metrizable space. | Let $m, n \in \Z_{>0}$ such that $m < n$.
Let $O_m$ and $O_n$ be arbitrary non-empty open sets of $T$.
Then:
:$O_m \cap O_n = O_m$
As $O_m$ and $O_n$ are arbitrary , it follows that there exist no $O_m$ and $O_n$ in $\tau$ such that $O_m \cap O_n = \O$.
Hence $T$ is not a $T_2$ (Hausdorff) space.
The result follows fro... | Let $\Z_{>0}$ be the [[Definition:Set|set]] of [[Definition:Strictly Positive Integer|strictly positive integers]].
Let $T = \struct {\Z_{>0}, \tau}$ denote the '''[[Definition:Right Order Space on Strictly Positive Integers|right order space on $\Z_{>0}$]]'''.
Then $T = \struct {\Z_{>0}, \tau}$ is not a [[Definitio... | Let $m, n \in \Z_{>0}$ such that $m < n$.
Let $O_m$ and $O_n$ be [[Definition:Arbitrary|arbitrary]] [[Definition:Non-Empty Set|non-empty]] [[Definition:Open Set (Topology)|open sets]] of $T$.
Then:
:$O_m \cap O_n = O_m$
As $O_m$ and $O_n$ are [[Definition:Arbitrary|arbitrary]] , it follows that there exist no $O_m$ ... | Right Order Topology on Strictly Positive Integers is not Metrizable | https://proofwiki.org/wiki/Right_Order_Topology_on_Strictly_Positive_Integers_is_not_Metrizable | https://proofwiki.org/wiki/Right_Order_Topology_on_Strictly_Positive_Integers_is_not_Metrizable | [
"Right Order Topologies",
"Examples of Metrizable Spaces"
] | [
"Definition:Set",
"Definition:Strictly Positive/Integer",
"Definition:Right Order Topology on Strictly Positive Integers",
"Definition:Metrizable Space"
] | [
"Definition:Arbitrary",
"Definition:Non-Empty Set",
"Definition:Open Set/Topology",
"Definition:Arbitrary",
"Definition:T2 Space",
"Metrizable Space is Hausdorff"
] |
proofwiki-18062 | Euclidean Metric and Chebyshev Distance on Real Metric Space give rise to Same Topological Space | For $n \in \N$, let $\R^n$ be an Euclidean space.
Let $d_2$ be the Euclidean metric on $\R^n$.
Let $d_\infty$ be the Chebyshev distance on $\R^n$.
Let $T_2 = \struct {\R^n, \tau_2}$ denote the topological space which is induced by $d_2$.
Let $T_\infty = \struct {\R^n, \tau_\infty}$ denote the topological space which is... | From P-Product Metrics on Real Vector Space are Topologically Equivalent, $\tau_2$ and $\tau_\infty$ are topologically equivalent metrics.
The result follows from Topologically Equivalent Metrics induce Equal Topologies.
{{qed}} | For $n \in \N$, let $\R^n$ be an [[Definition:Euclidean Space|Euclidean space]].
Let $d_2$ be the [[Definition:Euclidean Metric on Real Vector Space|Euclidean metric]] on $\R^n$.
Let $d_\infty$ be the [[Definition:Chebyshev Distance on Real Vector Space|Chebyshev distance]] on $\R^n$.
Let $T_2 = \struct {\R^n, \tau... | From [[P-Product Metrics on Real Vector Space are Topologically Equivalent]], $\tau_2$ and $\tau_\infty$ are [[Definition:Topologically Equivalent Metrics|topologically equivalent metrics]].
The result follows from [[Topologically Equivalent Metrics induce Equal Topologies]].
{{qed}} | Euclidean Metric and Chebyshev Distance on Real Metric Space give rise to Same Topological Space | https://proofwiki.org/wiki/Euclidean_Metric_and_Chebyshev_Distance_on_Real_Metric_Space_give_rise_to_Same_Topological_Space | https://proofwiki.org/wiki/Euclidean_Metric_and_Chebyshev_Distance_on_Real_Metric_Space_give_rise_to_Same_Topological_Space | [
"Chebyshev Distance",
"Euclidean Metric",
"Topologically Equivalent Metrics"
] | [
"Definition:Euclidean Space",
"Definition:Euclidean Metric/Real Vector Space",
"Definition:Chebyshev Distance/Real Vector Space",
"Definition:Topological Space",
"Definition:Topology Induced by Metric",
"Definition:Topological Space",
"Definition:Topology Induced by Metric"
] | [
"P-Product Metrics on Real Vector Space are Topologically Equivalent",
"Definition:Topologically Equivalent Metrics",
"Topologically Equivalent Metrics induce Equal Topologies"
] |
proofwiki-18063 | Topologically Equivalent Metrics induce Equal Topologies | Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be metric spaces on the same underlying set $A$.
Let $d_1$ and $d_2$ be topologically equivalent.
Let $\tau_1$ and $\tau_2$ denote the topologies on $A$ induced by $d_1$ and $d_2$ respectively.
Then $\tau_1$ and $\tau_2$ are equal. | Let $d_1$ and $d_2$ be topologically equivalent by hypothesis.
By definition of topological equivalence:
$d_1$ and $d_2$ are '''topologically equivalent''' {{iff}}:
:$U \subseteq A$ is $d_1$-open {{iff}} $U \subseteq A$ is $d_2$-open.
By definition of the induced topology:
:The '''topology on the metric space $M = \str... | Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be [[Definition:Metric Space|metric spaces]] on the same [[Definition:Underlying Set of Metric Space|underlying set]] $A$.
Let $d_1$ and $d_2$ be [[Definition:Topologically Equivalent Metrics|topologically equivalent]].
Let $\tau_1$ and $\tau_2$ denote the [[... | Let $d_1$ and $d_2$ be [[Definition:Topologically Equivalent Metrics|topologically equivalent]] by [[Definition:By Hypothesis|hypothesis]].
By definition of [[Definition:Topologically Equivalent Metrics|topological equivalence]]:
$d_1$ and $d_2$ are '''topologically equivalent''' {{iff}}:
:$U \subseteq A$ is [[Defi... | Topologically Equivalent Metrics induce Equal Topologies | https://proofwiki.org/wiki/Topologically_Equivalent_Metrics_induce_Equal_Topologies | https://proofwiki.org/wiki/Topologically_Equivalent_Metrics_induce_Equal_Topologies | [
"Topologically Equivalent Metrics"
] | [
"Definition:Metric Space",
"Definition:Underlying Set/Metric Space",
"Definition:Topologically Equivalent Metrics",
"Definition:Topology Induced by Metric",
"Definition:Set Equality"
] | [
"Definition:Topologically Equivalent Metrics",
"Definition:By Hypothesis",
"Definition:Topologically Equivalent Metrics",
"Definition:Open Set/Metric Space",
"Definition:Open Set/Metric Space",
"Definition:Topology Induced by Metric",
"Definition:Topology",
"Definition:Metric Space",
"Definition:Dis... |
proofwiki-18064 | Right Order Topology on Strictly Positive Integers as Neighborhood Space | Let $\Z_{>0}$ denote the set of (strictly) positive integers.
Let $n \in \Z_{>0}$.
Let $U \subseteq \Z_{>0}$ be defined as being a neighborhood of $n$ {{iff}}
:$\forall m \in \Z: m \ge n \implies m \in U$
Then the set $\NN$ of all $U$ for all $n \in \Z_{>0}$ forms a neighborhood space which is the same as the right ord... | First it is noted that a neighborhood of $n$ is exactly an element of the right order topology on $\Z_{>0}$.
It remains to be shown that $\NN$ actually forms a neighborhood space.
Let $\NN_n$ denote the set of all neighborhood of a given $n \in \Z_{>0}$.
Checking the neighborhood space axioms in turn: | Let $\Z_{>0}$ denote the [[Definition:Strictly Positive Integer|set of (strictly) positive integers]].
Let $n \in \Z_{>0}$.
Let $U \subseteq \Z_{>0}$ be defined as being a [[Definition:Neighborhood (Neighborhood Space)|neighborhood]] of $n$ {{iff}}
:$\forall m \in \Z: m \ge n \implies m \in U$
Then the set $\NN$ of... | First it is noted that a [[Definition:Neighborhood (Neighborhood Space)|neighborhood]] of $n$ is exactly an [[Definition:Element|element]] of the [[Definition:Right Order Topology on Strictly Positive Integers|right order topology on $\Z_{>0}$]].
It remains to be shown that $\NN$ actually forms a [[Definition:Neighbor... | Right Order Topology on Strictly Positive Integers as Neighborhood Space | https://proofwiki.org/wiki/Right_Order_Topology_on_Strictly_Positive_Integers_as_Neighborhood_Space | https://proofwiki.org/wiki/Right_Order_Topology_on_Strictly_Positive_Integers_as_Neighborhood_Space | [
"Right Order Topologies",
"Examples of Neighborhood Spaces"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Neighborhood (Neighborhood Space)",
"Definition:Neighborhood Space",
"Definition:Right Order Topology on Strictly Positive Integers"
] | [
"Definition:Neighborhood (Neighborhood Space)",
"Definition:Element",
"Definition:Right Order Topology on Strictly Positive Integers",
"Definition:Neighborhood Space",
"Definition:Set of Sets",
"Definition:Neighborhood (Neighborhood Space)",
"Axiom:Neighborhood Space Axioms",
"Definition:Set",
"Axio... |
proofwiki-18065 | Translation Mapping is Bijection | Let $\struct {G, +}$ be an abelian group.
Let $g \in G$.
Let $\tau_g: G \to G$ be the translation by $g$:
:$\forall h \in G: \map {\tau_g} h = h + \paren {-g}$
where $-g$ is the inverse of $g$ with respect to $+$ in $G$.
Then $\tau_g$ is a bijection. | === Proof of Injectivity ===
{{begin-eqn}}
{{eqn | q = \forall h_1, h_2 \in G
| l = \map {\tau_g} {h_1}
| r = \map {\tau_g} {h_2}
| c =
}}
{{eqn | ll= \leadsto
| l = h_1 + \paren {-g}
| r = h_2 + \paren {-g}
| c = Definition of $\tau_g$
}}
{{eqn | ll= \leadsto
| l = h_1
... | Let $\struct {G, +}$ be an [[Definition:Abelian Group|abelian group]].
Let $g \in G$.
Let $\tau_g: G \to G$ be the [[Definition:Translation in Abelian Group|translation by $g$]]:
:$\forall h \in G: \map {\tau_g} h = h + \paren {-g}$
where $-g$ is the [[Definition:Inverse Element|inverse]] of $g$ with respect to $+... | === Proof of Injectivity ===
{{begin-eqn}}
{{eqn | q = \forall h_1, h_2 \in G
| l = \map {\tau_g} {h_1}
| r = \map {\tau_g} {h_2}
| c =
}}
{{eqn | ll= \leadsto
| l = h_1 + \paren {-g}
| r = h_2 + \paren {-g}
| c = Definition of $\tau_g$
}}
{{eqn | ll= \leadsto
| l = h_1
... | Translation Mapping is Bijection | https://proofwiki.org/wiki/Translation_Mapping_is_Bijection | https://proofwiki.org/wiki/Translation_Mapping_is_Bijection | [
"Translation Mappings",
"Bijections"
] | [
"Definition:Abelian Group",
"Definition:Translation Mapping/Abelian Group",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Bijection"
] | [
"Cancellation Laws"
] |
proofwiki-18066 | Translation Mapping is Isometry | Let $\Gamma = \R^n$ denote the real Euclidean space of $n$ dimensions.
Let $\tau_\mathbf x$ be a translation on $\Gamma$:
:$\forall \mathbf y \in \R^n: \map {\tau_\mathbf x} {\mathbf y} = \mathbf y - \mathbf x$
where $\mathbf x$ is a vector in $\R^n$.
Then $\tau_\mathbf x$ is an isometry. | From Translation Mapping is Bijection, $\tau_\mathbf x$ is a bijection.
From Euclidean Metric on Real Number Space is Translation Invariant, $\tau_\mathbf x$ is distance-preserving on $\Gamma$.
The result follows by definition of isometry.
{{qed}} | Let $\Gamma = \R^n$ denote the [[Definition:Real Euclidean Space|real Euclidean space]] of $n$ [[Definition:Dimension of Vector Space|dimensions]].
Let $\tau_\mathbf x$ be a [[Definition:Translation in Euclidean Space|translation]] on $\Gamma$:
:$\forall \mathbf y \in \R^n: \map {\tau_\mathbf x} {\mathbf y} = \mathbf... | From [[Translation Mapping is Bijection]], $\tau_\mathbf x$ is a [[Definition:Bijection|bijection]].
From [[Euclidean Metric on Real Number Space is Translation Invariant]], $\tau_\mathbf x$ is [[Definition:Distance-Preserving Mapping|distance-preserving]] on $\Gamma$.
The result follows by definition of [[Definition... | Translation Mapping is Isometry | https://proofwiki.org/wiki/Translation_Mapping_is_Isometry | https://proofwiki.org/wiki/Translation_Mapping_is_Isometry | [
"Translation Mappings",
"Isometries (Metric Spaces)",
"Isometries (Euclidean Geometry)"
] | [
"Definition:Euclidean Space/Real",
"Definition:Dimension of Vector Space",
"Definition:Translation Mapping/Euclidean Space",
"Definition:Vector",
"Definition:Isometry (Metric Spaces)"
] | [
"Translation Mapping is Bijection",
"Definition:Bijection",
"Euclidean Metric on Real Number Space is Translation Invariant",
"Definition:Distance-Preserving Mapping",
"Definition:Isometry (Metric Spaces)"
] |
proofwiki-18067 | Existence of Translation between Each Pair of Points in Euclidean Space | Let $\R^n$ denote the real Euclidean space of $n$ dimensions.
Let $\mathbf a = \tuple {a_1, a_2, \ldots, a_n}$ and $\mathbf b = \tuple {b_1, b_2, \ldots, b_n}$ be points in $\R^n$.
There exists an isometry $f: \R^n \to \R^n$ such that $\map f {\mathbf a} = b$. | Let $\mathbf t = \mathbf a - \mathbf b$.
Then the translation $\tau_\mathbf t$ is such an isometry.
We have that:
:$\map {\tau_\mathbf t} {\mathbf a} = \mathbf a - \paren {\mathbf a - \mathbf b} = \mathbf b$
The result follows from Translation Mapping is Isometry.
{{qed}} | Let $\R^n$ denote the [[Definition:Real Euclidean Space|real Euclidean space]] of $n$ [[Definition:Dimension of Vector Space|dimensions]].
Let $\mathbf a = \tuple {a_1, a_2, \ldots, a_n}$ and $\mathbf b = \tuple {b_1, b_2, \ldots, b_n}$ be points in $\R^n$.
There exists an [[Definition:Isometry (Metric Spaces)|isome... | Let $\mathbf t = \mathbf a - \mathbf b$.
Then the [[Definition:Translation in Euclidean Space|translation]] $\tau_\mathbf t$ is such an [[Definition:Isometry (Metric Spaces)|isometry]].
We have that:
:$\map {\tau_\mathbf t} {\mathbf a} = \mathbf a - \paren {\mathbf a - \mathbf b} = \mathbf b$
The result follows from... | Existence of Translation between Each Pair of Points in Euclidean Space | https://proofwiki.org/wiki/Existence_of_Translation_between_Each_Pair_of_Points_in_Euclidean_Space | https://proofwiki.org/wiki/Existence_of_Translation_between_Each_Pair_of_Points_in_Euclidean_Space | [
"Translation Mappings",
"Euclidean Metric"
] | [
"Definition:Euclidean Space/Real",
"Definition:Dimension of Vector Space",
"Definition:Isometry (Metric Spaces)"
] | [
"Definition:Translation Mapping/Euclidean Space",
"Definition:Isometry (Metric Spaces)",
"Translation Mapping is Isometry"
] |
proofwiki-18068 | Mapping is Continuous iff Inverse Images of Open Sets are Open | Let $X$ and $Y$ be normed vector spaces.
Let $f : X \to Y$ be a mapping.
Then:
:$f$ is continuous on $X$
{{iff}}:
:for every $V$ open in $Y$, $\map {f^{-1}} V$ is open in $X$. | === Sufficient Condition ===
Let $c \in X$.
Let $\epsilon \in \R_{\mathop > 0}$.
Let $V := \map {B_\epsilon} {\map f c}$ be an open ball in $Y$.
By Open Ball is Open Set in Normed Vector Space, $V$ is an open set in $Y$.
Let $\map {f^{-1}} V = \map {f^{-1}} {\map {B_\epsilon} {\map f c}}$ be an open set in $X$.
Note th... | Let $X$ and $Y$ be [[Definition:Normed Vector Space|normed vector spaces]].
Let $f : X \to Y$ be a [[Definition:Mapping|mapping]].
Then:
:$f$ is [[Definition:Continuous Mapping (Normed Vector Space)/Space|continuous]] on $X$
{{iff}}:
:for every $V$ [[Definition:Open Set in Normed Vector Space|open]] in $Y$, $\map {f... | === Sufficient Condition ===
Let $c \in X$.
Let $\epsilon \in \R_{\mathop > 0}$.
Let $V := \map {B_\epsilon} {\map f c}$ be an [[Definition:Open Ball in Normed Vector Space|open ball]] in $Y$.
By [[Open Ball is Open Set in Normed Vector Space]], $V$ is an [[Definition:Open Set in Normed Vector Space|open set]] in $... | Mapping is Continuous iff Inverse Images of Open Sets are Open | https://proofwiki.org/wiki/Mapping_is_Continuous_iff_Inverse_Images_of_Open_Sets_are_Open | https://proofwiki.org/wiki/Mapping_is_Continuous_iff_Inverse_Images_of_Open_Sets_are_Open | [
"Continuous Mappings on Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Mapping",
"Definition:Continuous Mapping (Normed Vector Space)/Space",
"Definition:Open Set/Normed Vector Space",
"Definition:Open Set/Normed Vector Space"
] | [
"Definition:Open Ball/Normed Vector Space",
"Open Ball is Open Set/Normed Vector Space",
"Definition:Open Set/Normed Vector Space",
"Definition:Open Set/Normed Vector Space",
"Definition:Open Set/Normed Vector Space",
"Definition:Open Ball/Normed Vector Space",
"Definition:Continuous Mapping (Normed Vec... |
proofwiki-18069 | Mapping is Continuous iff Inverse Images of Open Sets are Open/Corollary | Let $X$ and $Y$ be normed vector spaces.
Let $f : X \to Y$ be a mapping.
Then:
:$f$ is continuous on $X$
{{iff}}:
:for every $F$ closed in $Y$, $\map {f^{-1}} F$ is closed in $X$. | === Sufficient Condition ===
Suppose that for every closed $F$ in $Y$, $\map {f^{-1}} F$ is closed in $X$.
Let $V$ be open in $Y$.
By definition, $Y \setminus V$ is closed in $Y$.
By assumption, $\map {f^{-1}} {Y \setminus V}$ is closed in $X$.
We have that:
{{begin-eqn}}
{{eqn | l = \map {f^{-1} } {Y \setminus V}
... | Let $X$ and $Y$ be [[Definition:Normed Vector Space|normed vector spaces]].
Let $f : X \to Y$ be a [[Definition:Mapping|mapping]].
Then:
:$f$ is [[Definition:Continuous Mapping (Normed Vector Space)/Space|continuous]] on $X$
{{iff}}:
:for every $F$ [[Definition:Closed Set in Normed Vector Space|closed]] in $Y$, $\ma... | === Sufficient Condition ===
Suppose that for every [[Definition:Closed Set of Normed Vector Space|closed]] $F$ in $Y$, $\map {f^{-1}} F$ is [[Definition:Closed Set of Normed Vector Space|closed]] in $X$.
Let $V$ be [[Definition:Open Set in Normed Vector Space|open]] in $Y$.
By definition, $Y \setminus V$ is [[Defin... | Mapping is Continuous iff Inverse Images of Open Sets are Open/Corollary | https://proofwiki.org/wiki/Mapping_is_Continuous_iff_Inverse_Images_of_Open_Sets_are_Open/Corollary | https://proofwiki.org/wiki/Mapping_is_Continuous_iff_Inverse_Images_of_Open_Sets_are_Open/Corollary | [
"Continuous Mappings on Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Mapping",
"Definition:Continuous Mapping (Normed Vector Space)/Space",
"Definition:Closed Set/Normed Vector Space",
"Definition:Closed Set/Normed Vector Space"
] | [
"Definition:Closed Set/Normed Vector Space",
"Definition:Closed Set/Normed Vector Space",
"Definition:Open Set/Normed Vector Space",
"Definition:Closed Set/Normed Vector Space/Definition 1",
"Definition:Closed Set/Normed Vector Space",
"Preimage of Set Difference under Mapping",
"Definition:Closed Set/N... |
proofwiki-18070 | Set of Isometries in Euclidean Space under Composition forms Group | Let $\struct {\R^n, d}$ be a real Euclidean space of $n$ dimensions.
Let $S$ be the set of all mappings $f: \R^n \to \R^n$ which preserve distance:
That is:
:$\map d {\map f a, \map f b} = \map d {a, b}$
Let $\struct {S, \circ}$ be the algebraic structure formed from $S$ and the composition operation $\circ$.
Then $\st... | From Euclidean Metric on Real Vector Space is Metric, $\R^n$ is a metric space.
Hence it is seen that a complex function $f: \C \to \C$ which preserves distance is in fact an isometry on $\C$.
Taking the group axioms in turn: | Let $\struct {\R^n, d}$ be a [[Definition:Real Euclidean Space|real Euclidean space]] of $n$ [[Definition:Dimension of Vector Space|dimensions]].
Let $S$ be the [[Definition:Set|set]] of all [[Definition:Mapping|mappings]] $f: \R^n \to \R^n$ which preserve [[Definition:Distance Function|distance]]:
That is:
:$\map d ... | From [[Euclidean Metric on Real Vector Space is Metric]], $\R^n$ is a [[Definition:Metric Space|metric space]].
Hence it is seen that a [[Definition:Complex Function|complex function]] $f: \C \to \C$ which preserves [[Definition:Distance Function|distance]] is in fact an [[Definition:Isometry (Euclidean Geometry)|isom... | Set of Isometries in Euclidean Space under Composition forms Group | https://proofwiki.org/wiki/Set_of_Isometries_in_Euclidean_Space_under_Composition_forms_Group | https://proofwiki.org/wiki/Set_of_Isometries_in_Euclidean_Space_under_Composition_forms_Group | [
"Euclidean Metric",
"Isometries (Euclidean Geometry)",
"Examples of Groups"
] | [
"Definition:Euclidean Space/Real",
"Definition:Dimension of Vector Space",
"Definition:Set",
"Definition:Mapping",
"Definition:Distance Function",
"Definition:Algebraic Structure/One Operation",
"Definition:Composition of Mappings",
"Definition:Group"
] | [
"Euclidean Metric on Real Vector Space is Metric",
"Definition:Metric Space",
"Definition:Complex Function",
"Definition:Distance Function",
"Definition:Isometry (Euclidean Geometry)",
"Axiom:Group Axioms",
"Definition:Isometry (Euclidean Geometry)",
"Definition:Isometry (Euclidean Geometry)",
"Defi... |
proofwiki-18071 | Composite of Continuous Mappings between Normed Vector Spaces is Continuous | Let $X, Y, Z$ be normed vector spaces.
Let $f : X \to Y$ and $g : Y \to Z$ be continuous mappings on $X$ and $Y$ respectively.
Let $g \circ f : X \to Z$ be a composite mapping where:
:$\forall x \in X : \map {\paren {g \circ f} } x := \map g {\map f x}$
Then $g \circ f$ is continuous on $X$. | Let $W$ be open in $Z$.
$g$ is continuous on $Y$.
By Mapping is Continuous iff Inverse Images of Open Sets are Open, $\map {g^{-1} } W$ is open in $Y$.
$f$ is continuous on $X$.
By Mapping is Continuous iff Inverse Images of Open Sets are Open, $\map {f^{-1} } {\map {g^{-1} } W}$ is open in $X$.
By Inverse of Composite... | Let $X, Y, Z$ be [[Definition:Normed Vector Space|normed vector spaces]].
Let $f : X \to Y$ and $g : Y \to Z$ be [[Definition:Continuous Mapping (Normed Vector Space)|continuous mappings]] on $X$ and $Y$ respectively.
Let $g \circ f : X \to Z$ be a [[Definition:Composition of Mappings|composite mapping]] where:
:$\f... | Let $W$ be [[Definition:Open Set in Normed Vector Space|open]] in $Z$.
$g$ is [[Definition:Continuous Mapping (Normed Vector Space)|continuous]] on $Y$.
By [[Mapping is Continuous iff Inverse Images of Open Sets are Open]], $\map {g^{-1} } W$ is [[Definition:Open Set in Normed Vector Space|open]] in $Y$.
$f$ is [[De... | Composite of Continuous Mappings between Normed Vector Spaces is Continuous | https://proofwiki.org/wiki/Composite_of_Continuous_Mappings_between_Normed_Vector_Spaces_is_Continuous | https://proofwiki.org/wiki/Composite_of_Continuous_Mappings_between_Normed_Vector_Spaces_is_Continuous | [
"Continuous Mappings on Normed Vector Spaces",
"Normed Vector Spaces",
"Continuity",
"Composite Mappings"
] | [
"Definition:Normed Vector Space",
"Definition:Continuous Mapping (Normed Vector Space)",
"Definition:Composition of Mappings",
"Definition:Continuous Mapping (Normed Vector Space)/Space"
] | [
"Definition:Open Set/Normed Vector Space",
"Definition:Continuous Mapping (Normed Vector Space)",
"Mapping is Continuous iff Inverse Images of Open Sets are Open",
"Definition:Open Set/Normed Vector Space",
"Definition:Continuous Mapping (Normed Vector Space)",
"Mapping is Continuous iff Inverse Images of... |
proofwiki-18072 | Dot Product of Unit Vectors | Let $\mathbf a$ and $\mathbf b$ be unit vectors.
Then their dot product $\mathbf a \cdot \mathbf b$ is:
:$\mathbf a \cdot \mathbf b = \cos \theta$
where $\theta$ is the angle between $\mathbf a$ and $\mathbf b$. | We have by definition of dot product :
:$\mathbf a \cdot \mathbf b = \norm {\mathbf a} \, \norm {\mathbf b} \cos \theta$
where $\norm {\mathbf a}$ denotes the length of $\mathbf a$.
From Length of Unit Vector is 1:
:$\norm {\mathbf a} = \norm {\mathbf b} = 1$
from which the result follows.
{{qed}} | Let $\mathbf a$ and $\mathbf b$ be [[Definition:Unit Vector|unit vectors]].
Then their [[Definition:Dot Product|dot product]] $\mathbf a \cdot \mathbf b$ is:
:$\mathbf a \cdot \mathbf b = \cos \theta$
where $\theta$ is the [[Definition:Angle Between Vectors|angle between $\mathbf a$ and $\mathbf b$]]. | We have by definition of [[Definition:Dot Product|dot product]] :
:$\mathbf a \cdot \mathbf b = \norm {\mathbf a} \, \norm {\mathbf b} \cos \theta$
where $\norm {\mathbf a}$ denotes the [[Definition:Vector Length|length]] of $\mathbf a$.
From [[Length of Unit Vector is 1]]:
:$\norm {\mathbf a} = \norm {\mathbf b} = 1... | Dot Product of Unit Vectors | https://proofwiki.org/wiki/Dot_Product_of_Unit_Vectors | https://proofwiki.org/wiki/Dot_Product_of_Unit_Vectors | [
"Dot Product",
"Unit Vectors"
] | [
"Definition:Unit Vector",
"Definition:Dot Product",
"Definition:Angle between Vectors"
] | [
"Definition:Dot Product",
"Definition:Vector Length",
"Length of Unit Vector is 1"
] |
proofwiki-18073 | Square of Sum of Vectors | Let $\mathbf a$ and $\mathbf b$ be vector quantities.
Then:
:$\paren {\mathbf a + \mathbf b}^2 = \mathbf a^2 + 2 \mathbf a \cdot \mathbf b + \mathbf b^2$
where:
:$\mathbf a \cdot \mathbf b$ denotes dot product
:$\mathbf a^2$ denotes the square of $\mathbf a$, that is: $\mathbf a \cdot \mathbf a$. | {{begin-eqn}}
{{eqn | l = \paren {\mathbf a + \mathbf b}^2
| r = \paren {\mathbf a + \mathbf b} \cdot \paren {\mathbf a + \mathbf b}
| c = {{Defof|Square of Vector Quantity}}
}}
{{eqn | r = \mathbf a \cdot \paren {\mathbf a + \mathbf b} + \mathbf b \cdot \paren {\mathbf a + \mathbf b}
| c = Dot Produc... | Let $\mathbf a$ and $\mathbf b$ be [[Definition:Vector Quantity|vector quantities]].
Then:
:$\paren {\mathbf a + \mathbf b}^2 = \mathbf a^2 + 2 \mathbf a \cdot \mathbf b + \mathbf b^2$
where:
:$\mathbf a \cdot \mathbf b$ denotes [[Definition:Dot Product|dot product]]
:$\mathbf a^2$ denotes the [[Definition:Square of ... | {{begin-eqn}}
{{eqn | l = \paren {\mathbf a + \mathbf b}^2
| r = \paren {\mathbf a + \mathbf b} \cdot \paren {\mathbf a + \mathbf b}
| c = {{Defof|Square of Vector Quantity}}
}}
{{eqn | r = \mathbf a \cdot \paren {\mathbf a + \mathbf b} + \mathbf b \cdot \paren {\mathbf a + \mathbf b}
| c = [[Dot Prod... | Square of Sum of Vectors | https://proofwiki.org/wiki/Square_of_Sum_of_Vectors | https://proofwiki.org/wiki/Square_of_Sum_of_Vectors | [
"Vector Addition",
"Dot Product"
] | [
"Definition:Vector Quantity",
"Definition:Dot Product",
"Definition:Square of Vector Quantity"
] | [
"Dot Product Distributes over Addition",
"Dot Product Distributes over Addition",
"Dot Product Operator is Commutative"
] |
proofwiki-18074 | Dot Product of Sum with Difference of Vectors | Let $\mathbf a$ and $\mathbf b$ be vector quantities.
Then:
:$\paren {\mathbf a + \mathbf b} \cdot \paren {\mathbf a - \mathbf b} = \mathbf a^2 - \mathbf b^2$
where:
:$\cdot$ denotes dot product
:$\mathbf a^2$ denotes the square of $\mathbf a$, that is: $\mathbf a \cdot \mathbf a$. | {{begin-eqn}}
{{eqn | l = \paren {\mathbf a + \mathbf b} \cdot \paren {\mathbf a - \mathbf b}
| r = \mathbf a \cdot \paren {\mathbf a - \mathbf b} + \mathbf b \cdot \paren {\mathbf a - \mathbf b}
| c = Dot Product Distributes over Addition
}}
{{eqn | r = \mathbf a \cdot \mathbf a - \mathbf a \cdot \mathbf b... | Let $\mathbf a$ and $\mathbf b$ be [[Definition:Vector Quantity|vector quantities]].
Then:
:$\paren {\mathbf a + \mathbf b} \cdot \paren {\mathbf a - \mathbf b} = \mathbf a^2 - \mathbf b^2$
where:
:$\cdot$ denotes [[Definition:Dot Product|dot product]]
:$\mathbf a^2$ denotes the [[Definition:Square of Vector Quantity... | {{begin-eqn}}
{{eqn | l = \paren {\mathbf a + \mathbf b} \cdot \paren {\mathbf a - \mathbf b}
| r = \mathbf a \cdot \paren {\mathbf a - \mathbf b} + \mathbf b \cdot \paren {\mathbf a - \mathbf b}
| c = [[Dot Product Distributes over Addition]]
}}
{{eqn | r = \mathbf a \cdot \mathbf a - \mathbf a \cdot \math... | Dot Product of Sum with Difference of Vectors | https://proofwiki.org/wiki/Dot_Product_of_Sum_with_Difference_of_Vectors | https://proofwiki.org/wiki/Dot_Product_of_Sum_with_Difference_of_Vectors | [
"Vector Addition",
"Dot Product"
] | [
"Definition:Vector Quantity",
"Definition:Dot Product",
"Definition:Square of Vector Quantity"
] | [
"Dot Product Distributes over Addition",
"Dot Product Distributes over Addition",
"Dot Product Operator is Commutative"
] |
proofwiki-18075 | Square of Vector Quantity in Coordinate Form | Let $\mathbf a$ be a vector in a vector space $\mathbf V$ of $n$ dimensions:
$\ds \mathbf a = \sum_{k \mathop = 1}^n a_k \mathbf e_k$
where $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ is the standard ordered basis of $\mathbf V$.
Then:
:$\ds \mathbf a^2 = \sum_{k \mathop = 1}^n a_k^2$
where $\mathbf a^2$ d... | By definition of square of $\mathbf a$:
:$\mathbf a^2 = \mathbf a \cdot \mathbf a$
By definition of dot product:
:$\ds \mathbf a \cdot \mathbf a = a_1 a_1 + a_2 a_2 + \cdots + a_n a_n = \sum_{k \mathop = 1}^n a_k^2$
{{qed}} | Let $\mathbf a$ be a [[Definition:Vector (Linear Algebra)|vector]] in a [[Definition:Vector Space|vector space]] $\mathbf V$ of [[Definition:Dimension of Vector Space|$n$ dimensions]]:
$\ds \mathbf a = \sum_{k \mathop = 1}^n a_k \mathbf e_k$
where $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ is the [[Defi... | By definition of [[Definition:Square of Vector Quantity|square]] of $\mathbf a$:
:$\mathbf a^2 = \mathbf a \cdot \mathbf a$
By definition of [[Definition:Dot Product|dot product]]:
:$\ds \mathbf a \cdot \mathbf a = a_1 a_1 + a_2 a_2 + \cdots + a_n a_n = \sum_{k \mathop = 1}^n a_k^2$
{{qed}} | Square of Vector Quantity in Coordinate Form | https://proofwiki.org/wiki/Square_of_Vector_Quantity_in_Coordinate_Form | https://proofwiki.org/wiki/Square_of_Vector_Quantity_in_Coordinate_Form | [
"Dot Product"
] | [
"Definition:Vector/Linear Algebra",
"Definition:Vector Space",
"Definition:Dimension of Vector Space",
"Definition:Standard Ordered Basis/Vector Space",
"Definition:Square of Vector Quantity"
] | [
"Definition:Square of Vector Quantity",
"Definition:Dot Product"
] |
proofwiki-18076 | Continuous Composition of Measurable Functions into Second Countable Space is Measurable | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\struct {X_i, \tau_i}$ for $i = 1, \ldots, n$ and $\struct {Y, \tau_Y}$ be topological spaces such that $X_1, \ldots, X_n$ are second countable.
Let $f_i: \struct {X, \Sigma} \to \struct {X_i, \map \BB {X_i, \tau_i} }$, $i = 1, \ldots, n$ be measurable functions wh... | By Mapping Measurable iff Measurable on Generator, it suffices to check that $h$ is measurable on open sets.
Thus let $U \in \tau_Y$ be given.
As $F$ is continuous by assumption, the pre-image $F^{-1} \sqbrk U$ is in $\tau$.
By:
:definition of basis
:Countable Product of Second-Countable Spaces is Second-Countable
the... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\struct {X_i, \tau_i}$ for $i = 1, \ldots, n$ and $\struct {Y, \tau_Y}$ be [[Definition:Topological Space|topological spaces]] such that $X_1, \ldots, X_n$ are [[Definition:Second-Countable Space|second countable]].
Let $f_i: \stru... | By [[Mapping Measurable iff Measurable on Generator]], it suffices to check that $h$ is [[Definition:Measurable Function|measurable]] on [[Definition:Open Set (Topology)|open sets]].
Thus let $U \in \tau_Y$ be given.
As $F$ is [[Definition:Everywhere Continuous Mapping (Topology)|continuous]] by assumption, the [[Def... | Continuous Composition of Measurable Functions into Second Countable Space is Measurable | https://proofwiki.org/wiki/Continuous_Composition_of_Measurable_Functions_into_Second_Countable_Space_is_Measurable | https://proofwiki.org/wiki/Continuous_Composition_of_Measurable_Functions_into_Second_Countable_Space_is_Measurable | [
"Measure Theory"
] | [
"Definition:Measurable Space",
"Definition:Topological Space",
"Definition:Second-Countable Space",
"Definition:Measurable Function",
"Definition:Borel Sigma-Algebra",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Product Topology/Finite Product",
"Definition:Cartesian Product/Fin... | [
"Mapping Measurable iff Measurable on Generator",
"Definition:Measurable Function",
"Definition:Open Set/Topology",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Preimage/Mapping/Subset",
"Definition:Basis (Topology)/Analytic Basis",
"Countable Product of Second-Countable Spaces is ... |
proofwiki-18077 | Point dividing Line Segment between Two Points in Given Ratio | Let $A$ and $B$ be points whose position vectors relative to an origin $O$ of a Euclidean space are $\mathbf a$ and $\mathbf b$.
Let $\mathbf r$ be the position vector of a point $R$ on $AB$ which divides $AB$ in the ratio $m : n$.
:320px
Then:
:$\mathbf r = \dfrac {n \mathbf a + m \mathbf b} {m + n}$ | Let the coordinates of $A$ be $\tuple {x_1, y_1}$.
Let the coordinates of $B$ be $\tuple {x_2, y_2}$.
Let the coordinates of $R$ be $\tuple {X, Y}$.
Then we have:
:$\dfrac {x_2 - X} {X - x_1} = \dfrac n m$
and so:
:$X \paren {m + n} = m x_2 + n x_1$
Similarly for $Y$, giving $R$ as:
:$\tuple {\dfrac {m x_2 + n x_1} {m ... | Let $A$ and $B$ be [[Definition:Point|points]] whose [[Definition:Position Vector|position vectors]] relative to an [[Definition:Origin|origin]] $O$ of a [[Definition:Euclidean Space|Euclidean space]] are $\mathbf a$ and $\mathbf b$.
Let $\mathbf r$ be the [[Definition:Position Vector|position vector]] of a [[Definiti... | Let the [[Definition:Cartesian Coordinates|coordinates]] of $A$ be $\tuple {x_1, y_1}$.
Let the [[Definition:Cartesian Coordinates|coordinates]] of $B$ be $\tuple {x_2, y_2}$.
Let the [[Definition:Cartesian Coordinates|coordinates]] of $R$ be $\tuple {X, Y}$.
Then we have:
:$\dfrac {x_2 - X} {X - x_1} = \dfrac n m$
... | Point dividing Line Segment between Two Points in Given Ratio/Proof 2 | https://proofwiki.org/wiki/Point_dividing_Line_Segment_between_Two_Points_in_Given_Ratio | https://proofwiki.org/wiki/Point_dividing_Line_Segment_between_Two_Points_in_Given_Ratio/Proof_2 | [
"Point dividing Line Segment between Two Points in Given Ratio",
"Division in Ratio",
"Vector Algebra",
"Straight Lines"
] | [
"Definition:Point",
"Definition:Position Vector",
"Definition:Coordinate System/Origin",
"Definition:Euclidean Space",
"Definition:Position Vector",
"Definition:Point",
"Definition:Ratio",
"File:Point-dividing-Line-Segment.png"
] | [
"Definition:Cartesian Coordinate System",
"Definition:Cartesian Coordinate System",
"Definition:Cartesian Coordinate System"
] |
proofwiki-18078 | Centroid of Weighted Pair of Points | Let $A$ and $B$ be two points in Euclidean space.
Let $A$ and $B$ have weights $p$ and $q$ respectively.
Let $G$ be the centroid of $A$ and $B$.
Then $G$ divides the line $AB$ in the ratio $q : p$.
That is:
:$AG = \dfrac q {p + q} AB$
:$BG = \dfrac p {p + q} AB$ | Let the position vectors of $A$ and $B$ be given by $\mathbf a$ and $\mathbf b$ repectively.
By definition of centroid:
:$\vec {O G} = \dfrac {p \mathbf a + q \mathbf b} {p + q}$
The result follows from Point dividing Line Segment between Two Points in Given Ratio.
{{qed}} | Let $A$ and $B$ be two [[Definition:Point|points]] in [[Definition:Euclidean Space|Euclidean space]].
Let $A$ and $B$ have [[Definition:Weight Function|weights]] $p$ and $q$ respectively.
Let $G$ be the [[Definition:Centroid of Weighted Set of Points|centroid]] of $A$ and $B$.
Then $G$ divides the [[Definition:Line... | Let the [[Definition:Position Vector|position vectors]] of $A$ and $B$ be given by $\mathbf a$ and $\mathbf b$ repectively.
By definition of [[Definition:Centroid of Weighted Set of Points|centroid]]:
:$\vec {O G} = \dfrac {p \mathbf a + q \mathbf b} {p + q}$
The result follows from [[Point dividing Line Segment be... | Centroid of Weighted Pair of Points | https://proofwiki.org/wiki/Centroid_of_Weighted_Pair_of_Points | https://proofwiki.org/wiki/Centroid_of_Weighted_Pair_of_Points | [
"Centroids"
] | [
"Definition:Point",
"Definition:Euclidean Space",
"Definition:Weight Function",
"Definition:Centroid/Weighted Set of Points",
"Definition:Line/Segment",
"Definition:Ratio"
] | [
"Definition:Position Vector",
"Definition:Centroid/Weighted Set of Points",
"Point dividing Line Segment between Two Points in Given Ratio"
] |
proofwiki-18079 | Centroid of Weighted Set of Points is Independent of Origin | Let $O'$ be a point whose position vector from $O$ is $\mathbf l$.
Let $S = \set {A_1, A_2, \ldots, A_n}$ be a set of $n$ points in Euclidean space whose position vectors are given by $\mathbf a_1, \mathbf a_2, \dotsc, \mathbf a_n$ repectively.
Let $W: S \to \R$ be a weight function on $S$.
Let $G$ be the centroid of $... | The position vectors of the elements of $S$ are given by:
:$\mathbf a_1 - \mathbf l, \mathbf a_2 - \mathbf l, \dotsc, \mathbf a_n - \mathbf l$
Hence the centroid of $S$ with weight function $W$ with respect to $O'$ ias:
:$\vec {OG'} = \dfrac {w_1 \paren {\mathbf a_1 - \mathbf l} + w_2 \paren {\mathbf a_2 - \mathbf l} +... | Let $O'$ be a [[Definition:Point|point]] whose [[Definition:Position Vector|position vector]] from $O$ is $\mathbf l$.
Let $S = \set {A_1, A_2, \ldots, A_n}$ be a [[Definition:Set|set]] of $n$ [[Definition:Point|points]] in [[Definition:Euclidean Space|Euclidean space]] whose [[Definition:Position Vector|position vect... | The [[Definition:Position Vector|position vectors]] of the [[Definition:Element|elements]] of $S$ are given by:
:$\mathbf a_1 - \mathbf l, \mathbf a_2 - \mathbf l, \dotsc, \mathbf a_n - \mathbf l$
Hence the [[Definition:Centroid of Weighted Set of Points|centroid]] of $S$ with [[Definition:Weight Function|weight funct... | Centroid of Weighted Set of Points is Independent of Origin | https://proofwiki.org/wiki/Centroid_of_Weighted_Set_of_Points_is_Independent_of_Origin | https://proofwiki.org/wiki/Centroid_of_Weighted_Set_of_Points_is_Independent_of_Origin | [
"Centroids"
] | [
"Definition:Point",
"Definition:Position Vector",
"Definition:Set",
"Definition:Point",
"Definition:Euclidean Space",
"Definition:Position Vector",
"Definition:Weight Function",
"Definition:Centroid/Weighted Set of Points",
"Definition:Weight Function",
"Definition:Centroid/Weighted Set of Points"... | [
"Definition:Position Vector",
"Definition:Element",
"Definition:Centroid/Weighted Set of Points",
"Definition:Weight Function"
] |
proofwiki-18080 | Centroid of Combined Systems of Weighted Points | Let $O'$ be a point whose position vector from $O$ is $\mathbf l$.
Let $S = \set {A_1, A_2, \ldots, A_n}$ be a set of $n$ points in Euclidean space.
Let $W_S: S \to \R$ be a weight function on $S$.
Let $T = \set {B_1, B_2, \ldots, B_m}$ be a set of $m$ points in Euclidean space.
Let $W_T: T \to \R$ be a weight function... | Let the position vectors of the points in $S = \set {A_1, A_2, \ldots, A_n}$ be given by $\mathbf a_1, \mathbf a_2, \dotsc, \mathbf a_n$ repectively.
Let the position vectors of the points in $T = \set {B_1, B_2, \ldots, B_m}$ be given by $\mathbf b_1, \mathbf b_2, \dotsc, \mathbf b_n$ repectively.
We have that:
:$\vec... | Let $O'$ be a [[Definition:Point|point]] whose [[Definition:Position Vector|position vector]] from $O$ is $\mathbf l$.
Let $S = \set {A_1, A_2, \ldots, A_n}$ be a [[Definition:Set|set]] of $n$ [[Definition:Point|points]] in [[Definition:Euclidean Space|Euclidean space]].
Let $W_S: S \to \R$ be a [[Definition:Weight F... | Let the [[Definition:Position Vector|position vectors]] of the [[Definition:Point|points]] in $S = \set {A_1, A_2, \ldots, A_n}$ be given by $\mathbf a_1, \mathbf a_2, \dotsc, \mathbf a_n$ repectively.
Let the [[Definition:Position Vector|position vectors]] of the [[Definition:Point|points]] in $T = \set {B_1, B_2, \l... | Centroid of Combined Systems of Weighted Points | https://proofwiki.org/wiki/Centroid_of_Combined_Systems_of_Weighted_Points | https://proofwiki.org/wiki/Centroid_of_Combined_Systems_of_Weighted_Points | [
"Centroids"
] | [
"Definition:Point",
"Definition:Position Vector",
"Definition:Set",
"Definition:Point",
"Definition:Euclidean Space",
"Definition:Weight Function",
"Definition:Set",
"Definition:Point",
"Definition:Euclidean Space",
"Definition:Weight Function",
"Definition:Centroid/Weighted Set of Points",
"D... | [
"Definition:Position Vector",
"Definition:Point",
"Definition:Position Vector",
"Definition:Point",
"Definition:Centroid/Weighted Set of Points",
"Definition:Point",
"Definition:Weight Function"
] |
proofwiki-18081 | Unit Vector in Direction of Vector | Let $\mathbf v$ be a vector quantity.
The '''unit vector''' $\mathbf {\hat v}$ in the direction of $\mathbf v$ is:
:$\mathbf {\hat v} = \dfrac {\mathbf v} {\norm {\mathbf v} }$
where $\norm {\mathbf v}$ is the magnitude of $\mathbf v$. | From Vector Quantity as Scalar Product of Unit Vector Quantity:
:$\mathbf v = \norm {\mathbf v} \mathbf {\hat v}$
whence the result.
{{qed}} | Let $\mathbf v$ be a [[Definition:Vector Quantity|vector quantity]].
The '''[[Definition:Unit Vector|unit vector]]''' $\mathbf {\hat v}$ in the [[Definition:Direction|direction]] of $\mathbf v$ is:
:$\mathbf {\hat v} = \dfrac {\mathbf v} {\norm {\mathbf v} }$
where $\norm {\mathbf v}$ is the [[Definition:Magnitude|mag... | From [[Vector Quantity as Scalar Product of Unit Vector Quantity]]:
:$\mathbf v = \norm {\mathbf v} \mathbf {\hat v}$
whence the result.
{{qed}} | Unit Vector in Direction of Vector | https://proofwiki.org/wiki/Unit_Vector_in_Direction_of_Vector | https://proofwiki.org/wiki/Unit_Vector_in_Direction_of_Vector | [
"Unit Vectors"
] | [
"Definition:Vector Quantity",
"Definition:Unit Vector",
"Definition:Direction",
"Definition:Magnitude"
] | [
"Vector Quantity as Scalar Product of Unit Vector Quantity"
] |
proofwiki-18082 | Continuous Mappings preserve Convergent Sequences | Let $X, Y$ be normed vector spaces.
Let $c \in X$.
Let $f : X \to Y$ be a mapping.
Then $f$ is continuous at $c$ iff for every sequence $\sequence {x_n}_{n \mathop \in \N} \in X$ such that $\sequence {x_n}_{n \mathop \in \N}$ converges to $c$, $\sequence {\map f {x_n}}_{n \mathop \in \N}$ converges to $\map f c$. | === Necessary Condition ===
Let $f$ be continuous at $c$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$ such that $\sequence {x_n}_{n \mathop \in \N}$ converges to $c$.
Let $\epsilon \in \R_{\mathop > 0}$.
By definition of continuous mapping:
:$\exists \delta \in \R_{\mathop > 0} : \forall x \in X : \no... | Let $X, Y$ be [[Definition:Normed Vector Space|normed vector spaces]].
Let $c \in X$.
Let $f : X \to Y$ be a [[Definition:Mapping|mapping]].
Then $f$ is [[Definition:Continuous at Point of Normed Vector Space|continuous]] at $c$ iff for every [[Definition:Sequence|sequence]] $\sequence {x_n}_{n \mathop \in \N} \in ... | === Necessary Condition ===
Let $f$ be [[Definition:Continuous at Point of Normed Vector Space|continuous]] at $c$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $X$ such that $\sequence {x_n}_{n \mathop \in \N}$ [[Definition:Convergent Sequence in Normed Vector Space|converges]] t... | Continuous Mappings preserve Convergent Sequences | https://proofwiki.org/wiki/Continuous_Mappings_preserve_Convergent_Sequences | https://proofwiki.org/wiki/Continuous_Mappings_preserve_Convergent_Sequences | [
"Continuous Mappings",
"Convergent Sequences"
] | [
"Definition:Normed Vector Space",
"Definition:Mapping",
"Definition:Continuous Mapping (Normed Vector Space)/Point",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Convergent Sequence/Normed Vector Space"
] | [
"Definition:Continuous Mapping (Normed Vector Space)/Point",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Continuous Mapping (Normed Vector Space)/Point",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Continuous Mapping (Normed Vector Space)/... |
proofwiki-18083 | Zero Product of Numbers implies Factors are Zero | On all the number systems:
:natural numbers $\N$
:integers $\Z$
:rational numbers $\Q$
:real numbers $\R$
:complex numbers $\C$
the following holds.
Let $a \times b = 0$.
Then either $a = 0$ or $b = 0$. | From Natural Numbers have No Proper Zero Divisors
:$\forall a, b \in \N: a \times b = 0 \implies a = 0 \text { or } b = 0$
We have:
:Integers form Integral Domain
:Rational Numbers form Integral Domain
:Real Numbers form Integral Domain
:Complex Numbers form Integral Domain
Hence by definition of integral domain:
:$a \... | On all the number systems:
:[[Definition:Natural Numbers|natural numbers]] $\N$
:[[Definition:Integer|integers]] $\Z$
:[[Definition:Rational Number|rational numbers]] $\Q$
:[[Definition:Real Number|real numbers]] $\R$
:[[Definition:Complex Number|complex numbers]] $\C$
the following holds.
Let $a \times b = 0$.
Then ... | From [[Natural Numbers have No Proper Zero Divisors]]
:$\forall a, b \in \N: a \times b = 0 \implies a = 0 \text { or } b = 0$
We have:
:[[Integers form Integral Domain]]
:[[Rational Numbers form Integral Domain]]
:[[Real Numbers form Integral Domain]]
:[[Complex Numbers form Integral Domain]]
Hence by definition of ... | Zero Product of Numbers implies Factors are Zero | https://proofwiki.org/wiki/Zero_Product_of_Numbers_implies_Factors_are_Zero | https://proofwiki.org/wiki/Zero_Product_of_Numbers_implies_Factors_are_Zero | [
"Zero Divisors",
"Numbers"
] | [
"Definition:Natural Numbers",
"Definition:Integer",
"Definition:Rational Number",
"Definition:Real Number",
"Definition:Complex Number"
] | [
"Natural Numbers have No Proper Zero Divisors",
"Integers form Integral Domain",
"Rational Numbers form Integral Domain",
"Real Numbers form Integral Domain",
"Complex Numbers form Integral Domain",
"Definition:Integral Domain"
] |
proofwiki-18084 | Real Numbers form only Ordered Field which is Complete | The set of real numbers $\R$ is the only ordered field which also satisfies the Continuum Property. | From Real Numbers form Totally Ordered Field we have that $\R$ forms an totally ordered field.
From the Continuum Property we have that $\R$ is complete.
It remains to be shown that any ordered field which also satisfies the Continuum Property is isomorphic to $\R$.
{{ProofWanted}} | The [[Definition:Real Number|set of real numbers]] $\R$ is the only [[Definition:Ordered Field|ordered field]] which also satisfies the [[Continuum Property]]. | From [[Real Numbers form Totally Ordered Field]] we have that $\R$ forms an [[Definition:Totally Ordered Field|totally ordered field]].
From the [[Continuum Property]] we have that $\R$ is [[Definition:Complete Metric Space|complete]].
It remains to be shown that any [[Definition:Ordered Field|ordered field]] which a... | Real Numbers form only Ordered Field which is Complete | https://proofwiki.org/wiki/Real_Numbers_form_only_Ordered_Field_which_is_Complete | https://proofwiki.org/wiki/Real_Numbers_form_only_Ordered_Field_which_is_Complete | [
"Real Numbers",
"Ordered Fields",
"Continuum Property"
] | [
"Definition:Real Number",
"Definition:Ordered Field",
"Continuum Property"
] | [
"Real Numbers form Totally Ordered Field",
"Definition:Totally Ordered Field",
"Continuum Property",
"Definition:Complete Metric Space",
"Definition:Ordered Field",
"Continuum Property",
"Definition:Isomorphism (Abstract Algebra)/Field Isomorphism"
] |
proofwiki-18085 | Sine of Integer Multiple of Argument/Formulation 4 | {{begin-eqn}}
{{eqn | l = \map \sin {n \theta}
| r = \paren {2 \cos \theta } \map \sin {\paren {n - 1 } \theta} - \map \sin {\paren {n - 2 } \theta}
}}
{{end-eqn}} | {{begin-eqn}}
To proceed, we will require the following lemma: | {{begin-eqn}}
{{eqn | l = \map \sin {n \theta}
| r = \paren {2 \cos \theta } \map \sin {\paren {n - 1 } \theta} - \map \sin {\paren {n - 2 } \theta}
}}
{{end-eqn}} | {{begin-eqn}}
To proceed, we will require the following lemma: | Sine of Integer Multiple of Argument/Formulation 4 | https://proofwiki.org/wiki/Sine_of_Integer_Multiple_of_Argument/Formulation_4 | https://proofwiki.org/wiki/Sine_of_Integer_Multiple_of_Argument/Formulation_4 | [
"Sine of Integer Multiple of Argument"
] | [] | [] |
proofwiki-18086 | Continuous Mappings preserve Compact Subsets | Let $X, Y$ be normed vector spaces.
Let $K \subseteq X$ be a compact subset.
Suppose $f : X \to Y$ is a continuous mapping at each $x \in K$.
Then $\map f K$ is a compact subset of $Y$. | Let $\sequence {y_n}_{n \mathop \in \N}$ be a sequence contained in $\map f K$.
Then:
:$\forall n \in \N : \exists x_n \in K : y_n = \map f {x_n}$
$K$ is compact.
By definition, there is a convergent subsequence $\sequence {x_{n_k}}_{k \mathop \in \N}$ convergent with the limit $L \in K$.
$f$ is a continuous mapping.
B... | Let $X, Y$ be [[Definition:Normed Vector Space|normed vector spaces]].
Let $K \subseteq X$ be a [[Definition:Compact Space/Normed Vector Space/Subspace|compact subset]].
Suppose $f : X \to Y$ is a [[Definition:Continuous Mapping (Normed Vector Space)|continuous mapping]] at each $x \in K$.
Then $\map f K$ is a [[De... | Let $\sequence {y_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] contained in $\map f K$.
Then:
:$\forall n \in \N : \exists x_n \in K : y_n = \map f {x_n}$
$K$ is [[Definition:Compact Space/Normed Vector Space/Subspace|compact]].
By [[Definition:Compact Subset of Normed Vector Space|definition]], the... | Continuous Mappings preserve Compact Subsets | https://proofwiki.org/wiki/Continuous_Mappings_preserve_Compact_Subsets | https://proofwiki.org/wiki/Continuous_Mappings_preserve_Compact_Subsets | [
"Compact Normed Vector Spaces",
"Continuous Mappings on Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Compact Space/Normed Vector Space/Subspace",
"Definition:Continuous Mapping (Normed Vector Space)",
"Definition:Compact Space/Normed Vector Space/Subspace"
] | [
"Definition:Sequence",
"Definition:Compact Space/Normed Vector Space/Subspace",
"Definition:Compact Space/Normed Vector Space",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Limit of Sequence/Normed Vector Space",
"Definition:Cont... |
proofwiki-18087 | Sine of Integer Multiple of Argument/Formulation 5 | {{begin-eqn}}
{{eqn | l = \sin n \theta
| r = \paren {\sin \frac {n \pi} 2} \paren {\sin \theta} + \paren {2 \cos \theta} \paren {\map \sin {\paren {n - 1} \theta} - \map \sin {\paren {n - 3} \theta} + \map \sin {\paren {n - 5} \theta} - \cdots}
| c =
}}
{{eqn | r = \paren {\sin \frac {n \pi} 2} \paren {\s... | The proof proceeds by induction.
For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:
:$\ds \sin n \theta = \paren {\sin \frac {n \pi} 2} \paren {\sin \theta} + 2 \cos \theta \paren {\sum_{k \mathop = 0}^n \paren {\sin \frac {k \pi} 2} \map \sin {\paren {n - k} \theta} }$ | {{begin-eqn}}
{{eqn | l = \sin n \theta
| r = \paren {\sin \frac {n \pi} 2} \paren {\sin \theta} + \paren {2 \cos \theta} \paren {\map \sin {\paren {n - 1} \theta} - \map \sin {\paren {n - 3} \theta} + \map \sin {\paren {n - 5} \theta} - \cdots}
| c =
}}
{{eqn | r = \paren {\sin \frac {n \pi} 2} \paren {\s... | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \sin n \theta = \paren {\sin \frac {n \pi} 2} \paren {\sin \theta} + 2 \cos \theta \paren {\sum_{k \mathop = 0}^n \paren {\sin \frac {k \pi} 2} \map \si... | Sine of Integer Multiple of Argument/Formulation 5 | https://proofwiki.org/wiki/Sine_of_Integer_Multiple_of_Argument/Formulation_5 | https://proofwiki.org/wiki/Sine_of_Integer_Multiple_of_Argument/Formulation_5 | [
"Sine of Integer Multiple of Argument"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-18088 | Equal Surfaces do not Intersect | Let $R$ be a region of space, which may be the interior of a body.
Let there exist a point-function $F$ on $R$ giving rise to a scalar field.
Let $S_1$ and $S_2$ be equal surfaces in $R$ upon which the value of $F$ on $S_1$ is different from the value of $F$ on $S_2$.
Then $S_1$ and $S_2$ do not intersect. | Let:
:$\forall p \in S_1: \map F p = C_1$
:$\forall p \in S_2: \map F p = C_2$
By hypothesis, $C_1 \ne C_2$.
{{AimForCont}} there exists a point $P$ in $R$ such that both $P \in S_1$ and $P \in S_2$.
Then $\map F p = C_1$ and also $\map F p = C_2$.
This contradicts the fact that $F$ is a function.
Hence the result by P... | Let $R$ be a [[Definition:Region|region]] of [[Definition:Ordinary Space|space]], which may be the interior of a [[Definition:Body|body]].
Let there exist a [[Definition:Point-Function|point-function]] $F$ on $R$ giving rise to a [[Definition:Scalar Field (Physics)|scalar field]].
Let $S_1$ and $S_2$ be [[Definition:... | Let:
:$\forall p \in S_1: \map F p = C_1$
:$\forall p \in S_2: \map F p = C_2$
[[Definition:By Hypothesis|By hypothesis]], $C_1 \ne C_2$.
{{AimForCont}} there exists a [[Definition:Point|point]] $P$ in $R$ such that both $P \in S_1$ and $P \in S_2$.
Then $\map F p = C_1$ and also $\map F p = C_2$.
This [[Definition... | Equal Surfaces do not Intersect | https://proofwiki.org/wiki/Equal_Surfaces_do_not_Intersect | https://proofwiki.org/wiki/Equal_Surfaces_do_not_Intersect | [
"Equal Surfaces"
] | [
"Definition:Region",
"Definition:Ordinary Space",
"Definition:Body",
"Definition:Point-Function",
"Definition:Scalar Field (Physics)",
"Definition:Equal Surface",
"Definition:Intersection (Geometry)"
] | [
"Definition:By Hypothesis",
"Definition:Point",
"Definition:Contradiction",
"Definition:Function",
"Proof by Contradiction"
] |
proofwiki-18089 | Dot Product with Zero Vector is Zero | :$\mathbf u \cdot \mathbf 0 = 0$
where $\mathbf 0$ denotes the zero vector. | By definition of dot product:
{{begin-eqn}}
{{eqn | l = \mathbf u \cdot \mathbf 0
| r = \norm {\mathbf u} \norm {\mathbf 0} \cos \theta
| c = {{Defof|Dot Product|subdef = Real Euclidean Space}}
}}
{{eqn | r = \norm {\mathbf u} \times 0 \times \cos \theta
| c =
}}
{{eqn | r = 0
| c =
}}
{{end-e... | :$\mathbf u \cdot \mathbf 0 = 0$
where $\mathbf 0$ denotes the [[Definition:Zero Vector|zero vector]]. | By definition of [[Definition:Dot Product|dot product]]:
{{begin-eqn}}
{{eqn | l = \mathbf u \cdot \mathbf 0
| r = \norm {\mathbf u} \norm {\mathbf 0} \cos \theta
| c = {{Defof|Dot Product|subdef = Real Euclidean Space}}
}}
{{eqn | r = \norm {\mathbf u} \times 0 \times \cos \theta
| c =
}}
{{eqn | r... | Dot Product with Zero Vector is Zero | https://proofwiki.org/wiki/Dot_Product_with_Zero_Vector_is_Zero | https://proofwiki.org/wiki/Dot_Product_with_Zero_Vector_is_Zero | [
"Dot Product"
] | [
"Definition:Zero Vector"
] | [
"Definition:Dot Product"
] |
proofwiki-18090 | Dot Product of Like Vectors | Let $\mathbf a$ and $\mathbf b$ be vector quantities such that $\mathbf a$ and $\mathbf b$ are like.
Let $\mathbf a \cdot \mathbf b$ denote the dot product of $\mathbf a$ and $\mathbf b$.
Then:
:$\mathbf a \cdot \mathbf b = \norm {\mathbf a} \norm {\mathbf b}$
where $\norm {\, \cdot \,}$ denotes the magnitude of a vect... | By definition of dot product:
:$\mathbf a \cdot \mathbf b = \norm {\mathbf a} \norm {\mathbf b} \cos \theta$
where $\theta$ is the angle between $\mathbf a$ and $\mathbf b$.
When $\mathbf a$ and $\mathbf b$ are like, by definition $\theta = 0$.
The result follows by Cosine of Zero is One, which gives that $\cos 0 \degr... | Let $\mathbf a$ and $\mathbf b$ be [[Definition:Vector Quantity|vector quantities]] such that $\mathbf a$ and $\mathbf b$ are [[Definition:Like Vector Quantities|like]].
Let $\mathbf a \cdot \mathbf b$ denote the [[Definition:Dot Product|dot product]] of $\mathbf a$ and $\mathbf b$.
Then:
:$\mathbf a \cdot \mathbf b... | By definition of [[Definition:Dot Product|dot product]]:
:$\mathbf a \cdot \mathbf b = \norm {\mathbf a} \norm {\mathbf b} \cos \theta$
where $\theta$ is the [[Definition:Angle|angle]] between $\mathbf a$ and $\mathbf b$.
When $\mathbf a$ and $\mathbf b$ are [[Definition:Like Vector Quantities|like]], by definition $\... | Dot Product of Like Vectors | https://proofwiki.org/wiki/Dot_Product_of_Like_Vectors | https://proofwiki.org/wiki/Dot_Product_of_Like_Vectors | [
"Dot Product"
] | [
"Definition:Vector Quantity",
"Definition:Like Vector Quantities",
"Definition:Dot Product",
"Definition:Magnitude",
"Definition:Vector Quantity"
] | [
"Definition:Dot Product",
"Definition:Angle",
"Definition:Like Vector Quantities",
"Cosine of Zero is One"
] |
proofwiki-18091 | Dot Product of Elements of Standard Ordered Basis | Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis of Cartesian $3$-space $S$.
Then:
:$\mathbf i \cdot \mathbf j = \mathbf j \cdot \mathbf k = \mathbf k \cdot \mathbf i = 0$
where $\cdot$ denotes the dot product. | By definition, the Cartesian $3$-space is a frame of reference consisting of a rectangular coordinate system.
By definition of rectangular coordinate system, the coordinate axes are perpendicular to each other.
By definition of Component of Vector in $3$-space, the vectors $\mathbf i$, $\mathbf j$ and $\mathbf k$ are t... | Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the [[Definition:Standard Ordered Basis|standard ordered basis]] of [[Definition:Cartesian 3-Space|Cartesian $3$-space]] $S$.
Then:
:$\mathbf i \cdot \mathbf j = \mathbf j \cdot \mathbf k = \mathbf k \cdot \mathbf i = 0$
where $\cdot$ denotes the [[Definition:Dot Prod... | By definition, the [[Definition:Cartesian 3-Space|Cartesian $3$-space]] is a [[Definition:Frame of Reference|frame of reference]] consisting of a [[Definition:Rectangular Coordinate System|rectangular coordinate system]].
By definition of [[Definition:Rectangular Coordinate System|rectangular coordinate system]], the ... | Dot Product of Elements of Standard Ordered Basis | https://proofwiki.org/wiki/Dot_Product_of_Elements_of_Standard_Ordered_Basis | https://proofwiki.org/wiki/Dot_Product_of_Elements_of_Standard_Ordered_Basis | [
"Dot Product",
"Standard Ordered Bases"
] | [
"Definition:Standard Ordered Basis",
"Definition:Cartesian 3-Space",
"Definition:Dot Product"
] | [
"Definition:Cartesian 3-Space",
"Definition:Frame of Reference",
"Definition:Rectangular Coordinate System",
"Definition:Rectangular Coordinate System",
"Definition:Axis/Coordinate Axes",
"Definition:Right Angle/Perpendicular",
"Definition:Vector Quantity/Component/Cartesian 3-Space",
"Definition:Vect... |
proofwiki-18092 | Self-Product of Standard Ordered Basis Element equals 1 | Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis of Cartesian $3$-space $S$.
Then:
:$\mathbf i^2 = \mathbf j^2 = \mathbf k^2 = 1$
where $\mathbf i^2$ and so on denotes the square of a vector quantity:
:$\mathbf i^2 := \mathbf i \cdot \mathbf i$ | By definition, the Cartesian $3$-space is a frame of reference consisting of a rectangular coordinate system.
By definition of Component of Vector in $3$-space, the vectors $\mathbf i$, $\mathbf j$ and $\mathbf k$ are the unit vectors in the direction of the $x$-axis, $y$-axis and $z$-axis respectively.
Hence $\mathbf ... | Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the [[Definition:Standard Ordered Basis|standard ordered basis]] of [[Definition:Cartesian 3-Space|Cartesian $3$-space]] $S$.
Then:
:$\mathbf i^2 = \mathbf j^2 = \mathbf k^2 = 1$
where $\mathbf i^2$ and so on denotes the [[Definition:Square of Vector Quantity|square o... | By definition, the [[Definition:Cartesian 3-Space|Cartesian $3$-space]] is a [[Definition:Frame of Reference|frame of reference]] consisting of a [[Definition:Rectangular Coordinate System|rectangular coordinate system]].
By definition of [[Definition:Component of Vector in 3-Space|Component of Vector in $3$-space]], ... | Self-Product of Standard Ordered Basis Element equals 1 | https://proofwiki.org/wiki/Self-Product_of_Standard_Ordered_Basis_Element_equals_1 | https://proofwiki.org/wiki/Self-Product_of_Standard_Ordered_Basis_Element_equals_1 | [
"Dot Product",
"Standard Ordered Bases"
] | [
"Definition:Standard Ordered Basis",
"Definition:Cartesian 3-Space",
"Definition:Square of Vector Quantity"
] | [
"Definition:Cartesian 3-Space",
"Definition:Frame of Reference",
"Definition:Rectangular Coordinate System",
"Definition:Vector Quantity/Component/Cartesian 3-Space",
"Definition:Vector Quantity",
"Definition:Unit Vector",
"Definition:Axis/X-Axis",
"Definition:Axis/Y-Axis",
"Definition:Axis/Z-Axis",... |
proofwiki-18093 | Equation of Plane/Vector Form | Let $P$ be a plane which passes through a point $C$ whose position vector relative to the origin $O$ is $\mathbf c$.
Let $\mathbf p$ be the vector perpendicular to $P$ from $O$.
Let $\mathbf r$ be the position vector of an arbitrary point on $P$.
Then $P$ can be represented by the equation:
:$\mathbf p \cdot \paren {\m... | :600px
It is seen that $\mathbf r - \mathbf c$ lies entirely within the plane $P$.
As $P$ is perpendicular to $\mathbf p$, it follows that $\mathbf r - \mathbf c$ is perpendicular to $\mathbf p$.
Hence by Dot Product of Perpendicular Vectors:
:$\mathbf p \cdot \paren {\mathbf r - \mathbf c} = 0$
{{qed}} | Let $P$ be a [[Definition:Plane|plane]] which passes through a [[Definition:Point|point]] $C$ whose [[Definition:Position Vector|position vector]] relative to the [[Definition:Origin|origin]] $O$ is $\mathbf c$.
Let $\mathbf p$ be the [[Definition:Vector Quantity|vector]] [[Definition:Line Perpendicular to Plane|perpe... | :[[File:Vector-equation-of-plane.png|600px]]
It is seen that $\mathbf r - \mathbf c$ lies entirely within the [[Definition:Plane|plane]] $P$.
As $P$ is [[Definition:Line Perpendicular to Plane|perpendicular]] to $\mathbf p$, it follows that $\mathbf r - \mathbf c$ is [[Definition:Line Perpendicular to Plane|perpendic... | Equation of Plane/Vector Form | https://proofwiki.org/wiki/Equation_of_Plane/Vector_Form | https://proofwiki.org/wiki/Equation_of_Plane/Vector_Form | [
"Vector Equation of Plane",
"Equations of Planes"
] | [
"Definition:Plane Surface",
"Definition:Point",
"Definition:Position Vector",
"Definition:Coordinate System/Origin",
"Definition:Vector Quantity",
"Definition:Right Angle/Perpendicular/Plane",
"Definition:Position Vector",
"Definition:Point",
"Definition:Equation",
"Definition:Dot Product"
] | [
"File:Vector-equation-of-plane.png",
"Definition:Plane Surface",
"Definition:Right Angle/Perpendicular/Plane",
"Definition:Right Angle/Perpendicular/Plane",
"Dot Product of Perpendicular Vectors"
] |
proofwiki-18094 | Resultant in Terms of Dot Product | Let $\mathbf a$ and $\mathbf b$ be vector quantities.
Let their resultant be $\mathbf v$:
:$\mathbf v = \mathbf a + \mathbf b$
Then:
:$\mathbf v^2 = \mathbf a^2 + 2 \mathbf a \cdot \mathbf b + \mathbf b^2$
where:
:$\mathbf v^2$ denotes the square of $\mathbf v$
:$\mathbf a \cdot \mathbf b$ denotes the dot product of $\... | {{begin-eqn}}
{{eqn | l = \mathbf v^2
| r = \paren {\mathbf a + \mathbf b}^2
| c =
}}
{{eqn | r = \paren {\mathbf a + \mathbf b} \paren {\mathbf a + \mathbf b}
| c =
}}
{{eqn | r = \mathbf a \cdot \mathbf a + \mathbf a \cdot \mathbf b + \mathbf b \cdot \mathbf a + \mathbf b \cdot \mathbf b
| c... | Let $\mathbf a$ and $\mathbf b$ be [[Definition:Vector Quantity|vector quantities]].
Let their [[Definition:Resultant of Vectors|resultant]] be $\mathbf v$:
:$\mathbf v = \mathbf a + \mathbf b$
Then:
:$\mathbf v^2 = \mathbf a^2 + 2 \mathbf a \cdot \mathbf b + \mathbf b^2$
where:
:$\mathbf v^2$ denotes the [[Definiti... | {{begin-eqn}}
{{eqn | l = \mathbf v^2
| r = \paren {\mathbf a + \mathbf b}^2
| c =
}}
{{eqn | r = \paren {\mathbf a + \mathbf b} \paren {\mathbf a + \mathbf b}
| c =
}}
{{eqn | r = \mathbf a \cdot \mathbf a + \mathbf a \cdot \mathbf b + \mathbf b \cdot \mathbf a + \mathbf b \cdot \mathbf b
| c... | Resultant in Terms of Dot Product | https://proofwiki.org/wiki/Resultant_in_Terms_of_Dot_Product | https://proofwiki.org/wiki/Resultant_in_Terms_of_Dot_Product | [
"Vector Addition",
"Dot Product"
] | [
"Definition:Vector Quantity",
"Definition:Vector Sum",
"Definition:Square of Vector Quantity",
"Definition:Dot Product"
] | [] |
proofwiki-18095 | Components of Vector in Terms of Dot Product | Let $\mathbf a$ be a vector quantity embedded in Cartesian $3$-space $S$.
Let $\mathbf i$, $\mathbf j$ and $\mathbf k$ be the unit vectors in the positive directions of the $x$-axis, $y$-axis and $z$-axis respectively.
Let $\mathbf a$ be expressed in component form:
:$\mathbf a = x \mathbf i + y \mathbf j + z \mathbf k... | Let $\mathbf a$ be a vector as described.
From the definition of cartesian space (by axes) and from the fact that $\mathbf i, \mathbf j, \mathbf k$ are unit vectors, it follows that $\tuple {\mathbf i, \mathbf j,\mathbf k}$ is an orthonormal basis.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf a \cdot \mathbf i
| r = ... | Let $\mathbf a$ be a [[Definition:Vector Quantity|vector quantity]] embedded in [[Definition:Cartesian 3-Space|Cartesian $3$-space]] $S$.
Let $\mathbf i$, $\mathbf j$ and $\mathbf k$ be the [[Definition:Unit Vector|unit vectors]] in the [[Definition:Positive Direction|positive directions]] of the [[Definition:X-Axis|$... | Let $\mathbf a$ be a vector as described.
From the definition of [[Definition:Cartesian 3-Space/Definition by Axes|cartesian space (by axes)]] and from the fact that $\mathbf i, \mathbf j, \mathbf k$ are [[Definition:Unit Vector|unit vectors]], it follows that $\tuple {\mathbf i, \mathbf j,\mathbf k}$ is an [[Definit... | Components of Vector in Terms of Dot Product | https://proofwiki.org/wiki/Components_of_Vector_in_Terms_of_Dot_Product | https://proofwiki.org/wiki/Components_of_Vector_in_Terms_of_Dot_Product | [
"Dot Product"
] | [
"Definition:Vector Quantity",
"Definition:Cartesian 3-Space",
"Definition:Unit Vector",
"Definition:Axis/Positive Direction",
"Definition:Axis/X-Axis",
"Definition:Axis/Y-Axis",
"Definition:Axis/Z-Axis",
"Definition:Vector Quantity/Component/Cartesian 3-Space"
] | [
"Definition:Cartesian 3-Space/Definition by Axes",
"Definition:Unit Vector",
"Definition:Orthonormal Basis",
"Dot Product Operator is Bilinear",
"Dot Product Operator is Bilinear",
"Dot Product Operator is Bilinear"
] |
proofwiki-18096 | Electromotive Force in Closed Path in Electric Field is Zero | Let $\mathbf E$ be an electric field acting over a region of space $R$.
Let $\Gamma$ be a closed contour in $R$.
Then the electromotive force in $\Gamma$ is zero. | In an electrostatic context, electric fields and magnetic fields are constant in time.
From Maxwell-Faraday Equation, we have
{{begin-eqn}}
{{eqn | l = \nabla \times \mathbf E
| r = -\frac {\partial \mathbf B} {\partial t}
}}
{{eqn | r = 0
| c = Magnetic field is static
}}
{{end-eqn}}
Therefore, by Definiti... | Let $\mathbf E$ be an [[Definition:Electric Field|electric field]] acting over a [[Definition:Region|region]] of [[Definition:Ordinary Space|space]] $R$.
Let $\Gamma$ be a [[Definition:Closed Contour|closed contour]] in $R$.
Then the [[Definition:Electromotive Force|electromotive force]] in $\Gamma$ is zero. | In an [[Definition:Electrostatics|electrostatic]] context, [[Definition:Electric Field|electric fields]] and [[Definition:Magnetic Field|magnetic fields]] are [[Definition:Constant|constant]] in [[Definition:Time|time]].
From [[Maxwell-Faraday Equation]], we have
{{begin-eqn}}
{{eqn | l = \nabla \times \mathbf E
... | Electromotive Force in Closed Path in Electric Field is Zero | https://proofwiki.org/wiki/Electromotive_Force_in_Closed_Path_in_Electric_Field_is_Zero | https://proofwiki.org/wiki/Electromotive_Force_in_Closed_Path_in_Electric_Field_is_Zero | [
"Electrostatics"
] | [
"Definition:Electric Field",
"Definition:Region",
"Definition:Ordinary Space",
"Definition:Contour/Closed",
"Definition:Electromotive Force"
] | [
"Definition:Electrostatics",
"Definition:Electric Field",
"Definition:Magnetic Field",
"Definition:Constant",
"Definition:Time",
"Maxwell-Faraday Equation",
"Definition:Magnetic Field",
"Definition:Conservative Vector Field/Definition 1",
"Definition:Conservative Vector Field",
"Definition:Conserv... |
proofwiki-18097 | Extreme Value Theorem/Normed Vector Space | Let $X$ be a normed vector space.
Let $K \subseteq X$ be a compact subset.
Suppose $f : X \to \R$ is a continuous mapping at each $x \in K$.
Then:
:$\ds \exists c \in K : \map f c = \sup_{x \mathop \in K} \map f x = \max_{x \mathop \in K} \map f x$
:$\ds \exists d \in K : \map f d = \inf_{x \mathop \in K} \map f x = \m... | Let $K$ be compact.
By Continuous Mappings preserve Compact Subsets, $\map f K$ is compact.
By Compact Subset of Normed Vector Space is Closed and Bounded, $K$ is bounded.
Hence, $\map f K$ is bounded.
$K$ is nonempty, so $\map f K$ is non-empty.
By Characterizing Property of Supremum of Subset of Real Numbers, non-emp... | Let $X$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $K \subseteq X$ be a [[Definition:Compact Space/Normed Vector Space/Subspace|compact subset]].
Suppose $f : X \to \R$ is a [[Definition:Continuous Mapping (Normed Vector Space)|continuous mapping]] at each $x \in K$.
Then:
:$\ds \exists c \in... | Let $K$ be [[Definition:Compact Space/Normed Vector Space/Subspace|compact]].
By [[Continuous Mappings preserve Compact Subsets]], $\map f K$ is [[Definition:Compact Space/Normed Vector Space/Subspace|compact]].
By [[Compact Subset of Normed Vector Space is Closed and Bounded]], $K$ is [[Definition:Bounded Normed Vec... | Extreme Value Theorem/Normed Vector Space | https://proofwiki.org/wiki/Extreme_Value_Theorem/Normed_Vector_Space | https://proofwiki.org/wiki/Extreme_Value_Theorem/Normed_Vector_Space | [
"Continuous Mappings",
"Suprema",
"Infima",
"Max and Min Operations"
] | [
"Definition:Normed Vector Space",
"Definition:Compact Space/Normed Vector Space/Subspace",
"Definition:Continuous Mapping (Normed Vector Space)"
] | [
"Definition:Compact Space/Normed Vector Space/Subspace",
"Continuous Mappings preserve Compact Subsets",
"Definition:Compact Space/Normed Vector Space/Subspace",
"Compact Subset of Normed Vector Space is Closed and Bounded",
"Definition:Bounded Subset of Normed Vector Space",
"Definition:Bounded Mapping/R... |
proofwiki-18098 | Sine of Integer Multiple of Argument/Formulation 7 | {{begin-eqn}}
{{eqn | l = \sin n \theta
| r = \paren {\sin^2 \frac {n \pi} 2} \paren {\sin \theta} + \paren {2 \sin \theta } \paren {\paren {0} \map \cos {\paren {n - 0} \theta} + \paren {1} \map \cos {\paren {n - 1} \theta} + \paren {0} \map \cos {\paren {n - 2} \theta} + \paren {1} \map \cos {\paren {n - 3} \th... | The proof proceeds by induction.
For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:
:$\ds \sin n \theta = \paren {\sin^2 \frac {n \pi} 2} \paren {\sin \theta} + 2 \sin \theta \paren {\sum_{k \mathop = 0}^{n - 1} \paren {\sin^2 \frac {k \pi} 2} \map \cos {\paren {n - k} \theta} }$ | {{begin-eqn}}
{{eqn | l = \sin n \theta
| r = \paren {\sin^2 \frac {n \pi} 2} \paren {\sin \theta} + \paren {2 \sin \theta } \paren {\paren {0} \map \cos {\paren {n - 0} \theta} + \paren {1} \map \cos {\paren {n - 1} \theta} + \paren {0} \map \cos {\paren {n - 2} \theta} + \paren {1} \map \cos {\paren {n - 3} \th... | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \sin n \theta = \paren {\sin^2 \frac {n \pi} 2} \paren {\sin \theta} + 2 \sin \theta \paren {\sum_{k \mathop = 0}^{n - 1} \paren {\sin^2 \frac {k \pi} 2... | Sine of Integer Multiple of Argument/Formulation 7 | https://proofwiki.org/wiki/Sine_of_Integer_Multiple_of_Argument/Formulation_7 | https://proofwiki.org/wiki/Sine_of_Integer_Multiple_of_Argument/Formulation_7 | [
"Sine of Integer Multiple of Argument"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-18099 | Cosine of Integer Multiple of Argument/Formulation 5 | {{begin-eqn}}
{{eqn | l = \cos n \theta
| r = \map \sin {\frac {\paren {n + 1} \pi} 2} + \paren {\sin \frac {n \pi} 2} \cos \theta + \paren {2 \cos \theta} \paren {\map \cos {\paren {n - 1} \theta} - \map \cos {\paren {n - 3} \theta} + \map \cos {\paren {n - 5} \theta} - \cdots}
| c =
}}
{{eqn | r = \map \... | The proof proceeds by induction.
For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:
:$\ds \cos n \theta = \map \sin {\frac {\paren {n + 1} \pi} 2} + \paren {\sin \frac {n \pi} 2} \cos \theta + 2 \cos \theta \paren {\sum_{k \mathop = 0}^{n - 1} \paren {\sin \frac {k \pi} 2} \map \cos {\paren {n - k} \theta} }$ | {{begin-eqn}}
{{eqn | l = \cos n \theta
| r = \map \sin {\frac {\paren {n + 1} \pi} 2} + \paren {\sin \frac {n \pi} 2} \cos \theta + \paren {2 \cos \theta} \paren {\map \cos {\paren {n - 1} \theta} - \map \cos {\paren {n - 3} \theta} + \map \cos {\paren {n - 5} \theta} - \cdots}
| c =
}}
{{eqn | r = \map \... | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \cos n \theta = \map \sin {\frac {\paren {n + 1} \pi} 2} + \paren {\sin \frac {n \pi} 2} \cos \theta + 2 \cos \theta \paren {\sum_{k \mathop = 0}^{n - 1... | Cosine of Integer Multiple of Argument/Formulation 5 | https://proofwiki.org/wiki/Cosine_of_Integer_Multiple_of_Argument/Formulation_5 | https://proofwiki.org/wiki/Cosine_of_Integer_Multiple_of_Argument/Formulation_5 | [
"Cosine of Integer Multiple of Argument"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.