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-20000 | Open Ball is Simply Connected | Let $\struct {V, \norm {\,\cdot\,} }$ be a normed vector space over $\R$ or $\C$.
Let $d: V \times V \to \R_{\ge 0}$ be the metric induced by the norm $\norm {\,\cdot\,}$ on $V$.
Let $\tau$ be the the topology on $V$ induced by the metric $d$.
Let $v \in V$ and $\epsilon \in \R_{>0}$.
Let $\map {B_\epsilon} v$ be the o... | Normed Vector Space is Hausdorff Topological Vector Space shows that $\struct {V, \tau}$ is a topological vector space.
The result now follows from Open Ball is Convex Set and Convex Set is Simply Connected.
{{qed}}
Category:Normed Vector Spaces
Category:Open Balls
Category:Simply Connected Spaces
hzf6v9nvwl00p0jon7rg5... | Let $\struct {V, \norm {\,\cdot\,} }$ be a [[Definition:Normed Vector Space|normed vector space]] over $\R$ or $\C$.
Let $d: V \times V \to \R_{\ge 0}$ be the [[Definition:Metric Induced by Norm|metric induced]] by the [[Definition:Norm on Vector Space|norm]] $\norm {\,\cdot\,}$ on $V$.
Let $\tau$ be the the [[Defini... | [[Normed Vector Space is Hausdorff Topological Vector Space]] shows that $\struct {V, \tau}$ is a [[Definition:Topological Vector Space|topological vector space]].
The result now follows from [[Open Ball is Convex Set]] and [[Convex Set is Simply Connected]].
{{qed}}
[[Category:Normed Vector Spaces]]
[[Category:Open... | Open Ball is Simply Connected | https://proofwiki.org/wiki/Open_Ball_is_Simply_Connected | https://proofwiki.org/wiki/Open_Ball_is_Simply_Connected | [
"Normed Vector Spaces",
"Open Balls",
"Simply Connected Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Metric Induced by Norm",
"Definition:Norm/Vector Space",
"Definition:Topology Induced by Metric",
"Definition:Open Ball",
"Definition:Topological Subspace",
"Definition:Simply Connected"
] | [
"Normed Vector Space is Hausdorff Topological Vector Space",
"Definition:Topological Vector Space",
"Open Ball is Convex Set",
"Convex Set is Simply Connected",
"Category:Normed Vector Spaces",
"Category:Open Balls",
"Category:Simply Connected Spaces"
] |
proofwiki-20001 | Basis Element of Furstenberg Topology is Clopen | Let $\tau$ be the Furstenberg topology on the set of integers $\Z$.
Let $a, b \in \Z$ such that $a \ne 0$.
Then $a \Z + b$ is clopen in $\struct {\Z, \tau}$. | $a \Z + b \in \tau$ by {{Defof|Furstenberg Topology}}.
It remains to show:
:$\Z \setminus \paren {a \Z + b} \in \tau$
As $a \Z = \paren {-a} \Z$, we may assume $a > 0$.
If $a = 1$, then $\Z \setminus \Z = \O \in \tau$.
Thus we assume that $a \ge 2$.
Then:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \Z \setm... | Let $\tau$ be the [[Definition:Furstenberg Topology|Furstenberg topology]] on the [[Definition:Integer|set of integers]] $\Z$.
Let $a, b \in \Z$ such that $a \ne 0$.
Then $a \Z + b$ is [[Definition:Clopen Set|clopen]] in $\struct {\Z, \tau}$. | $a \Z + b \in \tau$ by {{Defof|Furstenberg Topology}}.
It remains to show:
:$\Z \setminus \paren {a \Z + b} \in \tau$
As $a \Z = \paren {-a} \Z$, we may assume $a > 0$.
If $a = 1$, then $\Z \setminus \Z = \O \in \tau$.
Thus we assume that $a \ge 2$.
Then:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \Z... | Basis Element of Furstenberg Topology is Clopen | https://proofwiki.org/wiki/Basis_Element_of_Furstenberg_Topology_is_Clopen | https://proofwiki.org/wiki/Basis_Element_of_Furstenberg_Topology_is_Clopen | [
"Furstenberg Topology"
] | [
"Definition:Furstenberg Topology",
"Definition:Integer",
"Definition:Clopen Set"
] | [
"Category:Furstenberg Topology"
] |
proofwiki-20002 | Cosine of 2 is Strictly Negative | :$\cos 2 < 0$ | Recall the definition of the cosine:
{{Definition:Real Cosine Function}}
Thus:
{{begin-eqn}}
{{eqn | l = \cos 2
| r = 1 - \paren {\frac {2^2} {2!} - \frac {2^4} {4!} } - \paren {\frac {2^6} {6!} - \frac {2^8} {8!} } - \cdots
}}
{{eqn | r = 1 - \sum_{n \mathop = 1}^\infty A_n
}}
{{end-eqn}}
where:
{{begin-eqn}}
{... | :$\cos 2 < 0$ | Recall the definition of the [[Definition:Real Cosine Function|cosine]]:
{{Definition:Real Cosine Function}}
Thus:
{{begin-eqn}}
{{eqn | l = \cos 2
| r = 1 - \paren {\frac {2^2} {2!} - \frac {2^4} {4!} } - \paren {\frac {2^6} {6!} - \frac {2^8} {8!} } - \cdots
}}
{{eqn | r = 1 - \sum_{n \mathop = 1}^\infty A_n... | Cosine of 2 is Strictly Negative | https://proofwiki.org/wiki/Cosine_of_2_is_Strictly_Negative | https://proofwiki.org/wiki/Cosine_of_2_is_Strictly_Negative | [
"Cosine Function"
] | [] | [
"Definition:Cosine/Real Function",
"Category:Cosine Function"
] |
proofwiki-20003 | Interior of Simply Closed Contour Extends to Simply Connected Domain | Let $C$ be a simple closed contour in $U$, where $U \subseteq \C$ is an open set.
Let $\Int C \subseteq U$, where $\Int C$ denotes the interior of $C$.
Then there exists a simply connected domain $V$ such that $\Int C \subseteq V \subseteq U$, and $C$ is a contour in $V$. | Let $\mathbb S^1$ denote the unit circle in $\R^2$ whose center is at the origin $\mathbf 0$ of $\R^2$.
Let:
:$\map {B_1} {\mathbf 0 }$ denote the open ball in $\R^2$ with radius $1$ and center $\mathbf 0$
:$\map {B_1^-} {\mathbf 0}$ denote the closed ball in $\R^2$ with radius $1$ and center $\mathbf 0$.
We use the Jo... | Let $C$ be a [[Definition:Simple Contour (Complex Plane)|simple]] [[Definition:Closed Contour (Complex Plane)|closed contour]] in $U$, where $U \subseteq \C$ is an [[Definition:Open Set (Complex Analysis)|open set]].
Let $\Int C \subseteq U$, where $\Int C$ denotes the [[Definition:Interior of Simple Closed Contour|in... | Let $\mathbb S^1$ denote the [[Definition:Unit Circle|unit circle]] in $\R^2$ whose [[Definition:Center of Circle|center]] is at the [[Definition:Origin|origin]] $\mathbf 0$ of $\R^2$.
Let:
:$\map {B_1} {\mathbf 0 }$ denote the [[Definition:Open Ball in Normed Vector Space|open ball]] in $\R^2$ with [[Definition:Radiu... | Interior of Simply Closed Contour Extends to Simply Connected Domain | https://proofwiki.org/wiki/Interior_of_Simply_Closed_Contour_Extends_to_Simply_Connected_Domain | https://proofwiki.org/wiki/Interior_of_Simply_Closed_Contour_Extends_to_Simply_Connected_Domain | [
"Complex Contour Integrals"
] | [
"Definition:Contour/Simple/Complex Plane",
"Definition:Contour/Closed/Complex Plane",
"Definition:Open Set/Complex Analysis",
"Definition:Interior of Simple Closed Contour",
"Definition:Connected Domain (Complex Analysis)/Simply Connected Domain",
"Definition:Contour/Complex Plane"
] | [
"Definition:Unit Circle",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Open Ball/Normed Vector Space",
"Definition:Open Ball/Radius",
"Definition:Open Ball/Center",
"Definition:Closed Ball/Normed Vector Space",
"Definition:Open Ball/Radius",
"Definition:Open Ball/Ce... |
proofwiki-20004 | Non-Empty Open Set of Furstenberg Topology is Infinite | Let $\struct {\Z, \tau}$ be the topological space formed by the Furstenberg topology on the set of integers $\Z$.
Let $U \in \tau$ such that $U \ne \O$.
Then $U$ is an infinite set. | {{ProofWanted}}
Category:Furstenberg Topology
bsxfvjbqv2zdugidjfr0ebjr5tsn8a2 | Let $\struct {\Z, \tau}$ be the [[Definition:Topological Space|topological space]] formed by the [[Definition:Furstenberg Topology|Furstenberg topology]] on the [[Definition:Integer|set of integers]] $\Z$.
Let $U \in \tau$ such that $U \ne \O$.
Then $U$ is an [[Definition:Infinite Set|infinite set]]. | {{ProofWanted}}
[[Category:Furstenberg Topology]]
bsxfvjbqv2zdugidjfr0ebjr5tsn8a2 | Non-Empty Open Set of Furstenberg Topology is Infinite | https://proofwiki.org/wiki/Non-Empty_Open_Set_of_Furstenberg_Topology_is_Infinite | https://proofwiki.org/wiki/Non-Empty_Open_Set_of_Furstenberg_Topology_is_Infinite | [
"Furstenberg Topology"
] | [
"Definition:Topological Space",
"Definition:Furstenberg Topology",
"Definition:Integer",
"Definition:Infinite Set"
] | [
"Category:Furstenberg Topology"
] |
proofwiki-20005 | Finite Hausdorff Measure Implies Zero Higher Dimensional Measure | Let $n \in \N_{>0}$.
Let $F \subseteq \R^n$ be a subset of the real Euclidean space.
Let $s \in \R_{\ge 0}$.
Let $\map {\HH^s} \cdot$ denote the $s$-dimensional Hausdorff measure.
Then:
:$\map {\HH^s} F < +\infty \implies \forall t \in \R_{>s} : \map {\HH^t} F = 0$ | For each $\delta$-cover $\sequence {U_i}$ of $F$:
{{begin-eqn}}
{{eqn | l = \sum \size {U_i}^t
| r = \sum \size {U_i}^s \size {U_i}^{t - s}
}}
{{eqn | o = \le
| r = \delta^{t - s} \sum \size {U_i}^s
}}
{{end-eqn}}
Thus:
:$\map {\HH^t_\delta} F \le \delta^{t - s} \map {\HH^s_\delta} F$
Therefore:
{{begin-eqn... | Let $n \in \N_{>0}$.
Let $F \subseteq \R^n$ be a [[Definition:Subset|subset]] of the [[Definition:Real Euclidean Space|real Euclidean space]].
Let $s \in \R_{\ge 0}$.
Let $\map {\HH^s} \cdot$ denote the [[Definition:Hausdorff Measure|$s$-dimensional Hausdorff measure]].
Then:
:$\map {\HH^s} F < +\infty \implies \f... | For each [[Definition:Delta-Cover|$\delta$-cover]] $\sequence {U_i}$ of $F$:
{{begin-eqn}}
{{eqn | l = \sum \size {U_i}^t
| r = \sum \size {U_i}^s \size {U_i}^{t - s}
}}
{{eqn | o = \le
| r = \delta^{t - s} \sum \size {U_i}^s
}}
{{end-eqn}}
Thus:
:$\map {\HH^t_\delta} F \le \delta^{t - s} \map {\HH^s_\delt... | Finite Hausdorff Measure Implies Zero Higher Dimensional Measure | https://proofwiki.org/wiki/Finite_Hausdorff_Measure_Implies_Zero_Higher_Dimensional_Measure | https://proofwiki.org/wiki/Finite_Hausdorff_Measure_Implies_Zero_Higher_Dimensional_Measure | [
"Hausdorff Measures",
"Fractals"
] | [
"Definition:Subset",
"Definition:Euclidean Space/Real",
"Definition:Hausdorff Measure"
] | [
"Definition:Cover of Set/Delta-Cover",
"Category:Hausdorff Measures",
"Category:Fractals"
] |
proofwiki-20006 | Strictly Positive Hausdorff Measure implies Infinite Lower Dimensional Measure | Let $n \in \N_{>0}$.
Let $F \subseteq \R^n$ be a subset of the real Euclidean space.
Let $\map {\HH^s} \cdot$ denote the $s$-dimensional Hausdorff measure.
Let $s \in \R_{\ge 0}$.
Then:
:$\map {\HH^s} F > 0 \implies \forall t \in \hointr 0 s : \map {\HH^t} F = +\infty$ | Let:
:$\exists t \in \hointr 0 s : \map {\HH^t} F < +\infty$
Then by Finite Hausdorff Measure Implies Zero Higher Dimensional Measure:
:$\map {\HH^s} F = 0$
Hence the result by Proof by Contraposition.
{{qed}}
Category:Measure Theory
Category:Fractals
pmp0ltjih7enimrhedgf40avfbbjz1b | Let $n \in \N_{>0}$.
Let $F \subseteq \R^n$ be a [[Definition:Subset|subset]] of the [[Definition:Real Euclidean Space|real Euclidean space]].
Let $\map {\HH^s} \cdot$ denote the [[Definition:Hausdorff Measure|$s$-dimensional Hausdorff measure]].
Let $s \in \R_{\ge 0}$.
Then:
:$\map {\HH^s} F > 0 \implies \forall ... | Let:
:$\exists t \in \hointr 0 s : \map {\HH^t} F < +\infty$
Then by [[Finite Hausdorff Measure Implies Zero Higher Dimensional Measure]]:
:$\map {\HH^s} F = 0$
Hence the result by [[Proof by Contraposition]].
{{qed}}
[[Category:Measure Theory]]
[[Category:Fractals]]
pmp0ltjih7enimrhedgf40avfbbjz1b | Strictly Positive Hausdorff Measure implies Infinite Lower Dimensional Measure | https://proofwiki.org/wiki/Strictly_Positive_Hausdorff_Measure_implies_Infinite_Lower_Dimensional_Measure | https://proofwiki.org/wiki/Strictly_Positive_Hausdorff_Measure_implies_Infinite_Lower_Dimensional_Measure | [
"Measure Theory",
"Fractals"
] | [
"Definition:Subset",
"Definition:Euclidean Space/Real",
"Definition:Hausdorff Measure"
] | [
"Finite Hausdorff Measure Implies Zero Higher Dimensional Measure",
"Proof by Contraposition",
"Category:Measure Theory",
"Category:Fractals"
] |
proofwiki-20007 | Hausdorff-Besicovitch Dimension is Well-Defined | Let $F \subseteq \R^n$ be a subset of the $n$-dimensional Euclidean space.
Let $\map {\HH^s} F$ be the $s$-dimensional Hausdorff measure on $\R^n$ of $F$ for each $s \in \R_{\ge 0}$.
Then:
{{begin-eqn}}
{{eqn | o =
| r = \inf \set {s \in \R_{\ge 0} : \map {\HH^s} F = 0}
}}
{{eqn | r = \sup \set {s \in \R_{\ge 0} ... | {{ProofWanted}}
Category:Hausdorff-Besicovitch Dimension
br8947i6jj969phdfqr0ryvavqdl3wl | Let $F \subseteq \R^n$ be a [[Definition:Subset|subset]] of the [[Definition:Dimension of Vector Space|$n$-dimensional]] [[Definition:Real Euclidean Space|Euclidean space]].
Let $\map {\HH^s} F$ be the [[Definition:Hausdorff Measure|$s$-dimensional Hausdorff measure]] on $\R^n$ of $F$ for each $s \in \R_{\ge 0}$.
The... | {{ProofWanted}}
[[Category:Hausdorff-Besicovitch Dimension]]
br8947i6jj969phdfqr0ryvavqdl3wl | Hausdorff-Besicovitch Dimension is Well-Defined | https://proofwiki.org/wiki/Hausdorff-Besicovitch_Dimension_is_Well-Defined | https://proofwiki.org/wiki/Hausdorff-Besicovitch_Dimension_is_Well-Defined | [
"Hausdorff-Besicovitch Dimension"
] | [
"Definition:Subset",
"Definition:Dimension of Vector Space",
"Definition:Euclidean Space/Real",
"Definition:Hausdorff Measure",
"Definition:Fractal Dimension/Hausdorff-Besicovitch Dimension",
"Definition:Well-Defined/Mapping"
] | [
"Category:Hausdorff-Besicovitch Dimension"
] |
proofwiki-20008 | Higher Dimensional Hausdorff Measure than Euclidean Space is Zero | Let $\R^n$ be the $n$-dimensional Euclidean space.
Let $s \in \R_{>n}$.
Then:
:$\map {\HH ^s} {\R^n} = 0$
where $\map {\HH^s} \cdot$ denotes the $s$-dimensional Hausdorff measure on $\R^n$ | Let $s \in \R_{>n}$.
Consider the $n$-cube:
:$Q := {\closedint 0 1}^n$
Then:
:$\ds \R^n = \bigcup _{x \mathop \in \Z^n} Q + x$
As $\map {\HH ^s} \cdot$ is countably subadditive and translation invariant, it suffices to show:
:$\map {\HH ^s} Q = 0$
Let $N \in \N_{>0}$.
Let $\CC_N$ be the set of $n$-cubes:
:$\ds \closedi... | Let $\R^n$ be the [[Definition:Dimension of Vector Space|$n$-dimensional]] [[Definition:Real Euclidean Space|Euclidean space]].
Let $s \in \R_{>n}$.
Then:
:$\map {\HH ^s} {\R^n} = 0$
where $\map {\HH^s} \cdot$ denotes the [[Definition:Hausdorff Measure|$s$-dimensional Hausdorff measure]] on $\R^n$ | Let $s \in \R_{>n}$.
Consider the [[Definition:N-Cube (Euclidean Space)|$n$-cube]]:
:$Q := {\closedint 0 1}^n$
Then:
:$\ds \R^n = \bigcup _{x \mathop \in \Z^n} Q + x$
As $\map {\HH ^s} \cdot$ is [[Definition:Countably Subadditive Function|countably subadditive]] and translation invariant, it suffices to show:
:$\map... | Higher Dimensional Hausdorff Measure than Euclidean Space is Zero | https://proofwiki.org/wiki/Higher_Dimensional_Hausdorff_Measure_than_Euclidean_Space_is_Zero | https://proofwiki.org/wiki/Higher_Dimensional_Hausdorff_Measure_than_Euclidean_Space_is_Zero | [
"Hausdorff Measures",
"Fractals"
] | [
"Definition:Dimension of Vector Space",
"Definition:Euclidean Space/Real",
"Definition:Hausdorff Measure"
] | [
"Definition:N-Cube (Euclidean Space)",
"Definition:Countably Subadditive Function",
"Definition:Set",
"Definition:N-Cube (Euclidean Space)",
"Diameter of N-Cube",
"Definition:Diameter of Subset of Metric Space",
"Category:Hausdorff Measures",
"Category:Fractals"
] |
proofwiki-20009 | Orientation of Contour is Well-Defined | Let $C$ be a contour in the complex plane $\C$.
Let $D \subseteq \C$ be a connected domain.
Let $\Img C \subseteq \partial D$, where $\Img C$ denotes the image of $C$, and $\partial D$ denotes the boundary of $D$.
The orientation of $C$ with respect to $D$, if it exists, does not depend on the choice of parameterizatio... | By definition of contour, $C$ is a concatenation of a finite sequence of directed smooth curves $C_1, \ldots, C_n$.
Let each directed smooth curve $C_k$ be parameterized by the smooth path $\gamma_k: \closedint {a_k}{b_k} \to \C$ for all $k \in \set {1, \ldots, n}$.
Let $\sigma_k: \closedint {c_k}{d_k}$ be a reparamete... | Let $C$ be a [[Definition:Contour (Complex Plane)|contour]] in the [[Definition:Complex Plane|complex plane]] $\C$.
Let $D \subseteq \C$ be a [[Definition:Connected Domain (Complex Analysis)|connected domain]].
Let $\Img C \subseteq \partial D$, where $\Img C$ denotes the [[Definition:Image of Contour (Complex Plane)... | By definition of [[Definition:Contour (Complex Plane)|contour]], $C$ is a [[Definition:Concatenation of Contours (Complex Plane)|concatenation]] of a [[Definition:Finite Sequence|finite sequence]] of [[Definition:Directed Smooth Curve (Complex Plane)|directed smooth curves]] $C_1, \ldots, C_n$.
Let each [[Definition:D... | Orientation of Contour is Well-Defined | https://proofwiki.org/wiki/Orientation_of_Contour_is_Well-Defined | https://proofwiki.org/wiki/Orientation_of_Contour_is_Well-Defined | [
"Orientation of Complex Contour"
] | [
"Definition:Contour/Complex Plane",
"Definition:Complex Number/Complex Plane",
"Definition:Connected Domain (Complex Analysis)",
"Definition:Contour/Image/Complex Plane",
"Definition:Boundary (Topology)",
"Definition:Orientation of Contour (Complex Plane)",
"Definition:Contour/Parameterization/Complex P... | [
"Definition:Contour/Complex Plane",
"Definition:Concatenation of Contours/Complex Plane",
"Definition:Finite Sequence",
"Definition:Directed Smooth Curve/Complex Plane",
"Definition:Directed Smooth Curve/Complex Plane",
"Definition:Directed Smooth Curve/Parameterization/Complex Plane",
"Definition:Smoot... |
proofwiki-20010 | Reversed Contour Reverses Orientation | Let $C$ be a contour in the complex plane $\C$.
Let $D \subseteq \C$ be a connected domain.
Let $\Img C \subseteq \partial D$, where $\Img C$ denotes the image of $C$, and $\partial D$ denotes the boundary of $D$.
Let $-C$ be the reversed contour of $C$.
If $C$ is positively oriented {{WRT}} $D$, then $-C$ is negativel... | By definition of contour, $C$ is a concatenation of a finite sequence of directed smooth curves $C_1, \ldots, C_n$.
Let each directed smooth curve $C_k$ be parameterized by the smooth path $\gamma_k: \closedint {a_k} {b_k} \to \C$ for all $k \in \set {1, \ldots, n}$.
Let $\gamma: \closedint a b \to \C$ be the parameter... | Let $C$ be a [[Definition:Contour (Complex Plane)|contour]] in the [[Definition:Complex Plane|complex plane]] $\C$.
Let $D \subseteq \C$ be a [[Definition:Connected Domain (Complex Analysis)|connected domain]].
Let $\Img C \subseteq \partial D$, where $\Img C$ denotes the [[Definition:Image of Contour (Complex Plane)... | By definition of [[Definition:Contour (Complex Plane)|contour]], $C$ is a [[Definition:Concatenation of Contours (Complex Plane)|concatenation]] of a [[Definition:Finite Sequence|finite sequence]] of [[Definition:Directed Smooth Curve (Complex Plane)|directed smooth curves]] $C_1, \ldots, C_n$.
Let each [[Definition:D... | Reversed Contour Reverses Orientation | https://proofwiki.org/wiki/Reversed_Contour_Reverses_Orientation | https://proofwiki.org/wiki/Reversed_Contour_Reverses_Orientation | [
"Reversed Contour Reverses Orientation",
"Orientation of Complex Contour"
] | [
"Definition:Contour/Complex Plane",
"Definition:Complex Number/Complex Plane",
"Definition:Connected Domain (Complex Analysis)",
"Definition:Contour/Image/Complex Plane",
"Definition:Boundary (Topology)",
"Definition:Reversed Contour/Complex Plane",
"Definition:Orientation of Contour (Complex Plane)/Pos... | [
"Definition:Contour/Complex Plane",
"Definition:Concatenation of Contours/Complex Plane",
"Definition:Finite Sequence",
"Definition:Directed Smooth Curve/Complex Plane",
"Definition:Directed Smooth Curve/Complex Plane",
"Definition:Directed Smooth Curve/Parameterization/Complex Plane",
"Definition:Smoot... |
proofwiki-20011 | L-2 Space forms Hilbert Space | Let $\struct{ X, \Sigma, \mu }$ be a measure space.
Let $\map {L^2} \mu$ be the $L^2$ space of $\mu$.
Let $\innerprod \cdot \cdot$ be the inner product on $\map {L^2} \mu$.
Then $\map {L^2} \mu$ endowed with $\innerprod \cdot \cdot$ is a Hilbert space. | {{ProofWanted|We only need the completeness, rest is already up}} | Let $\struct{ X, \Sigma, \mu }$ be a [[Definition:Measure Space|measure space]].
Let $\map {L^2} \mu$ be the [[Definition:Lp Space|$L^2$ space]] of $\mu$.
Let $\innerprod \cdot \cdot$ be the [[Definition:L-2 Inner Product|inner product]] on $\map {L^2} \mu$.
Then $\map {L^2} \mu$ endowed with $\innerprod \cdot \cdo... | {{ProofWanted|We only need the completeness, rest is already up}} | L-2 Space forms Hilbert Space | https://proofwiki.org/wiki/L-2_Space_forms_Hilbert_Space | https://proofwiki.org/wiki/L-2_Space_forms_Hilbert_Space | [
"Examples of Hilbert Spaces",
"Lp Spaces"
] | [
"Definition:Measure Space",
"Definition:Lp Space",
"Definition:L-2 Inner Product",
"Definition:Hilbert Space"
] | [] |
proofwiki-20012 | Real Vector Space with Dot Product is Hilbert Space | Let $\R^d$ be a real vector space with $d$ dimensions.
Let $\innerprod \cdot \cdot$ be the dot product on $\R^d$.
Then $\R^d$ endowed with $\innerprod \cdot \cdot$ is a Hilbert space. | Let us explore the inner product norm of $\innerprod \cdot \cdot$:
:$\ds \norm x = \sqrt{ \sum_{i \mathop = 1}^d x_i^2 }$
and subsequently the metric induced by $\norm \cdot$:
:$\ds \map d {x, y} = \norm {x - y} = \sqrt{ \sum_{i \mathop = 1}^d \paren{ x_i - y_i }^2 }$
which is seen to be the definition of the Euclidean... | Let $\R^d$ be a [[Definition:Real Vector Space|real vector space]] with $d$ dimensions.
Let $\innerprod \cdot \cdot$ be the [[Definition:Dot Product|dot product]] on $\R^d$.
Then $\R^d$ endowed with $\innerprod \cdot \cdot$ is a [[Definition:Hilbert Space|Hilbert space]]. | Let us explore the [[Definition:Inner Product Norm|inner product norm]] of $\innerprod \cdot \cdot$:
:$\ds \norm x = \sqrt{ \sum_{i \mathop = 1}^d x_i^2 }$
and subsequently the [[Definition:Metric Induced by Norm|metric induced]] by $\norm \cdot$:
:$\ds \map d {x, y} = \norm {x - y} = \sqrt{ \sum_{i \mathop = 1}^d \... | Real Vector Space with Dot Product is Hilbert Space | https://proofwiki.org/wiki/Real_Vector_Space_with_Dot_Product_is_Hilbert_Space | https://proofwiki.org/wiki/Real_Vector_Space_with_Dot_Product_is_Hilbert_Space | [
"Examples of Hilbert Spaces"
] | [
"Definition:Real Vector Space",
"Definition:Dot Product",
"Definition:Hilbert Space"
] | [
"Definition:Inner Product Norm",
"Definition:Metric Induced by Norm",
"Definition:Euclidean Metric/Real Vector Space",
"Euclidean Space is Complete Metric Space",
"Definition:Complete Metric Space",
"Definition:Hilbert Space"
] |
proofwiki-20013 | Complex Vector Space with Dot Product is Hilbert Space | Let $\C^d$ be a complex vector space with $d$ dimensions.
Let $\innerprod \cdot \cdot$ be the dot product on $\C^d$.
{{mistake|This dot product is only defined on $\C$, not on $\C^d$}}
Then $\C^d$ endowed with $\innerprod \cdot \cdot$ is a Hilbert space. | Let us explore the inner product norm of $\innerprod \cdot \cdot$:
:$\ds \norm z = \sqrt{ \sum_{i \mathop = 1}^d x_i^2 + y_i^2 }$
and subsequently the metric induced by $\norm \cdot$:
:$\ds \map d {z, z'} = \norm {z - z'} = \sqrt{ \sum_{i \mathop = 1}^d \paren{ x_i - x'_i }^2 + \paren{ y_i - y'_i }^2 }$
{{ProofWanted|E... | Let $\C^d$ be a [[Definition:Complex Vector Space|complex vector space]] with $d$ dimensions.
Let $\innerprod \cdot \cdot$ be the [[Definition:Complex Dot Product|dot product]] on $\C^d$.
{{mistake|This dot product is only defined on $\C$, not on $\C^d$}}
Then $\C^d$ endowed with $\innerprod \cdot \cdot$ is a [[Defin... | Let us explore the [[Definition:Inner Product Norm|inner product norm]] of $\innerprod \cdot \cdot$:
:$\ds \norm z = \sqrt{ \sum_{i \mathop = 1}^d x_i^2 + y_i^2 }$
and subsequently the [[Definition:Metric Induced by Norm|metric induced]] by $\norm \cdot$:
:$\ds \map d {z, z'} = \norm {z - z'} = \sqrt{ \sum_{i \matho... | Complex Vector Space with Dot Product is Hilbert Space | https://proofwiki.org/wiki/Complex_Vector_Space_with_Dot_Product_is_Hilbert_Space | https://proofwiki.org/wiki/Complex_Vector_Space_with_Dot_Product_is_Hilbert_Space | [
"Examples of Hilbert Spaces"
] | [
"Definition:Complex Vector Space",
"Definition:Dot Product/Complex",
"Definition:Hilbert Space"
] | [
"Definition:Inner Product Norm",
"Definition:Metric Induced by Norm"
] |
proofwiki-20014 | Sum of Degenerate Linear Transformation is Degenerate | Let $U, V$ be vector spaces over a field $K$.
Let $S: U \to V$ be a degenerate linear transformation.
Let $T: U \to V$ be a degenerate linear transformation.
Then $S + T$ is a degenerate linear transformation. | Let $\set {s_1, \ldots, s_m}$ be a generator of $\Img S$.
Let $\set {t_1, \ldots, t_n}$ be a generator of $\Img T$.
Then $\set {s_1, \ldots, s_m, t_1, \ldots, t_n}$ is a generator of $\Img {S + T}$.
By Cardinality of Generator of Vector Space is not Less than Dimension:
:$\map \dim {\Img {S + T}} \le m + n$
{{qed}} | Let $U, V$ be [[Definition:Vector Space|vector spaces]] over a [[Definition:Field (Abstract Algebra)|field]] $K$.
Let $S: U \to V$ be a [[Definition:Degenerate Linear Transformation|degenerate]] [[Definition:Linear Transformation on Vector Space|linear transformation]].
Let $T: U \to V$ be a [[Definition:Degenerate L... | Let $\set {s_1, \ldots, s_m}$ be a [[Definition:Generator of Vector Space|generator]] of $\Img S$.
Let $\set {t_1, \ldots, t_n}$ be a [[Definition:Generator of Vector Space|generator]] of $\Img T$.
Then $\set {s_1, \ldots, s_m, t_1, \ldots, t_n}$ is a [[Definition:Generator of Vector Space|generator]] of $\Img {S + T... | Sum of Degenerate Linear Transformation is Degenerate | https://proofwiki.org/wiki/Sum_of_Degenerate_Linear_Transformation_is_Degenerate | https://proofwiki.org/wiki/Sum_of_Degenerate_Linear_Transformation_is_Degenerate | [
"Linear Algebra"
] | [
"Definition:Vector Space",
"Definition:Field (Abstract Algebra)",
"Definition:Degenerate Linear Transformation",
"Definition:Linear Transformation/Vector Space",
"Definition:Degenerate Linear Transformation",
"Definition:Linear Transformation/Vector Space",
"Definition:Degenerate Linear Transformation",... | [
"Definition:Generator of Vector Space",
"Definition:Generator of Vector Space",
"Definition:Generator of Vector Space",
"Cardinality of Generator of Vector Space is not Less than Dimension"
] |
proofwiki-20015 | Product with Degenerate Linear Transformation is Degenerate | Let $U, V, W$ be vector spaces over a field $K$.
Let $G: U \to V$ be a degenerate linear transformation.
Let $N: W \to U$ be a linear transformation.
Then $G \circ N$ is degenerate. | Recall that the dimension of subspace is not greater than its super space.
Thus the claim follows from:
:$\Img {G \circ N} \subseteq \Img G$
{{qed}} | Let $U, V, W$ be [[Definition:Vector Space|vector spaces]] over a [[Definition:Field (Abstract Algebra)|field]] $K$.
Let $G: U \to V$ be a [[Definition:Degenerate Linear Transformation|degenerate]] [[Definition:Linear Transformation on Vector Space|linear transformation]].
Let $N: W \to U$ be a [[Definition:Linear Tr... | Recall that [[Dimension of Proper Subspace is Less Than its Superspace|the dimension of subspace is not greater than its super space]].
Thus the claim follows from:
:$\Img {G \circ N} \subseteq \Img G$
{{qed}} | Product with Degenerate Linear Transformation is Degenerate | https://proofwiki.org/wiki/Product_with_Degenerate_Linear_Transformation_is_Degenerate | https://proofwiki.org/wiki/Product_with_Degenerate_Linear_Transformation_is_Degenerate | [
"Linear Algebra"
] | [
"Definition:Vector Space",
"Definition:Field (Abstract Algebra)",
"Definition:Degenerate Linear Transformation",
"Definition:Linear Transformation/Vector Space",
"Definition:Linear Transformation/Vector Space",
"Definition:Degenerate Linear Transformation"
] | [
"Dimension of Proper Subspace is Less Than its Superspace"
] |
proofwiki-20016 | Right Product with Degenerate Linear Transformation is Degenerate | Let $U, V, W$ be vector spaces over a field $K$.
Let $G: U \to V$ be a degenerate linear transformation.
Let $M: V \to W$ be a linear transformation.
Then $M \circ G$ is degenerate. | Let $\set {s_1, \ldots, s_n}$ be a generator of $\Img G$.
Then $\set {\map M {s_1}, \ldots, \map M {s_n} }$ is a generator of $\Img {M \circ G}$.
By Cardinality of Generator of Vector Space is not Less than Dimension:
:$\map \dim {\Img {M \circ G}} \le n$
{{qed}} | Let $U, V, W$ be [[Definition:Vector Space|vector spaces]] over a [[Definition:Field (Abstract Algebra)|field]] $K$.
Let $G: U \to V$ be a [[Definition:Degenerate Linear Transformation|degenerate]] [[Definition:Linear Transformation on Vector Space|linear transformation]].
Let $M: V \to W$ be a [[Definition:Linear Tr... | Let $\set {s_1, \ldots, s_n}$ be a [[Definition:Generator of Vector Space|generator]] of $\Img G$.
Then $\set {\map M {s_1}, \ldots, \map M {s_n} }$ is a [[Definition:Generator of Vector Space|generator]] of $\Img {M \circ G}$.
By [[Cardinality of Generator of Vector Space is not Less than Dimension]]:
:$\map \dim {\... | Right Product with Degenerate Linear Transformation is Degenerate | https://proofwiki.org/wiki/Right_Product_with_Degenerate_Linear_Transformation_is_Degenerate | https://proofwiki.org/wiki/Right_Product_with_Degenerate_Linear_Transformation_is_Degenerate | [
"Linear Algebra"
] | [
"Definition:Vector Space",
"Definition:Field (Abstract Algebra)",
"Definition:Degenerate Linear Transformation",
"Definition:Linear Transformation/Vector Space",
"Definition:Linear Transformation/Vector Space",
"Definition:Degenerate Linear Transformation"
] | [
"Definition:Generator of Vector Space",
"Definition:Generator of Vector Space",
"Cardinality of Generator of Vector Space is not Less than Dimension"
] |
proofwiki-20017 | Invariance of Pseudoinverse under Addition of Degenerate Transformation | Let $U, V$ be vector spaces over a field $K$.
Let $S: U \to V$ be a linear transformation.
Let $T: V \to U$ be a linear transformation.
Let $S$ and $T$ are pseudoinverse to each other.
Then $S + G_1$ and $T + G_2$ are pseudoinverse to each other, where:
:$G_1: U \to V$ is an arbitrary degenerate linear transformation
:... | Let:
{{begin-eqn}}
{{eqn | n = 1
| l = G_3
| o = :=
| r = T \circ S - I_U
}}
{{eqn | n = 2
| l = G_4
| o = :=
| r = S \circ T - I_V
}}
{{end-eqn}}
By {{Defof|Pseudoinverse of Linear Transformation}}, $G_3, G_4$ are degenerate.
Then:
{{begin-eqn}}
{{eqn | l = \paren {T + G_2} \circ \p... | Let $U, V$ be [[Definition:Vector Space|vector spaces]] over a [[Definition:Field (Abstract Algebra)|field]] $K$.
Let $S: U \to V$ be a [[Definition:Linear Transformation on Vector Space|linear transformation]].
Let $T: V \to U$ be a [[Definition:Linear Transformation on Vector Space|linear transformation]].
Let $S$... | Let:
{{begin-eqn}}
{{eqn | n = 1
| l = G_3
| o = :=
| r = T \circ S - I_U
}}
{{eqn | n = 2
| l = G_4
| o = :=
| r = S \circ T - I_V
}}
{{end-eqn}}
By {{Defof|Pseudoinverse of Linear Transformation}}, $G_3, G_4$ are [[Definition:Degenerate Linear Transformation|degenerate]].
Then:
... | Invariance of Pseudoinverse under Addition of Degenerate Transformation | https://proofwiki.org/wiki/Invariance_of_Pseudoinverse_under_Addition_of_Degenerate_Transformation | https://proofwiki.org/wiki/Invariance_of_Pseudoinverse_under_Addition_of_Degenerate_Transformation | [
"Linear Algebra"
] | [
"Definition:Vector Space",
"Definition:Field (Abstract Algebra)",
"Definition:Linear Transformation/Vector Space",
"Definition:Linear Transformation/Vector Space",
"Definition:Pseudoinverse of Linear Transformation",
"Definition:Pseudoinverse of Linear Transformation",
"Definition:Degenerate Linear Tran... | [
"Definition:Degenerate Linear Transformation",
"Definition:Degenerate Linear Transformation",
"Product with Degenerate Linear Transformation is Degenerate",
"Right Product with Degenerate Linear Transformation is Degenerate",
"Product with Degenerate Linear Transformation is Degenerate",
"Definition:Degen... |
proofwiki-20018 | Number of Permutations of All Elements | Let $S$ be a set of $n$ elements.
The number of permutations of $S$ is $n!$ | We are seeking to calculate the number of $r$-permutations of $S$, that is ${}^n P_r$, where $r = n$.
Hence:
{{begin-eqn}}
{{eqn | l = {}^n P_n
| r = \dfrac {n!} {\paren {n - n}!}
| c = Number of Permutations
}}
{{eqn | r = n!
| c = {{Defof|Factorial}}
}}
{{end-eqn}}
{{qed}} | Let $S$ be a [[Definition:Set|set]] of $n$ [[Definition:Element|elements]].
The number of [[Definition:Permutation (Ordered Selection)|permutations of $S$]] is $n!$ | We are seeking to calculate the number of [[Definition:Permutation (Ordered Selection)|$r$-permutations of $S$]], that is ${}^n P_r$, where $r = n$.
Hence:
{{begin-eqn}}
{{eqn | l = {}^n P_n
| r = \dfrac {n!} {\paren {n - n}!}
| c = [[Number of Permutations]]
}}
{{eqn | r = n!
| c = {{Defof|Factoria... | Number of Permutations of All Elements/Proof 1 | https://proofwiki.org/wiki/Number_of_Permutations_of_All_Elements | https://proofwiki.org/wiki/Number_of_Permutations_of_All_Elements/Proof_1 | [
"Number of Permutations"
] | [
"Definition:Set",
"Definition:Element",
"Definition:Permutation/Ordered Selection"
] | [
"Definition:Permutation/Ordered Selection",
"Number of Permutations"
] |
proofwiki-20019 | Number of Permutations of All Elements | Let $S$ be a set of $n$ elements.
The number of permutations of $S$ is $n!$ | We pick the elements of $S$ in an arbitrary order.
There are $n$ elements of $S$, so there are $n$ options for the first element.
Then there are $n - 1$ elements left in $S$ that we have not picked, so there are $n - 1$ options for the second element.
Then there are $n - 2$ elements left, so there are $n - 2$ options f... | Let $S$ be a [[Definition:Set|set]] of $n$ [[Definition:Element|elements]].
The number of [[Definition:Permutation (Ordered Selection)|permutations of $S$]] is $n!$ | We pick the [[Definition:Element|elements]] of $S$ in an arbitrary order.
There are $n$ [[Definition:Element|elements]] of $S$, so there are $n$ options for the first [[Definition:Element|element]].
Then there are $n - 1$ [[Definition:Element|elements]] left in $S$ that we have not picked, so there are $n - 1$ option... | Number of Permutations of All Elements/Proof 2 | https://proofwiki.org/wiki/Number_of_Permutations_of_All_Elements | https://proofwiki.org/wiki/Number_of_Permutations_of_All_Elements/Proof_2 | [
"Number of Permutations"
] | [
"Definition:Set",
"Definition:Element",
"Definition:Permutation/Ordered Selection"
] | [
"Definition:Element",
"Definition:Element",
"Definition:Element",
"Definition:Element",
"Definition:Element",
"Definition:Element",
"Definition:Element",
"Definition:Element",
"Product Rule for Counting"
] |
proofwiki-20020 | Number of Permutations of All Elements | Let $S$ be a set of $n$ elements.
The number of permutations of $S$ is $n!$ | From the definition, it can be seen that a bijection $f: S \to S$ is an $n$-permutation.
Hence, from Cardinality of Set of Bijections the number of $n$-permutations on a set of $n$ elements is:
:${}^n P_n = \dfrac {n!} {\paren {n - n}!} = n!$
{{qed}} | Let $S$ be a [[Definition:Set|set]] of $n$ [[Definition:Element|elements]].
The number of [[Definition:Permutation (Ordered Selection)|permutations of $S$]] is $n!$ | From the definition, it can be seen that a [[Definition:Bijection|bijection]] $f: S \to S$ is an [[Definition:Permutation (Ordered Selection)|$n$-permutation]].
Hence, from [[Cardinality of Set of Bijections]] the number of [[Definition:Permutation (Ordered Selection)|$n$-permutations]] on a [[Definition:Set|set]] of ... | Number of Permutations of All Elements/Proof 3 | https://proofwiki.org/wiki/Number_of_Permutations_of_All_Elements | https://proofwiki.org/wiki/Number_of_Permutations_of_All_Elements/Proof_3 | [
"Number of Permutations"
] | [
"Definition:Set",
"Definition:Element",
"Definition:Permutation/Ordered Selection"
] | [
"Definition:Bijection",
"Definition:Permutation/Ordered Selection",
"Cardinality of Set of Bijections",
"Definition:Permutation/Ordered Selection",
"Definition:Set",
"Definition:Element"
] |
proofwiki-20021 | Real Semi-Inner Product is Complex Semi-Inner Product | Let $V$ be a vector space over a real subfield $\GF$.
Let $\innerprod \cdot \cdot: V \times V \to \GF$ be a real semi-inner product.
Then $\innerprod \cdot \cdot$ is a complex semi-inner product. | This follows immediately from:
* {{Defof|Real Semi-Inner Product}}
* {{Defof|Complex Semi-Inner Product}}
* Complex Number equals Conjugate iff Wholly Real
{{qed}} | Let $V$ be a [[Definition:Vector Space|vector space]] over a [[Definition:Real Subfield|real subfield]] $\GF$.
Let $\innerprod \cdot \cdot: V \times V \to \GF$ be a [[Definition:Real Semi-Inner Product|real semi-inner product]].
Then $\innerprod \cdot \cdot$ is a [[Definition:Complex Semi-Inner Product|complex semi-... | This follows immediately from:
* {{Defof|Real Semi-Inner Product}}
* {{Defof|Complex Semi-Inner Product}}
* [[Complex Number equals Conjugate iff Wholly Real]]
{{qed}} | Real Semi-Inner Product is Complex Semi-Inner Product | https://proofwiki.org/wiki/Real_Semi-Inner_Product_is_Complex_Semi-Inner_Product | https://proofwiki.org/wiki/Real_Semi-Inner_Product_is_Complex_Semi-Inner_Product | [
"Semi-Inner Product Spaces"
] | [
"Definition:Vector Space",
"Definition:Real Subfield",
"Definition:Semi-Inner Product/Real Field",
"Definition:Semi-Inner Product/Complex Field"
] | [
"Complex Number equals Conjugate iff Wholly Real"
] |
proofwiki-20022 | Real Semi-Inner Product Space is Complex Semi-Inner Product Space | Let $\struct{V, \innerprod \cdot \cdot}$ be a real semi-inner product space.
Then $\struct{V, \innerprod \cdot \cdot}$ is a complex semi-inner product Space. | This follows immediately from:
* {{Defof|Real Semi-Inner Product Space}}
* {{Defof|Complex Semi-Inner Product Space}}
* Real Semi-Inner Product is Complex Semi-Inner Product
{{qed}} | Let $\struct{V, \innerprod \cdot \cdot}$ be a [[Definition:Real Semi-Inner Product Space|real semi-inner product space]].
Then $\struct{V, \innerprod \cdot \cdot}$ is a [[Definition:Complex Semi-Inner Product Space|complex semi-inner product Space]]. | This follows immediately from:
* {{Defof|Real Semi-Inner Product Space}}
* {{Defof|Complex Semi-Inner Product Space}}
* [[Real Semi-Inner Product is Complex Semi-Inner Product]]
{{qed}} | Real Semi-Inner Product Space is Complex Semi-Inner Product Space | https://proofwiki.org/wiki/Real_Semi-Inner_Product_Space_is_Complex_Semi-Inner_Product_Space | https://proofwiki.org/wiki/Real_Semi-Inner_Product_Space_is_Complex_Semi-Inner_Product_Space | [
"Semi-Inner Product Spaces"
] | [
"Definition:Semi-Inner Product Space/Real Field",
"Definition:Semi-Inner Product Space/Complex Field"
] | [
"Real Semi-Inner Product is Complex Semi-Inner Product"
] |
proofwiki-20023 | Real Inner Product is Complex Inner Product | Let $V$ be a vector space over a real subfield $\GF$.
Let $\innerprod \cdot \cdot: V \times V \to \GF$ be a real inner product.
Then $\innerprod \cdot \cdot$ is a complex inner product | This follows immediately from:
* {{Defof|Real Inner Product}}
* {{Defof|Complex Inner Product}}
* Complex Number equals Conjugate iff Wholly Real
{{qed}} | Let $V$ be a [[Definition:Vector Space|vector space]] over a [[Definition:Real Subfield|real subfield]] $\GF$.
Let $\innerprod \cdot \cdot: V \times V \to \GF$ be a [[Definition:Real Inner Product|real inner product]].
Then $\innerprod \cdot \cdot$ is a [[Definition:Complex Inner Product|complex inner product]] | This follows immediately from:
* {{Defof|Real Inner Product}}
* {{Defof|Complex Inner Product}}
* [[Complex Number equals Conjugate iff Wholly Real]]
{{qed}} | Real Inner Product is Complex Inner Product | https://proofwiki.org/wiki/Real_Inner_Product_is_Complex_Inner_Product | https://proofwiki.org/wiki/Real_Inner_Product_is_Complex_Inner_Product | [
"Inner Product Spaces"
] | [
"Definition:Vector Space",
"Definition:Real Subfield",
"Definition:Inner Product/Real Field",
"Definition:Inner Product/Complex Field"
] | [
"Complex Number equals Conjugate iff Wholly Real"
] |
proofwiki-20024 | Real Inner Product Space is Complex Inner Product Space | Let $\struct{V, \innerprod \cdot \cdot}$ be a real inner product space.
Then $\struct{V, \innerprod \cdot \cdot}$ is a complex inner product Space. | This follows immediately from:
* {{Defof|Real Inner Product Space}}
* {{Defof|Complex Inner Product Space}}
* Real Inner Product is Complex Inner Product
{{qed}} | Let $\struct{V, \innerprod \cdot \cdot}$ be a [[Definition:Real Inner Product Space|real inner product space]].
Then $\struct{V, \innerprod \cdot \cdot}$ is a [[Definition:Complex Inner Product Space|complex inner product Space]]. | This follows immediately from:
* {{Defof|Real Inner Product Space}}
* {{Defof|Complex Inner Product Space}}
* [[Real Inner Product is Complex Inner Product]]
{{qed}} | Real Inner Product Space is Complex Inner Product Space | https://proofwiki.org/wiki/Real_Inner_Product_Space_is_Complex_Inner_Product_Space | https://proofwiki.org/wiki/Real_Inner_Product_Space_is_Complex_Inner_Product_Space | [
"Inner Product Spaces"
] | [
"Definition:Inner Product Space/Real Field",
"Definition:Inner Product Space/Complex Field"
] | [
"Real Inner Product is Complex Inner Product"
] |
proofwiki-20025 | Number of Ways of Seating People at Circular Table | Let there be $n$ people to be seated at a round table.
Let $N$ be the number of different ways to seat those $n$ people.
Note that if person $A$ is seated between person $B$ and person $C$, then having $B$ on the left and $C$ on the right is considered different from having $B$ on the right and $C$ on the left.
Then:
:... | From Number of Permutations of All Elements, the number of different ways of arranging $n$ people in line is $n!$.
However, a round table has no beginning and end.
Hence the vital factor is the arrangement relative to an arbitrary given person.
So we fix one person, and arrange the other $n - 1$ people relative to that... | Let there be $n$ people to be seated at a [[Definition:Circle|round]] table.
Let $N$ be the number of different ways to seat those $n$ people.
Note that if person $A$ is seated between person $B$ and person $C$, then having $B$ on the left and $C$ on the right is considered different from having $B$ on the right and ... | From [[Number of Permutations of All Elements]], the number of different ways of arranging $n$ people in line is $n!$.
However, a [[Definition:Circle|round]] table has no beginning and end.
Hence the vital factor is the arrangement relative to an arbitrary given person.
So we fix one person, and arrange the other $n... | Number of Ways of Seating People at Circular Table | https://proofwiki.org/wiki/Number_of_Ways_of_Seating_People_at_Circular_Table | https://proofwiki.org/wiki/Number_of_Ways_of_Seating_People_at_Circular_Table | [
"Number of Ways of Seating People at Circular Table",
"Combinatorics"
] | [
"Definition:Circle"
] | [
"Number of Permutations of All Elements",
"Definition:Circle",
"Number of Permutations of All Elements"
] |
proofwiki-20026 | Number of Ways of Threading Beads on a Loop of Wire | Let there be $n$ beads to be threaded on a circular loop of wire.
Let $N$ be the number of different ways to thread those $n$ beads.
Then:
:$N = \dfrac {\paren {n - 1}!} 2$ | This is the same as Number of Ways of Seating People at Circular Table, except that having threaded the beads in one arrangement, you can get another arrangement by flipping the ring of beads over.
From Number of Ways of Seating People at Circular Table, the number of arrangements of $n$ beads on a ring, without flippi... | Let there be $n$ beads to be threaded on a [[Definition:Circle|circular]] loop of wire.
Let $N$ be the number of different [[Definition:Arrangement|ways to thread]] those $n$ beads.
Then:
:$N = \dfrac {\paren {n - 1}!} 2$ | This is the same as [[Number of Ways of Seating People at Circular Table]], except that having threaded the beads in one [[Definition:Arrangement|arrangement]], you can get another [[Definition:Arrangement|arrangement]] by flipping the ring of beads over.
From [[Number of Ways of Seating People at Circular Table]], th... | Number of Ways of Threading Beads on a Loop of Wire | https://proofwiki.org/wiki/Number_of_Ways_of_Threading_Beads_on_a_Loop_of_Wire | https://proofwiki.org/wiki/Number_of_Ways_of_Threading_Beads_on_a_Loop_of_Wire | [
"Combinatorics"
] | [
"Definition:Circle",
"Definition:Permutation/Ordered Selection"
] | [
"Number of Ways of Seating People at Circular Table",
"Definition:Permutation/Ordered Selection",
"Definition:Permutation/Ordered Selection",
"Number of Ways of Seating People at Circular Table",
"Definition:Permutation/Ordered Selection",
"Definition:Permutation/Ordered Selection",
"Definition:Permutat... |
proofwiki-20027 | Number of Arrangements of n Objects of m Types | Let $S$ be a collection of $n$ objects.
Let these $n$ objects be of $m$ different types, as follows:
Let there be:
:$k_1$ objects of type $1$
:$k_2$ objects of type $2$
:$\cdots$
:$k_m$ objects of type $m$
such that:
:for each $j \in \set {1, 2, \ldots, m}$, all objects of type $j$ are indistinguishable from each other... | Let $N$ be the number of different arrangements of $S$.
First suppose that all $n$ objects are distinct one from another.
Then from Number of Permutations of All Elements:
:$N = n!$
Now suppose that $k_j$ elements of $S$ are indistinguishable from each other.
From Number of Permutations of All Elements, there are $k_j!... | Let $S$ be a [[Definition:Collection|collection]] of $n$ [[Definition:Object|objects]].
Let these $n$ [[Definition:Object|objects]] be of $m$ different types, as follows:
Let there be:
:$k_1$ [[Definition:Object|objects]] of type $1$
:$k_2$ [[Definition:Object|objects]] of type $2$
:$\cdots$
:$k_m$ [[Definition:Obje... | Let $N$ be the number of different [[Definition:Arrangement|arrangements]] of $S$.
First suppose that all $n$ [[Definition:Object|objects]] are [[Definition:Distinct Elements|distinct]] one from another.
Then from [[Number of Permutations of All Elements]]:
:$N = n!$
Now suppose that $k_j$ [[Definition:Element|eleme... | Number of Arrangements of n Objects of m Types | https://proofwiki.org/wiki/Number_of_Arrangements_of_n_Objects_of_m_Types | https://proofwiki.org/wiki/Number_of_Arrangements_of_n_Objects_of_m_Types | [
"Number of Arrangements of n Objects of m Types",
"Combinatorics"
] | [
"Definition:Collection",
"Definition:Object",
"Definition:Object",
"Definition:Object",
"Definition:Object",
"Definition:Object",
"Definition:Object",
"Definition:Distinct/Indistinguishable",
"Definition:Permutation/Ordered Selection",
"Definition:Multinomial Coefficient"
] | [
"Definition:Permutation/Ordered Selection",
"Definition:Object",
"Definition:Distinct/Plural",
"Number of Permutations of All Elements",
"Definition:Element",
"Definition:Distinct/Indistinguishable",
"Number of Permutations of All Elements",
"Definition:Permutation/Ordered Selection",
"Definition:El... |
proofwiki-20028 | Space of Square Summable Mappings is Vector Space | Let $\GF$ be a subfield of $\C$.
Let $I$ be a set.
Let $\map {\ell^2} I$ be the space of square summable mappings over $I$.
Then $\map {\ell^2} I$ is a vector space. | By definition, $\map {\ell^2} I$ is a subset of the vector space $\GF^I$ of all mappings $f: I \to \GF$.
Let us apply the One-Step Vector Subspace Test.
Thus, let $f, g \in \map {\ell^2} I$ and $\lambda \in \GF$.
Then we must show that $f + \lambda g: I \to \GF$ is square summable.
First, note that:
:$\set{ i \in I: \m... | Let $\GF$ be a [[Definition:Subfield|subfield]] of $\C$.
Let $I$ be a [[Definition:Set|set]].
Let $\map {\ell^2} I$ be the [[Definition:Space of Square Summable Mappings|space of square summable mappings]] over $I$.
Then $\map {\ell^2} I$ is a [[Definition:Vector Space|vector space]]. | By definition, $\map {\ell^2} I$ is a [[Definition:Subset|subset]] of the [[Definition:Vector Space of All Mappings|vector space $\GF^I$ of all mappings]] $f: I \to \GF$.
Let us apply the [[One-Step Vector Subspace Test]].
Thus, let $f, g \in \map {\ell^2} I$ and $\lambda \in \GF$.
Then we must show that $f + \lamb... | Space of Square Summable Mappings is Vector Space | https://proofwiki.org/wiki/Space_of_Square_Summable_Mappings_is_Vector_Space | https://proofwiki.org/wiki/Space_of_Square_Summable_Mappings_is_Vector_Space | [
"Examples of Vector Spaces",
"Space of Square Summable Mappings"
] | [
"Definition:Subfield",
"Definition:Set",
"Definition:Space of Square Summable Mappings",
"Definition:Vector Space"
] | [
"Definition:Subset",
"Definition:Vector Space of All Mappings",
"One-Step Vector Subspace Test",
"Definition:Square Summable Mapping",
"Finite Union of Countable Sets is Countable",
"Subset of Countable Set is Countable",
"Definition:Countable Set",
"Definition:Square Summable Mapping",
"One-Step Ve... |
proofwiki-20029 | Number of Selections of 1 or More from Set | Let $S$ be a set of $n$ objects.
The total number $N$ of ways that at least $1$ object can be selected from $S$ is:
:$N = 2^n - 1$ | This is equivalent to counting the number of non-empty subsets of $S$.
From Cardinality of Power Set of Finite Set, the total number of subsets of $S$ is $2^n$.
This includes the empty set.
Excluding the empty set from the count gives the result.
{{qed}} | Let $S$ be a [[Definition:Set|set]] of $n$ [[Definition:Object|objects]].
The total number $N$ of ways that at least $1$ [[Definition:Object|object]] can be selected from $S$ is:
:$N = 2^n - 1$ | This is equivalent to counting the number of [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subsets]] of $S$.
From [[Cardinality of Power Set of Finite Set]], the total number of [[Definition:Subset|subsets]] of $S$ is $2^n$.
This includes the [[Definition:Empty Set|empty set]].
Excluding the [[Definitio... | Number of Selections of 1 or More from Set | https://proofwiki.org/wiki/Number_of_Selections_of_1_or_More_from_Set | https://proofwiki.org/wiki/Number_of_Selections_of_1_or_More_from_Set | [
"Number of Selections of 1 or More from Set",
"Combinations"
] | [
"Definition:Set",
"Definition:Object",
"Definition:Object"
] | [
"Definition:Non-Empty Set",
"Definition:Subset",
"Cardinality of Power Set of Finite Set",
"Definition:Subset",
"Definition:Empty Set",
"Definition:Empty Set"
] |
proofwiki-20030 | Number of Selections from n Objects of 2 Types | Let $S$ be a collection of $n$ objects, consisting of:
:$p$ objects of one type
:$q$ objects of another type.
The total number $N$ of different non-empty selections from $S$ is given by:
:$N = 2^n \paren {p + 1} \paren {q + 1} - 1$ | For each object, we have $2$ choices: whether to select or whether not.
From Cardinality of Power Set of Finite Set, this gives us $2^n$ options.
From the $p$ things of the first type, we may take $0, 1, 2, \ldots, p$ objects.
Hence we have $\paren {p + 1}$ choices.
Similarly, from the $q$ things of the second type, we... | Let $S$ be a [[Definition:Collection|collection]] of $n$ [[Definition:Object|objects]], consisting of:
:$p$ [[Definition:Object|objects]] of one type
:$q$ [[Definition:Object|objects]] of another type.
The total number $N$ of different [[Definition:Non-Empty Set|non-empty]] selections from $S$ is given by:
:$N = 2^n \... | For each [[Definition:Object|object]], we have $2$ choices: whether to select or whether not.
From [[Cardinality of Power Set of Finite Set]], this gives us $2^n$ options.
From the $p$ things of the first type, we may take $0, 1, 2, \ldots, p$ [[Definition:Object|objects]].
Hence we have $\paren {p + 1}$ choices.
S... | Number of Selections from n Objects of 2 Types | https://proofwiki.org/wiki/Number_of_Selections_from_n_Objects_of_2_Types | https://proofwiki.org/wiki/Number_of_Selections_from_n_Objects_of_2_Types | [
"Number of Arrangements of n Objects of m Types"
] | [
"Definition:Collection",
"Definition:Object",
"Definition:Object",
"Definition:Object",
"Definition:Non-Empty Set"
] | [
"Definition:Object",
"Cardinality of Power Set of Finite Set",
"Definition:Object",
"Definition:Non-Empty Set",
"Cardinality of Power Set of Finite Set"
] |
proofwiki-20031 | Inner Product on Space of Square Summable Mappings is Complex Inner Product | Let $\GF$ be a subfield of $\C$.
Let $I$ be a set.
Let $\map {\ell^2} I$ be the space of square summable mappings over $I$.
Let $\innerprod \cdot \cdot: \map {\ell^2} I \times \map {\ell^2} I \to \GF$ be the inner product on $\map {\ell^2} I$.
Then $\innerprod \cdot \cdot$ is a complex inner product. | By Space of Square Summable Mappings is $L^2$ Space, $\map {\ell^2} I$ is equal to $\map {L^2} {I, \powerset I, \mu}$.
Here, $\mu$ is the counting measure on the subsets of $I$.
By Inner Product/Examples/Lebesgue 2-Space, there is a complex inner product on $\map {\ell^2} I = \map {L^2} {I, \powerset I, \mu}$ defined b... | Let $\GF$ be a [[Definition:Subfield|subfield]] of $\C$.
Let $I$ be a [[Definition:Set|set]].
Let $\map {\ell^2} I$ be the [[Definition:Space of Square Summable Mappings|space of square summable mappings]] over $I$.
Let $\innerprod \cdot \cdot: \map {\ell^2} I \times \map {\ell^2} I \to \GF$ be the [[Definition:Inne... | By [[Space of Square Summable Mappings is L-2 Space|Space of Square Summable Mappings is $L^2$ Space]], $\map {\ell^2} I$ is equal to $\map {L^2} {I, \powerset I, \mu}$.
Here, $\mu$ is the [[Definition:Counting Measure|counting measure]] on the [[Definition:Subset|subsets]] of $I$.
By [[Inner Product/Examples/Lebesg... | Inner Product on Space of Square Summable Mappings is Complex Inner Product | https://proofwiki.org/wiki/Inner_Product_on_Space_of_Square_Summable_Mappings_is_Complex_Inner_Product | https://proofwiki.org/wiki/Inner_Product_on_Space_of_Square_Summable_Mappings_is_Complex_Inner_Product | [
"Examples of Inner Products"
] | [
"Definition:Subfield",
"Definition:Set",
"Definition:Space of Square Summable Mappings",
"Definition:Space of Square Summable Mappings/Inner Product",
"Definition:Inner Product/Complex Field"
] | [
"Space of Square Summable Mappings is L-2 Space",
"Definition:Counting Measure",
"Definition:Subset",
"Inner Product/Examples/Lebesgue 2-Space",
"Definition:Inner Product",
"Integral over Counting Measure is Sum over Values",
"Definition:Space of Square Summable Mappings/Inner Product",
"Category:Exam... |
proofwiki-20032 | Space of Square Summable Mappings is Hilbert Space | Let $\GF$ be a subfield of $\C$.
Let $I$ be a set.
Let $\map {\ell^2} I$ be the space of square summable mappings over $I$.
Let $\innerprod \cdot \cdot: \map {\ell^2} I \times \map {\ell^2} I \to \GF$ be the inner product on $\map {\ell^2} I$.
Then $\map {\ell^2} I$ endowed with $\innerprod \cdot \cdot$ is a Hilbert sp... | By Space of Square Summable Mappings is $L^2$ Space, $\map {\ell^2} I$ is equal to $\map {L^2} {I, \powerset I, \mu}$.
The result follows by $L^2$ Space forms Hilbert Space.
{{qed}} | Let $\GF$ be a [[Definition:Subfield|subfield]] of $\C$.
Let $I$ be a [[Definition:Set|set]].
Let $\map {\ell^2} I$ be the [[Definition:Space of Square Summable Mappings|space of square summable mappings]] over $I$.
Let $\innerprod \cdot \cdot: \map {\ell^2} I \times \map {\ell^2} I \to \GF$ be the [[Definition:Inne... | By [[Space of Square Summable Mappings is L-2 Space|Space of Square Summable Mappings is $L^2$ Space]], $\map {\ell^2} I$ is equal to $\map {L^2} {I, \powerset I, \mu}$.
The result follows by [[L-2 Space forms Hilbert Space|$L^2$ Space forms Hilbert Space]].
{{qed}} | Space of Square Summable Mappings is Hilbert Space | https://proofwiki.org/wiki/Space_of_Square_Summable_Mappings_is_Hilbert_Space | https://proofwiki.org/wiki/Space_of_Square_Summable_Mappings_is_Hilbert_Space | [
"Examples of Hilbert Spaces",
"Space of Square Summable Mappings"
] | [
"Definition:Subfield",
"Definition:Set",
"Definition:Space of Square Summable Mappings",
"Definition:Space of Square Summable Mappings/Inner Product",
"Definition:Hilbert Space"
] | [
"Space of Square Summable Mappings is L-2 Space",
"L-2 Space forms Hilbert Space"
] |
proofwiki-20033 | Space of Square Summable Mappings is L-2 Space | Let $\GF$ be a subfield of $\C$.
Let $I$ be a set.
Let $\map {\ell^2} I$ be the space of square summable mappings over $I$.
Then $\map {\ell^2} I$ is equal to the $L^2$ space $\map {L^2} {I, \powerset I, \mu}$, where $\mu$ is the counting measure on $I$. | First, let us unpack the definition of the Lebesgue space $\map {\LL^2} {I, \powerset I, \mu}$:
:$\ds \map {\LL^2} {I, \powerset I, \mu} = \set{ f: I \to \GF: f \in \map \MM { \powerset I }, \int \size f^2 \rd \mu < \infty}$
By Function Measurable with respect to Power Set, all mappings $f: I \to \GF$ are measurable.
B... | Let $\GF$ be a [[Definition:Subfield|subfield]] of $\C$.
Let $I$ be a [[Definition:Set|set]].
Let $\map {\ell^2} I$ be the [[Definition:Space of Square Summable Mappings|space of square summable mappings]] over $I$.
Then $\map {\ell^2} I$ is equal to the [[Definition:Lp Space|$L^2$ space]] $\map {L^2} {I, \powerset... | First, let us unpack the definition of the [[Definition:Lebesgue Space|Lebesgue space]] $\map {\LL^2} {I, \powerset I, \mu}$:
:$\ds \map {\LL^2} {I, \powerset I, \mu} = \set{ f: I \to \GF: f \in \map \MM { \powerset I }, \int \size f^2 \rd \mu < \infty}$
By [[Function Measurable with respect to Power Set]], all [[Def... | Space of Square Summable Mappings is L-2 Space | https://proofwiki.org/wiki/Space_of_Square_Summable_Mappings_is_L-2_Space | https://proofwiki.org/wiki/Space_of_Square_Summable_Mappings_is_L-2_Space | [
"Lp Spaces",
"Space of Square Summable Mappings"
] | [
"Definition:Subfield",
"Definition:Set",
"Definition:Space of Square Summable Mappings",
"Definition:Lp Space",
"Definition:Counting Measure"
] | [
"Definition:Lebesgue Space",
"Function Measurable with respect to Power Set",
"Definition:Mapping",
"Definition:Measurable Function",
"Integral over Counting Measure is Sum of Values",
"Definition:Almost-Everywhere Equality Relation",
"Definition:Counting Measure",
"Definition:Trivial Equivalence Rela... |
proofwiki-20034 | Equivalence of Definitions of Semiring of Sets/Definition 1 implies Definition 2 | Let $\SS$ be a system of sets satisfying the semiring of sets axioms:
{{:Axiom:Semiring of Sets Axioms/Axioms 1}}
Then $\SS$ satisfies the semiring of sets axioms:
{{:Axiom:Semiring of Sets Axioms/Axioms 2}} | Let $\SS$ be a system of sets satisfying the axioms:
{{:Axiom:Semiring of Sets Axioms/Axioms 1}}
It remains to be shown that $\SS$ satisfies the axiom
{{begin-axiom}}
{{axiom | n = 3'
| q = \forall A, B \in \SS
| tr = $\exists n \in \N$ and pairwise disjoint sets $A_1, A_2, A_3, \ldots, A_n \in \SS : \... | Let $\SS$ be a [[Definition:System of Sets|system of sets]] satisfying the [[Axiom:Semiring of Sets Axioms/Axioms 1|semiring of sets axioms]]:
{{:Axiom:Semiring of Sets Axioms/Axioms 1}}
Then $\SS$ satisfies the [[Axiom:Semiring of Sets Axioms/Axioms 2|semiring of sets axioms]]:
{{:Axiom:Semiring of Sets Axioms/Axiom... | Let $\SS$ be a [[Definition:System of Sets|system of sets]] satisfying the [[Axiom:Semiring of Sets Axioms/Axioms 1|axioms]]:
{{:Axiom:Semiring of Sets Axioms/Axioms 1}}
It remains to be shown that $\SS$ satisfies the [[Definition:Axiom|axiom]]
{{begin-axiom}}
{{axiom | n = 3'
| q = \forall A, B \in \SS
... | Equivalence of Definitions of Semiring of Sets/Definition 1 implies Definition 2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Semiring_of_Sets/Definition_1_implies_Definition_2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Semiring_of_Sets/Definition_1_implies_Definition_2 | [
"Equivalence of Definitions of Semiring of Sets"
] | [
"Definition:Set of Sets",
"Axiom:Semiring of Sets Axioms/Axioms 1",
"Axiom:Semiring of Sets Axioms/Axioms 2"
] | [
"Definition:Set of Sets",
"Axiom:Semiring of Sets Axioms/Axioms 1",
"Definition:Axiom",
"Definition:Pairwise Disjoint",
"Axiom:Semiring of Sets Axioms/Axioms 1",
"Intersection is Subset",
"Axiom:Semiring of Sets Axioms/Axioms 1",
"Definition:Finite Sequence",
"Definition:Pairwise Disjoint",
"Set D... |
proofwiki-20035 | Equivalence of Definitions of Semiring of Sets/Definition 2 implies Definition 1 | Let $\SS$ be a system of sets satisfying the semiring of sets axioms:
{{:Axiom:Semiring of Sets Axioms/Axioms 2}}
Then $\SS$ satisfies the semiring of sets axioms:
{{:Axiom:Semiring of Sets Axioms/Axioms 1}} | Let $\SS$ be a system of sets satisfying the axioms:
{{:Axiom:Semiring of Sets Axioms/Axioms 2}}
It remains to be shown that $\SS$ satisfies the axiom
{{begin-axiom}}
{{axiom | n = 3
| q = \forall A, A_1 \in \SS : A_1 \subseteq A
| tr = $\exists n \in \N$ and pairwise disjoint sets $A_2, A_3, \ldots, A... | Let $\SS$ be a [[Definition:System of Sets|system of sets]] satisfying the [[Axiom:Semiring of Sets Axioms/Axioms 2|semiring of sets axioms]]:
{{:Axiom:Semiring of Sets Axioms/Axioms 2}}
Then $\SS$ satisfies the [[Axiom:Semiring of Sets Axioms/Axioms 1|semiring of sets axioms]]:
{{:Axiom:Semiring of Sets Axioms/Axiom... | Let $\SS$ be a [[Definition:System of Sets|system of sets]] satisfying the [[Axiom:Semiring of Sets Axioms/Axioms 2|axioms]]:
{{:Axiom:Semiring of Sets Axioms/Axioms 2}}
It remains to be shown that $\SS$ satisfies the [[Definition:Axiom|axiom]]
{{begin-axiom}}
{{axiom | n = 3
| q = \forall A, A_1 \in \SS : A_... | Equivalence of Definitions of Semiring of Sets/Definition 2 implies Definition 1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Semiring_of_Sets/Definition_2_implies_Definition_1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Semiring_of_Sets/Definition_2_implies_Definition_1 | [
"Equivalence of Definitions of Semiring of Sets"
] | [
"Definition:Set of Sets",
"Axiom:Semiring of Sets Axioms/Axioms 2",
"Axiom:Semiring of Sets Axioms/Axioms 1"
] | [
"Definition:Set of Sets",
"Axiom:Semiring of Sets Axioms/Axioms 2",
"Definition:Axiom",
"Definition:Pairwise Disjoint",
"Axiom:Semiring of Sets Axioms/Axioms 2",
"Definition:Finite Sequence",
"Definition:Pairwise Disjoint",
"Definition:Disjoint Sets",
"Set Difference Union Second Set is Union",
"U... |
proofwiki-20036 | Set of Empty Set is Semiring of Sets | Let $\SS = \set{\O}$.
Then $\SS$ is a semiring of sets. | Taking the axioms of semiring of sets in turn: | Let $\SS = \set{\O}$.
Then $\SS$ is a [[Definition:Semiring of Sets|semiring of sets]]. | Taking the [[Axiom:Semiring of Sets Axioms/Axioms 1|axioms of semiring of sets]] in turn: | Set of Empty Set is Semiring of Sets | https://proofwiki.org/wiki/Set_of_Empty_Set_is_Semiring_of_Sets | https://proofwiki.org/wiki/Set_of_Empty_Set_is_Semiring_of_Sets | [
"Semirings of Sets"
] | [
"Definition:Semiring of Sets"
] | [
"Axiom:Semiring of Sets Axioms/Axioms 1",
"Axiom:Semiring of Sets Axioms/Axioms 1"
] |
proofwiki-20037 | Divisor of Integer/Examples/80 divides 9^2n - 1 | Let $n \in \Z_{\ge 0}$ be a non-negative integer.
Then:
:$80 \divides 9^{2 n} - 1$
where $\divides$ denotes divisibility. | The proof proceeds by induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the equivalent proposition:
:$80 \divides 9^{2 n} - 1$
where $\divides$ denotes divisbility
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn | l = 9^{2 \times 0} - 1
| r = 9^0 - 1
| c =
}}
{{eqn | r = 1 - 1
| c = Zeroth Power o... | Let $n \in \Z_{\ge 0}$ be a [[Definition:Non-Negative Integer|non-negative integer]].
Then:
:$80 \divides 9^{2 n} - 1$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]]. | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the equivalent [[Definition:Proposition|proposition]]:
:$80 \divides 9^{2 n} - 1$
where $\divides$ denotes [[Definition:Divisor of Integer|divisbility]]
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn ... | Divisor of Integer/Examples/80 divides 9^2n - 1/Proof 1 | https://proofwiki.org/wiki/Divisor_of_Integer/Examples/80_divides_9^2n_-_1 | https://proofwiki.org/wiki/Divisor_of_Integer/Examples/80_divides_9^2n_-_1/Proof_1 | [
"Divisor of Integer/Examples/80 divides 9^2n - 1",
"Examples of Divisors of Integers"
] | [
"Definition:Positive/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Divisor (Algebra)/Integer",
"Zeroth Power of Real Number equals One",
"Integer Divisor Results/Integer Divides Zero",
"Integer Divisor Results/Integer Divides Itself",
"Definition:Basis for the Induction",
"Definition:Inducti... |
proofwiki-20038 | Divisor of Integer/Examples/80 divides 9^2n - 1 | Let $n \in \Z_{\ge 0}$ be a non-negative integer.
Then:
:$80 \divides 9^{2 n} - 1$
where $\divides$ denotes divisibility. | From Integer Less One divides Power Less One, we have that:
:$\forall m, n \in \Z: \paren {m - 1} \divides \paren {m^n - 1}$
This result is the special case where $m = 9^2$.
{{qed}} | Let $n \in \Z_{\ge 0}$ be a [[Definition:Non-Negative Integer|non-negative integer]].
Then:
:$80 \divides 9^{2 n} - 1$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]]. | From [[Integer Less One divides Power Less One]], we have that:
:$\forall m, n \in \Z: \paren {m - 1} \divides \paren {m^n - 1}$
This result is the special case where $m = 9^2$.
{{qed}} | Divisor of Integer/Examples/80 divides 9^2n - 1/Proof 2 | https://proofwiki.org/wiki/Divisor_of_Integer/Examples/80_divides_9^2n_-_1 | https://proofwiki.org/wiki/Divisor_of_Integer/Examples/80_divides_9^2n_-_1/Proof_2 | [
"Divisor of Integer/Examples/80 divides 9^2n - 1",
"Examples of Divisors of Integers"
] | [
"Definition:Positive/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Integer Less One divides Power Less One"
] |
proofwiki-20039 | Divisor of Integer/Examples/63 divides 8^2n - 1 | Let $n \in \Z_{\ge 0}$ be a positive integer.
Then:
:$63 \divides 8^{2 n} - 1$
where $\divides$ denotes divisibility. | Proof by induction:
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$63 \divides 8^{2 n} - 1$
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn | l = 8^{2 \times 0} - 1
| r = 8^0 - 1
| c =
}}
{{eqn | r = 1 - 1
| c = Zeroth Power of Real Number equals One
}}
{{eqn | r = 0
| c =
}}
{{e... | Let $n \in \Z_{\ge 0}$ be a [[Definition:Positive Integer|positive integer]].
Then:
:$63 \divides 8^{2 n} - 1$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]]. | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$63 \divides 8^{2 n} - 1$
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn | l = 8^{2 \times 0} - 1
| r = 8^0 - 1
| c =
}}
{{eqn | r = 1 - 1
| c = [[Ze... | Divisor of Integer/Examples/63 divides 8^2n - 1/Proof 1 | https://proofwiki.org/wiki/Divisor_of_Integer/Examples/63_divides_8^2n_-_1 | https://proofwiki.org/wiki/Divisor_of_Integer/Examples/63_divides_8^2n_-_1/Proof_1 | [
"Divisor of Integer/Examples/63 divides 8^2n - 1",
"Examples of Divisors of Integers"
] | [
"Definition:Positive/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Zeroth Power of Real Number equals One",
"Integer Divisor Results/Integer Divides Zero",
"Integer Divisor Results/Integer Divides Itself",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction St... |
proofwiki-20040 | Divisor of Integer/Examples/63 divides 8^2n - 1 | Let $n \in \Z_{\ge 0}$ be a positive integer.
Then:
:$63 \divides 8^{2 n} - 1$
where $\divides$ denotes divisibility. | From Integer Less One divides Power Less One, we have that:
:$\forall m, n \in \Z: \paren {m - 1} \divides \paren {m^n - 1}$
This result is the special case where $m = 8^2$.
{{qed}} | Let $n \in \Z_{\ge 0}$ be a [[Definition:Positive Integer|positive integer]].
Then:
:$63 \divides 8^{2 n} - 1$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]]. | From [[Integer Less One divides Power Less One]], we have that:
:$\forall m, n \in \Z: \paren {m - 1} \divides \paren {m^n - 1}$
This result is the special case where $m = 8^2$.
{{qed}} | Divisor of Integer/Examples/63 divides 8^2n - 1/Proof 2 | https://proofwiki.org/wiki/Divisor_of_Integer/Examples/63_divides_8^2n_-_1 | https://proofwiki.org/wiki/Divisor_of_Integer/Examples/63_divides_8^2n_-_1/Proof_2 | [
"Divisor of Integer/Examples/63 divides 8^2n - 1",
"Examples of Divisors of Integers"
] | [
"Definition:Positive/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Integer Less One divides Power Less One"
] |
proofwiki-20041 | Divisor of Integer/Examples/6 divides 7^n - 1 | Let $n \in \Z_{\ge 0}$ be a positive integer.
Then:
:$6 \divides 7^n - 1$
where $\divides$ denotes divisibility. | From Integer Less One divides Power Less One, we have that:
:$\forall m, n \in \Z: \paren {m - 1} \divides \paren {m^n - 1}$
This result is the special case where $m = 7$.
{{qed}} | Let $n \in \Z_{\ge 0}$ be a [[Definition:Positive Integer|positive integer]].
Then:
:$6 \divides 7^n - 1$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]]. | From [[Integer Less One divides Power Less One]], we have that:
:$\forall m, n \in \Z: \paren {m - 1} \divides \paren {m^n - 1}$
This result is the special case where $m = 7$.
{{qed}} | Divisor of Integer/Examples/6 divides 7^n - 1/Proof 2 | https://proofwiki.org/wiki/Divisor_of_Integer/Examples/6_divides_7^n_-_1 | https://proofwiki.org/wiki/Divisor_of_Integer/Examples/6_divides_7^n_-_1/Proof_2 | [
"Divisor of Integer/Examples/6 divides 7^n - 1",
"Examples of Divisors of Integers"
] | [
"Definition:Positive/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Integer Less One divides Power Less One"
] |
proofwiki-20042 | Divisor of Integer/Examples/6 divides n (n+1) (n+2) | Let $n$ be an integer.
Then:
:$6 \divides n \paren {n + 1} \paren {n + 2}$ | From $3$ divides $n \paren {n + 1} \paren {n + 2}$:
:$3 \divides n \paren {n + 1} \paren {n + 2}$
From $2$ divides $n \paren {n + 1}$:
:$2 \divides n \paren {n + 1}$
and so:
:$2 \divides n \paren {n + 1} \paren {n + 2}$
Hence:
:$2 \times 3 = 6 \divides \paren {n + 1} \paren {n + 2}$
{{qed}} | Let $n$ be an [[Definition:Integer|integer]].
Then:
:$6 \divides n \paren {n + 1} \paren {n + 2}$ | From [[Divisor of Integer/Examples/3 divides n(n+1)(n+2)|$3$ divides $n \paren {n + 1} \paren {n + 2}$]]:
:$3 \divides n \paren {n + 1} \paren {n + 2}$
From [[Divisor of Integer/Examples/2 divides n(n+1)|$2$ divides $n \paren {n + 1}$]]:
:$2 \divides n \paren {n + 1}$
and so:
:$2 \divides n \paren {n + 1} \paren {n + ... | Divisor of Integer/Examples/6 divides n (n+1) (n+2)/Proof 1 | https://proofwiki.org/wiki/Divisor_of_Integer/Examples/6_divides_n_(n+1)_(n+2) | https://proofwiki.org/wiki/Divisor_of_Integer/Examples/6_divides_n_(n+1)_(n+2)/Proof_1 | [
"Divisor of Integer/Examples/6 divides n (n+1) (n+2)",
"Examples of Divisors of Integers"
] | [
"Definition:Integer"
] | [
"Divisor of Integer/Examples/3 divides n(n+1)(n+2)",
"Divisor of Integer/Examples/2 divides n(n+1)"
] |
proofwiki-20043 | Divisor of Integer/Examples/6 divides n (n+1) (n+2) | Let $n$ be an integer.
Then:
:$6 \divides n \paren {n + 1} \paren {n + 2}$ | Proof by induction:
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$6 \divides n \paren {n + 1} \paren {n + 2}$
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn | l = 0 \times 1 \times 2
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = 6
| o = \divides
| r = 0 \paren {0 + 1} \paren... | Let $n$ be an [[Definition:Integer|integer]].
Then:
:$6 \divides n \paren {n + 1} \paren {n + 2}$ | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$6 \divides n \paren {n + 1} \paren {n + 2}$
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn | l = 0 \times 1 \times 2
| r = 0
| c =
}}
{{eqn | ll= \leadsto... | Divisor of Integer/Examples/6 divides n (n+1) (n+2)/Proof 2 | https://proofwiki.org/wiki/Divisor_of_Integer/Examples/6_divides_n_(n+1)_(n+2) | https://proofwiki.org/wiki/Divisor_of_Integer/Examples/6_divides_n_(n+1)_(n+2)/Proof_2 | [
"Divisor of Integer/Examples/6 divides n (n+1) (n+2)",
"Examples of Divisors of Integers"
] | [
"Definition:Integer"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Integer Divisor Results/Integer Divides Zero",
"Integer Divisor Results/Integer Divides Itself",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Divisor of Integer/Examples/6 divide... |
proofwiki-20044 | Unique Point of Minimal Distance to Closed Linear Subspace of Hilbert Space iff Orthogonal | Let $H$ be a Hilbert space, and let $h \in H$.
Let $K \subseteq H$ be a closed linear subspace of $H$.
Then the unique point $k_0 \in K$ from Unique Point of Minimal Distance to Closed Convex Subset of Hilbert Space such that:
:$\norm {h - k_0} = \map d {h, K}$
where $d$ denotes distance to a set, is characterised by:
... | === Necessary Condition ===
Suppose that:
:$\ds \norm {h - k_0} = \map d {h, K} = \inf_{k \in K} \norm {h - k}$
Let $k \in K$.
Since $K$ is a linear subspace, we have $k_0 + k \in K$.
Hence, we have:
:$\norm {h - k_0}^2 \le \norm {h - \paren {k_0 + k} }^2 = \norm {\paren {h - k_0} - k}^2$
From Square of Inner Product... | Let $H$ be a [[Definition:Hilbert Space|Hilbert space]], and let $h \in H$.
Let $K \subseteq H$ be a [[Definition:Closed Set (Topology)|closed]] [[Definition:Linear Subspace|linear subspace]] of $H$.
Then the unique point $k_0 \in K$ from [[Unique Point of Minimal Distance to Closed Convex Subset of Hilbert Space]] ... | === Necessary Condition ===
Suppose that:
:$\ds \norm {h - k_0} = \map d {h, K} = \inf_{k \in K} \norm {h - k}$
Let $k \in K$.
Since $K$ is a [[Definition:Linear Subspace|linear subspace]], we have $k_0 + k \in K$.
Hence, we have:
:$\norm {h - k_0}^2 \le \norm {h - \paren {k_0 + k} }^2 = \norm {\paren {h - k_0} -... | Unique Point of Minimal Distance to Closed Linear Subspace of Hilbert Space iff Orthogonal | https://proofwiki.org/wiki/Unique_Point_of_Minimal_Distance_to_Closed_Linear_Subspace_of_Hilbert_Space_iff_Orthogonal | https://proofwiki.org/wiki/Unique_Point_of_Minimal_Distance_to_Closed_Linear_Subspace_of_Hilbert_Space_iff_Orthogonal | [
"Hilbert Spaces",
"Convex Sets (Vector Spaces)"
] | [
"Definition:Hilbert Space",
"Definition:Closed Set/Topology",
"Definition:Linear Subspace",
"Unique Point of Minimal Distance to Closed Convex Subset of Hilbert Space",
"Definition:Distance/Sets/Metric Spaces",
"Definition:Orthogonal (Linear Algebra)"
] | [
"Definition:Linear Subspace",
"Square of Inner Product Norm of Sum",
"Inner Product is Sesquilinear",
"Definition:Linear Subspace"
] |
proofwiki-20045 | Hyperbolic Sine of Zero is Zero | :$\map \sinh 0 = 0$
where $\sinh$ denotes the hyperbolic sine. | {{begin-eqn}}
{{eqn | l = \map \sinh 0
| r = \dfrac {e^0 - e^{-0} } 2
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \dfrac {1 - 1} 2
| c = {{Defof|Integer Power}}
}}
{{eqn | r = 0
| c =
}}
{{end-eqn}}
{{qed}} | :$\map \sinh 0 = 0$
where $\sinh$ denotes the [[Definition:Hyperbolic Sine|hyperbolic sine]]. | {{begin-eqn}}
{{eqn | l = \map \sinh 0
| r = \dfrac {e^0 - e^{-0} } 2
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \dfrac {1 - 1} 2
| c = {{Defof|Integer Power}}
}}
{{eqn | r = 0
| c =
}}
{{end-eqn}}
{{qed}} | Hyperbolic Sine of Zero is Zero | https://proofwiki.org/wiki/Hyperbolic_Sine_of_Zero_is_Zero | https://proofwiki.org/wiki/Hyperbolic_Sine_of_Zero_is_Zero | [
"Hyperbolic Sine Function"
] | [
"Definition:Hyperbolic Sine"
] | [] |
proofwiki-20046 | Hyperbolic Cosine of Zero is One | :$\map \cosh 0 = 1$
where $\cosh$ denotes the hyperbolic cosine. | {{begin-eqn}}
{{eqn | l = \map \cosh 0
| r = \dfrac {e^0 + e^{-0} } 2
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \dfrac {1 + 1} 2
| c = {{Defof|Integer Power}}
}}
{{eqn | r = 1
| c =
}}
{{end-eqn}}
{{qed}} | :$\map \cosh 0 = 1$
where $\cosh$ denotes the [[Definition:Hyperbolic Cosine|hyperbolic cosine]]. | {{begin-eqn}}
{{eqn | l = \map \cosh 0
| r = \dfrac {e^0 + e^{-0} } 2
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \dfrac {1 + 1} 2
| c = {{Defof|Integer Power}}
}}
{{eqn | r = 1
| c =
}}
{{end-eqn}}
{{qed}} | Hyperbolic Cosine of Zero is One | https://proofwiki.org/wiki/Hyperbolic_Cosine_of_Zero_is_One | https://proofwiki.org/wiki/Hyperbolic_Cosine_of_Zero_is_One | [
"Hyperbolic Cosine Function",
"Hyperbolic Cosine Function"
] | [
"Definition:Hyperbolic Cosine"
] | [] |
proofwiki-20047 | Binomial Theorem/Approximations/1st Order | When $x$ is sufficiently small that $x^2$ can be neglected then:
:$\paren {1 + x}^\alpha \approx 1 + \alpha x$
and the error is of the order of $\dfrac {\alpha \paren {\alpha - 1} } 2 x^2$ | {{ProofWanted|Not sure how this is formulated yet, I need a more precise definition and source work}} | When $x$ is sufficiently small that $x^2$ can be neglected then:
:$\paren {1 + x}^\alpha \approx 1 + \alpha x$
and the [[Definition:Error|error]] is of the order of $\dfrac {\alpha \paren {\alpha - 1} } 2 x^2$ | {{ProofWanted|Not sure how this is formulated yet, I need a more precise definition and source work}} | Binomial Theorem/Approximations/1st Order | https://proofwiki.org/wiki/Binomial_Theorem/Approximations/1st_Order | https://proofwiki.org/wiki/Binomial_Theorem/Approximations/1st_Order | [
"Binomial Theorem"
] | [
"Definition:Error"
] | [] |
proofwiki-20048 | Binomial Theorem/Approximations/2nd Order | When $x$ is sufficiently small that $x^3$ can be neglected, then:
:$\paren {1 + x}^\alpha \approx 1 + \alpha x + \dfrac {\alpha \paren {\alpha - 1} } 2 x^2$
and the error is of the order of:
:$\dfrac {\alpha \paren {\alpha - 1} \paren {\alpha - 2} } 6 x^3$ | {{ProofWanted|Not sure how this is formulated yet, I need a more precise definition and source work}} | When $x$ is sufficiently small that $x^3$ can be neglected, then:
:$\paren {1 + x}^\alpha \approx 1 + \alpha x + \dfrac {\alpha \paren {\alpha - 1} } 2 x^2$
and the [[Definition:Error|error]] is of the order of:
:$\dfrac {\alpha \paren {\alpha - 1} \paren {\alpha - 2} } 6 x^3$ | {{ProofWanted|Not sure how this is formulated yet, I need a more precise definition and source work}} | Binomial Theorem/Approximations/2nd Order | https://proofwiki.org/wiki/Binomial_Theorem/Approximations/2nd_Order | https://proofwiki.org/wiki/Binomial_Theorem/Approximations/2nd_Order | [
"Binomial Theorem"
] | [
"Definition:Error"
] | [] |
proofwiki-20049 | Normed Vector Space with Schauder Basis is Separable | Let $\struct {X, \norm {\, \cdot \,}}$ be a normed vector space over $\R$.
Suppose $X$ admits a Schauder basis.
Then $X$ is separable. | Let $\set {\mathbf E_i : i \in \N}$ be a Schauder basis.
Then:
:$\ds \forall x \in X : \exists \sequence {x_i'}_{i \mathop \in \N} \in \R : x = \sum_{i \mathop = 0}^\infty x_i' \mathbf E_i$
Let $\ds \mathbf e_i = \frac {\mathbf E_i}{\norm {\mathbf E_i}}$ and $\ds x_i=x_i' \norm{\mathbf E_i}$.
Then $\ds x = \sum_{i \mat... | Let $\struct {X, \norm {\, \cdot \,}}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\R$.
Suppose $X$ admits a [[Definition:Schauder Basis|Schauder basis]].
Then $X$ is [[Definition:Separable Normed Vector Space|separable]]. | Let $\set {\mathbf E_i : i \in \N}$ be a [[Definition:Schauder Basis|Schauder basis]].
Then:
:$\ds \forall x \in X : \exists \sequence {x_i'}_{i \mathop \in \N} \in \R : x = \sum_{i \mathop = 0}^\infty x_i' \mathbf E_i$
Let $\ds \mathbf e_i = \frac {\mathbf E_i}{\norm {\mathbf E_i}}$ and $\ds x_i=x_i' \norm{\mathbf ... | Normed Vector Space with Schauder Basis is Separable | https://proofwiki.org/wiki/Normed_Vector_Space_with_Schauder_Basis_is_Separable | https://proofwiki.org/wiki/Normed_Vector_Space_with_Schauder_Basis_is_Separable | [
"Schauder Bases",
"Normed Vector Spaces",
"Separable Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Schauder Basis",
"Definition:Separable Space/Normed Vector Space"
] | [
"Definition:Schauder Basis",
"Definition:Countable Set",
"Definition:Generated Submodule/Linear Span",
"Rational Numbers form Subset of Real Numbers",
"Rational Numbers are Everywhere Dense in Set of Real Numbers/Normed Vector Space",
"Definition:Schauder Basis",
"Definition:Convergent Series/Normed Vec... |
proofwiki-20050 | Greatest Term of Binomial Expansion/Examples/Arbitrary Example 1 | Consider the expression:
:$E = \paren {1 + 2 x}^{10 \frac 1 2}$
Let $x = \dfrac 3 7$.
Then the greatest term in the power series expansion of $E$ by means of the General Binomial Theorem is:
{{begin-eqn}}
{{eqn | o =
| r = \dfrac {21 \times 19 \times 17 \times 15 \times 13} {5!} \paren {\dfrac 3 7}^5
}}
{{eqn | ... | Let us perform the expansion:
:$\paren {1 + 2 x}^{\frac {21} 2} = 1 + \dfrac {21} 2 \paren {2 x} + \dfrac {\paren {\frac {21} 2} \paren {\frac {19} 2} } {2!} \paren {2 x}^2 + \dfrac {\paren {\frac {21} 2} \paren {\frac {19} 2} \paren {\frac {17} 2} } {3!} \paren {2 x}^3 + \cdots$
Consider the $\paren {r + 1}$th term:
:... | Consider the [[Definition:Expression|expression]]:
:$E = \paren {1 + 2 x}^{10 \frac 1 2}$
Let $x = \dfrac 3 7$.
Then the greatest term in the [[Definition:Power Series Expansion|power series expansion]] of $E$ by means of the [[General Binomial Theorem]] is:
{{begin-eqn}}
{{eqn | o =
| r = \dfrac {21 \times... | Let us perform the expansion:
:$\paren {1 + 2 x}^{\frac {21} 2} = 1 + \dfrac {21} 2 \paren {2 x} + \dfrac {\paren {\frac {21} 2} \paren {\frac {19} 2} } {2!} \paren {2 x}^2 + \dfrac {\paren {\frac {21} 2} \paren {\frac {19} 2} \paren {\frac {17} 2} } {3!} \paren {2 x}^3 + \cdots$
Consider the $\paren {r + 1}$th term:
... | Greatest Term of Binomial Expansion/Examples/Arbitrary Example 1 | https://proofwiki.org/wiki/Greatest_Term_of_Binomial_Expansion/Examples/Arbitrary_Example_1 | https://proofwiki.org/wiki/Greatest_Term_of_Binomial_Expansion/Examples/Arbitrary_Example_1 | [
"Binomial Theorem"
] | [
"Definition:Expression",
"Definition:Power Series Expansion",
"Binomial Theorem/General Binomial Theorem"
] | [] |
proofwiki-20051 | Weierstrass Approximation Theorem/Complex Case | Let $\Bbb I = \closedint a b$ be a closed real interval.
Let $f: \Bbb I \to \C$ be a continuous complex function.
Let $\epsilon \in \R_{>0}$.
Then there exists a complex polynomial function $p : \Bbb I \to \C$ such that:
:$\norm {p - f}_\infty < \epsilon$
where $\norm f_\infty$ denotes the supremum norm of $f$ on $\Bb... | Let $\Re f : \Bbb \to \R$ be the real function defined by:
:$\map {\Re f} x = \map \Re {\map f x}$
where $\map \Re z$ denotes the real part of a complex number $z \in \C$.
Let $\Im f : \Bbb \to \R$ be the real function defined by:
:$\map {\Im f} x = \map \Im {\map f x}$
where $\map \Im z$ denotes the imaginary part of ... | Let $\Bbb I = \closedint a b$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $f: \Bbb I \to \C$ be a [[Definition:Continuous Complex Function|continuous complex function]].
Let $\epsilon \in \R_{>0}$.
Then there exists a [[Definition:Complex Polynomial Function|complex polynomial function]] $p ... | Let $\Re f : \Bbb \to \R$ be the [[Definition:Real Function|real function]] defined by:
:$\map {\Re f} x = \map \Re {\map f x}$
where $\map \Re z$ denotes the [[Definition:Real Part|real part]] of a [[Definition:Complex Number|complex number]] $z \in \C$.
Let $\Im f : \Bbb \to \R$ be the [[Definition:Real Function|re... | Weierstrass Approximation Theorem/Complex Case | https://proofwiki.org/wiki/Weierstrass_Approximation_Theorem/Complex_Case | https://proofwiki.org/wiki/Weierstrass_Approximation_Theorem/Complex_Case | [
"Weierstrass Approximation Theorem"
] | [
"Definition:Real Interval/Closed",
"Definition:Continuous Complex Function",
"Definition:Polynomial Function/Complex",
"Definition:Supremum Norm"
] | [
"Definition:Real Function",
"Definition:Complex Number/Real Part",
"Definition:Complex Number",
"Definition:Real Function",
"Definition:Complex Number/Imaginary Part",
"Real and Imaginary Part Projections are Continuous",
"Composite of Continuous Mappings is Continuous",
"Definition:Continuous Real Fu... |
proofwiki-20052 | Complex Numbers with Complex Modulus form Normed Vector Space | Let $\C$ be the set of complex numbers.
Let $\size {\, \cdot \,}$ be the complex modulus.
Then $\struct {\C, \size {\, \cdot \,}}$ is a normed vector space. | We have that:
:Complex Numbers form Vector Space over Themselves
:Complex Modulus is Norm
By definition, $\struct {\C, \size {\, \cdot \,}}$ is a normed vector space.
{{qed}} | Let $\C$ be the [[Definition:Set|set]] of [[Definition:Complex Numbers|complex numbers]].
Let $\size {\, \cdot \,}$ be the [[Definition:Complex Modulus|complex modulus]].
Then $\struct {\C, \size {\, \cdot \,}}$ is a [[Definition:Normed Vector Space|normed vector space]]. | We have that:
:[[Complex Numbers form Vector Space over Themselves]]
:[[Complex Modulus is Norm]]
By definition, $\struct {\C, \size {\, \cdot \,}}$ is a [[Definition:Normed Vector Space|normed vector space]].
{{qed}} | Complex Numbers with Complex Modulus form Normed Vector Space | https://proofwiki.org/wiki/Complex_Numbers_with_Complex_Modulus_form_Normed_Vector_Space | https://proofwiki.org/wiki/Complex_Numbers_with_Complex_Modulus_form_Normed_Vector_Space | [
"Examples of Normed Vector Spaces",
"Complex Numbers"
] | [
"Definition:Set",
"Definition:Complex Number",
"Definition:Complex Modulus",
"Definition:Normed Vector Space"
] | [
"Complex Numbers form Vector Space over Themselves",
"Complex Modulus is Norm",
"Definition:Normed Vector Space"
] |
proofwiki-20053 | Gaussian Rationals are Everywhere Dense in Complex Numbers | Let $\struct {\C, \cmod {\, \cdot \,}}$ be the normed vector space of complex numbers.
Let $\Q \sqbrk i = \set {a + i b: a, b \in \Q}$ be the set of Gaussian rational numbers.
Then $\Q \sqbrk i$ is everywhere dense in $\struct {\C, \cmod {\, \cdot \,}}$. | Let $z = x + i y \in \C$ be a complex number.
Let $q = a + i b \in \Q \sqbrk i$ be a Gaussian rational number.
Then:
{{begin-eqn}}
{{eqn | l = \cmod {z - q}
| r = \cmod {\paren {x + i y} - \paren {a + i b} }
}}
{{eqn | r = \cmod {x - a + i \paren {y - b} }
}}
{{eqn | o = \le
| r = \cmod {x - a} + \cmod {i \... | Let $\struct {\C, \cmod {\, \cdot \,}}$ be the [[Complex Numbers with Complex Modulus form Normed Vector Space|normed vector space of complex numbers]].
Let $\Q \sqbrk i = \set {a + i b: a, b \in \Q}$ be the [[Definition:Set|set]] of [[Definition:Gaussian Rational|Gaussian rational numbers]].
Then $\Q \sqbrk i$ is [... | Let $z = x + i y \in \C$ be a [[Definition:Complex Number|complex number]].
Let $q = a + i b \in \Q \sqbrk i$ be a [[Definition:Gaussian Rational|Gaussian rational number]].
Then:
{{begin-eqn}}
{{eqn | l = \cmod {z - q}
| r = \cmod {\paren {x + i y} - \paren {a + i b} }
}}
{{eqn | r = \cmod {x - a + i \paren {... | Gaussian Rationals are Everywhere Dense in Complex Numbers | https://proofwiki.org/wiki/Gaussian_Rationals_are_Everywhere_Dense_in_Complex_Numbers | https://proofwiki.org/wiki/Gaussian_Rationals_are_Everywhere_Dense_in_Complex_Numbers | [
"Denseness",
"Complex Numbers",
"Gaussian Rationals"
] | [
"Complex Numbers with Complex Modulus form Normed Vector Space",
"Definition:Set",
"Definition:Gaussian Rational",
"Definition:Everywhere Dense/Normed Vector Space"
] | [
"Definition:Complex Number",
"Definition:Gaussian Rational",
"Rational Numbers are Everywhere Dense in Set of Real Numbers/Normed Vector Space",
"Definition:Everywhere Dense/Normed Vector Space",
"Category:Denseness",
"Category:Complex Numbers",
"Category:Gaussian Rationals"
] |
proofwiki-20054 | Square Root of Function under Derivative | :$\ds \int \dfrac {\map {f'} x} {\sqrt {\map f x} } \rd x = 2 \sqrt {\map f x} + C$ | Let $u = \map f x$.
Then:
:$\dfrac {\d u} {\d x} = \map {f'} x$
Hence:
{{begin-eqn}}
{{eqn | l = \int \dfrac {\d u} {\d x} \dfrac 1 {\sqrt {\map f x} } \rd x
| r = \int \dfrac {\d u} {\sqrt u}
| c = Integration by Substitution
}}
{{eqn | r = 2 \sqrt u + C
| c = Primitive of Power
}}
{{eqn | r = 2 \sqr... | :$\ds \int \dfrac {\map {f'} x} {\sqrt {\map f x} } \rd x = 2 \sqrt {\map f x} + C$ | Let $u = \map f x$.
Then:
:$\dfrac {\d u} {\d x} = \map {f'} x$
Hence:
{{begin-eqn}}
{{eqn | l = \int \dfrac {\d u} {\d x} \dfrac 1 {\sqrt {\map f x} } \rd x
| r = \int \dfrac {\d u} {\sqrt u}
| c = [[Integration by Substitution]]
}}
{{eqn | r = 2 \sqrt u + C
| c = [[Primitive of Power]]
}}
{{eqn |... | Square Root of Function under Derivative | https://proofwiki.org/wiki/Square_Root_of_Function_under_Derivative | https://proofwiki.org/wiki/Square_Root_of_Function_under_Derivative | [
"Primitives"
] | [] | [
"Integration by Substitution",
"Primitive of Power",
"Primitive of Power"
] |
proofwiki-20055 | Normed Vector Space over Complex Numbers with Schauder Basis is Separable | Let $\struct {X, \norm {\, \cdot \,}}$ be a normed vector space over $\C$.
Suppose $X$ admits a Schauder basis.
Then $X$ is separable. | Let $\set {\mathbf E_i : i \in \N}$ be a Schauder basis.
Then:
:$\ds \forall x \in X : \exists \sequence {x_i'}_{i \mathop \in \N} \in \C : x = \sum_{i \mathop = 0}^\infty x_i' \mathbf E_i$
Let $\ds \mathbf e_i = \frac {\mathbf E_i}{\norm {\mathbf E_i}}$ and $\ds x_i=x_i' \norm{\mathbf E_i}$.
Then $\ds x = \sum_{i \mat... | Let $\struct {X, \norm {\, \cdot \,}}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\C$.
Suppose $X$ admits a [[Definition:Schauder Basis|Schauder basis]].
Then $X$ is [[Definition:Separable Normed Vector Space|separable]]. | Let $\set {\mathbf E_i : i \in \N}$ be a [[Definition:Schauder Basis|Schauder basis]].
Then:
:$\ds \forall x \in X : \exists \sequence {x_i'}_{i \mathop \in \N} \in \C : x = \sum_{i \mathop = 0}^\infty x_i' \mathbf E_i$
Let $\ds \mathbf e_i = \frac {\mathbf E_i}{\norm {\mathbf E_i}}$ and $\ds x_i=x_i' \norm{\mathbf ... | Normed Vector Space over Complex Numbers with Schauder Basis is Separable | https://proofwiki.org/wiki/Normed_Vector_Space_over_Complex_Numbers_with_Schauder_Basis_is_Separable | https://proofwiki.org/wiki/Normed_Vector_Space_over_Complex_Numbers_with_Schauder_Basis_is_Separable | [
"Schauder Bases",
"Normed Vector Spaces",
"Separable Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Schauder Basis",
"Definition:Separable Space/Normed Vector Space"
] | [
"Definition:Schauder Basis",
"Definition:Countable Set",
"Definition:Generated Submodule/Linear Span",
"Gaussian Rational Numbers form Subset of Complex Numbers",
"Gaussian Rationals are Everywhere Dense in Complex Numbers",
"Definition:Schauder Basis",
"Definition:Limit of Sequence in Normed Vector Spa... |
proofwiki-20056 | Orthogonal Projection is Mapping | Let $H$ be a Hilbert space.
Let $K$ be a closed linear subspace of $H$.
Let $P_K: H \to H$ be the orthogonal projection on $K$.
Then $P_K$ is a mapping. | For $P_K$ to be a mapping we need to show that:
:$\forall h \in H: \map{P_K} h$ exists and is unique
By definition of $\map{P_K} h$, this amounts to:
:There is a unique $k \in K$ such that $\norm{ h - k } = \map d {h, K}$
This is precisely the statement of Unique Point of Minimal Distance to Closed Convex Subset of Hil... | Let $H$ be a [[Definition:Hilbert Space|Hilbert space]].
Let $K$ be a [[Definition:Closed Linear Subspace|closed linear subspace]] of $H$.
Let $P_K: H \to H$ be the [[Definition:Orthogonal Projection|orthogonal projection]] on $K$.
Then $P_K$ is a [[Definition:Mapping|mapping]]. | For $P_K$ to be a [[Definition:Mapping|mapping]] we need to show that:
:$\forall h \in H: \map{P_K} h$ exists and is unique
By [[Definition:Orthogonal Projection|definition of $\map{P_K} h$]], this amounts to:
:There is a unique $k \in K$ such that $\norm{ h - k } = \map d {h, K}$
This is precisely the statement of... | Orthogonal Projection is Mapping | https://proofwiki.org/wiki/Orthogonal_Projection_is_Mapping | https://proofwiki.org/wiki/Orthogonal_Projection_is_Mapping | [
"Orthogonal Projections"
] | [
"Definition:Hilbert Space",
"Definition:Closed Linear Subspace",
"Definition:Orthogonal Projection",
"Definition:Mapping"
] | [
"Definition:Mapping",
"Definition:Orthogonal Projection",
"Unique Point of Minimal Distance to Closed Convex Subset of Hilbert Space"
] |
proofwiki-20057 | Normal Vectors Form Space around Simple Complex Contour | Let $C$ be a simple contour in the complex plane $\C$ with parameterization $\gamma: \closedint a b \to \C$.
Let $t \in \openint a b$ such that $\gamma$ is complex-differentiable at $t$.
Then there exists $r, R \in \R_{>0}$ such that:
:for all $s \in \openint { t-R }{ t+R }$ and for all $\epsilon \in \openint 0 r$: $\m... | Suppose there exists no $r, R \in \R_{>0}$ such that for all $s \in \openint { t-R }{ t+R }$ and for all $\epsilon \in \openint 0 r$, we have $\map \gamma s + \epsilon i \map {\gamma'} s \notin \Img C$.
It follows that for all $n \in \N$, there exists $t_n \in \openint { t - \dfrac 1 n }{ t + \dfrac 1 n }$ and $\epsilo... | Let $C$ be a [[Definition:Simple Contour (Complex Plane)|simple contour]] in the [[Definition:Complex Plane|complex plane]] $\C$ with [[Definition:Parameterization of Contour (Complex Plane)|parameterization]] $\gamma: \closedint a b \to \C$.
Let $t \in \openint a b$ such that $\gamma$ is [[Definition:Complex-Differen... | Suppose there exists no $r, R \in \R_{>0}$ such that for all $s \in \openint { t-R }{ t+R }$ and for all $\epsilon \in \openint 0 r$, we have $\map \gamma s + \epsilon i \map {\gamma'} s \notin \Img C$.
It follows that for all $n \in \N$, there exists $t_n \in \openint { t - \dfrac 1 n }{ t + \dfrac 1 n }$ and $\epsil... | Normal Vectors Form Space around Simple Complex Contour | https://proofwiki.org/wiki/Normal_Vectors_Form_Space_around_Simple_Complex_Contour | https://proofwiki.org/wiki/Normal_Vectors_Form_Space_around_Simple_Complex_Contour | [
"Orientation of Complex Contour"
] | [
"Definition:Contour/Simple/Complex Plane",
"Definition:Complex Number/Complex Plane",
"Definition:Contour/Parameterization/Complex Plane",
"Definition:Differentiable Mapping/Complex Function/Point",
"Definition:Contour/Image/Complex Plane"
] | [
"Bolzano-Weierstrass Theorem",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Subsequence",
"Definition:Continuous Complex Function/Using Limit",
"Combination Theorem for Limits of Functions/Complex",
"Limit of Subsequence equals Limit of Sequence",
"Definition:Continuous Complex Function/Us... |
proofwiki-20058 | Fourier Transform of 1-Lebesgue Space Function is Bounded | Let $n \in \N_{>0}$.
Let $\map {L^1} {\R^n}$ be the complex-valued Lebesgue $1$-space with respect to the Lebesgue measure.
Let $f \in \map {L^1} {\R^n}$.
Let $\hat f$ be the Fourier transform of $f$.
Then:
:$\ds \sup_{\mathbf s \mathop \in \R^n} \cmod {\map {\hat f} {\mathbf s} } \le \norm f_{\map {L^1} {\R^n} }$ | For each $\mathbf s \in \R^n$:
{{begin-eqn}}
{{eqn | l = \cmod { \map {\hat f} {\mathbf s} }
| r = \cmod { \int_{\R^n} \map f {\mathbf x} e^{-2 \pi i \mathbf x \cdot \mathbf s} \rd \mathbf x }
| c = {{Defof|Fourier Transform}}
}}
{{eqn | o = \le
| r = \int_{\R^n} \cmod { \map f {\mathbf x} e^{-2 \pi i... | Let $n \in \N_{>0}$.
Let $\map {L^1} {\R^n}$ be the [[Definition:Complex-Valued Function|complex-valued]] [[Definition:Lebesgue Space|Lebesgue $1$-space]] with respect to the [[Definition:Lebesgue Measure|Lebesgue measure]].
Let $f \in \map {L^1} {\R^n}$.
Let $\hat f$ be the [[Definition:Fourier Transform|Fourier tr... | For each $\mathbf s \in \R^n$:
{{begin-eqn}}
{{eqn | l = \cmod { \map {\hat f} {\mathbf s} }
| r = \cmod { \int_{\R^n} \map f {\mathbf x} e^{-2 \pi i \mathbf x \cdot \mathbf s} \rd \mathbf x }
| c = {{Defof|Fourier Transform}}
}}
{{eqn | o = \le
| r = \int_{\R^n} \cmod { \map f {\mathbf x} e^{-2 \pi i... | Fourier Transform of 1-Lebesgue Space Function is Bounded | https://proofwiki.org/wiki/Fourier_Transform_of_1-Lebesgue_Space_Function_is_Bounded | https://proofwiki.org/wiki/Fourier_Transform_of_1-Lebesgue_Space_Function_is_Bounded | [
"Fourier Transforms"
] | [
"Definition:Complex-Valued Function",
"Definition:Lebesgue Space",
"Definition:Lebesgue Measure",
"Definition:Fourier Transform"
] | [
"Triangle Inequality for Integrals",
"Modulus and Argument of Complex Exponential",
"Category:Fourier Transforms"
] |
proofwiki-20059 | Primitive of p x + q over a x squared plus 2 b x plus c | :$\ds \int \dfrac {p x + q} {a x^2 + 2 b x + c} \rd x = \dfrac p {2 a} \ln \size {a x^2 + 2 b x + c} + \paren {q - \dfrac {p b} a} \int \dfrac {\d x} {a^2 + 2 b x + c} + C$ | {{begin-eqn}}
{{eqn | l = \int \dfrac {p x + q} {a x^2 + 2 b x + c} \rd x
| r = p \int \dfrac x {a x^2 + 2 b x + c} \rd x + q \int \dfrac {\rd x} {a x^2 + 2 b x + c}
| c = Linear Combination of Primitives
}}
{{eqn | r = \frac p {2 a} \ln \size {a x^2 + 2 b x + c} - \frac {p b} a \int \frac {\d x} {a x^2 + ... | :$\ds \int \dfrac {p x + q} {a x^2 + 2 b x + c} \rd x = \dfrac p {2 a} \ln \size {a x^2 + 2 b x + c} + \paren {q - \dfrac {p b} a} \int \dfrac {\d x} {a^2 + 2 b x + c} + C$ | {{begin-eqn}}
{{eqn | l = \int \dfrac {p x + q} {a x^2 + 2 b x + c} \rd x
| r = p \int \dfrac x {a x^2 + 2 b x + c} \rd x + q \int \dfrac {\rd x} {a x^2 + 2 b x + c}
| c = [[Linear Combination of Primitives]]
}}
{{eqn | r = \frac p {2 a} \ln \size {a x^2 + 2 b x + c} - \frac {p b} a \int \frac {\d x} {a x^... | Primitive of p x + q over a x squared plus 2 b x plus c | https://proofwiki.org/wiki/Primitive_of_p_x_+_q_over_a_x_squared_plus_2_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_p_x_+_q_over_a_x_squared_plus_2_b_x_plus_c | [
"Primitive of p x + q over a x squared plus 2 b x plus c",
"Primitives involving a x squared plus b x plus c"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of x over a x squared plus b x plus c"
] |
proofwiki-20060 | Orthogonal Projection onto Orthocomplement | Let $H$ be a Hilbert space.
Let $A$ be a closed linear subspace of $H$.
Let $P_A: H \to H$ be the orthogonal projection onto $A$.
Let $P_{A^\perp}: H \to H$ be the orthogonal projection onto $A^\perp$, the orthocomplement of $A$.
Let $I: H \to H$ be the identity operator on $H$.
Then:
:$P_{A^\perp} = I - P_A$ | Let $h \in H$.
By definition of orthogonal projection, $P_A h = a \in A$ such that $h - a \in A^\perp$.
Then the claim is that for all $h \in H$:
:$\paren {I - P_A } h = h - P_A h$
is in $A^\perp$, and:
:$h - \paren {h - P_A h} \in \paren {A^\perp}^\perp$
That $h - P_A h \in A^\perp$ is part of the definition of $P_A$.... | Let $H$ be a [[Definition:Hilbert Space|Hilbert space]].
Let $A$ be a [[Definition:Closed Linear Subspace|closed linear subspace]] of $H$.
Let $P_A: H \to H$ be the [[Definition:Orthogonal Projection|orthogonal projection]] onto $A$.
Let $P_{A^\perp}: H \to H$ be the [[Definition:Orthogonal Projection|orthogonal pro... | Let $h \in H$.
By definition of [[Definition:Orthogonal Projection|orthogonal projection]], $P_A h = a \in A$ such that $h - a \in A^\perp$.
Then the claim is that for all $h \in H$:
:$\paren {I - P_A } h = h - P_A h$
is in $A^\perp$, and:
:$h - \paren {h - P_A h} \in \paren {A^\perp}^\perp$
That $h - P_A h \in ... | Orthogonal Projection onto Orthocomplement | https://proofwiki.org/wiki/Orthogonal_Projection_onto_Orthocomplement | https://proofwiki.org/wiki/Orthogonal_Projection_onto_Orthocomplement | [
"Orthogonal Projections",
"Orthocomplements"
] | [
"Definition:Hilbert Space",
"Definition:Closed Linear Subspace",
"Definition:Orthogonal Projection",
"Definition:Orthogonal Projection",
"Definition:Orthogonal (Linear Algebra)/Orthogonal Complement",
"Definition:Identity Mapping"
] | [
"Definition:Orthogonal Projection"
] |
proofwiki-20061 | Cesàro Summation Operator is Continuous Linear Transformation | Let $\ell^\infty$ be the space of bounded sequences.
Let $A : \ell^\infty \to \ell^\infty$ be the Cesàro summation operator.
Then $A$ is a continuous linear transformation. | === Well-Definedness ===
Let $\mathbf x = \sequence {x_n}_{n \mathop \in \N} \in \ell^\infty$.
Then:
{{begin-eqn}}
{{eqn | q = \forall n \in \N
| l = \size {x_n}
| o = \le
| r = \sup_{n \mathop \in \N} \size {x_n}
}}
{{eqn | r = \norm {\mathbf x}_\infty
| c = {{Defof|Supremum Norm}}
}}
{{end-... | Let $\ell^\infty$ be the [[Definition:Space of Bounded Sequences|space of bounded sequences]].
Let $A : \ell^\infty \to \ell^\infty$ be the [[Definition:Cesàro Summation Operator|Cesàro summation operator]].
Then $A$ is a [[Definition:Continuous Linear Transformation Space|continuous linear transformation]]. | === Well-Definedness ===
Let $\mathbf x = \sequence {x_n}_{n \mathop \in \N} \in \ell^\infty$.
Then:
{{begin-eqn}}
{{eqn | q = \forall n \in \N
| l = \size {x_n}
| o = \le
| r = \sup_{n \mathop \in \N} \size {x_n}
}}
{{eqn | r = \norm {\mathbf x}_\infty
| c = {{Defof|Supremum Norm}}
}}
{{e... | Cesàro Summation Operator is Continuous Linear Transformation | https://proofwiki.org/wiki/Cesàro_Summation_Operator_is_Continuous_Linear_Transformation | https://proofwiki.org/wiki/Cesàro_Summation_Operator_is_Continuous_Linear_Transformation | [
"Cesàro Summation Operator",
"Operator Theory",
"Linear Transformations",
"Continuous Transformations"
] | [
"Definition:Space of Bounded Sequences",
"Definition:Cesàro Summation Operator",
"Definition:Continuous Linear Transformation Space"
] | [
"Triangle Inequality/Complex Numbers/General Result"
] |
proofwiki-20062 | Primitive of p x + q over Root of a x squared plus 2 b x plus c | :$\ds \int \dfrac {p x + q} {\sqrt {a x^2 + 2 b x + c} } \rd x = \dfrac p a \sqrt {a x^2 + 2 b x + c} + \paren {q - \dfrac {p b} a} \int \dfrac {\d x} {\sqrt {a x^2 + 2 b x + c} }$ | {{begin-eqn}}
{{eqn | l = \int \dfrac {p x + q} {\sqrt {a x^2 + 2 b x + c} } \rd x
| r = \dfrac p {2 a} \int \dfrac {2 a x + \frac {2 a q} p} {\sqrt {a x^2 + 2 b x + c} } \rd x
| c =
}}
{{eqn | r = \dfrac p {2 a} \int \dfrac {2 a x + 2 b - 2 b + \frac {2 a q} p} {\sqrt {a x^2 + 2 b x + c} } \rd x
| c... | :$\ds \int \dfrac {p x + q} {\sqrt {a x^2 + 2 b x + c} } \rd x = \dfrac p a \sqrt {a x^2 + 2 b x + c} + \paren {q - \dfrac {p b} a} \int \dfrac {\d x} {\sqrt {a x^2 + 2 b x + c} }$ | {{begin-eqn}}
{{eqn | l = \int \dfrac {p x + q} {\sqrt {a x^2 + 2 b x + c} } \rd x
| r = \dfrac p {2 a} \int \dfrac {2 a x + \frac {2 a q} p} {\sqrt {a x^2 + 2 b x + c} } \rd x
| c =
}}
{{eqn | r = \dfrac p {2 a} \int \dfrac {2 a x + 2 b - 2 b + \frac {2 a q} p} {\sqrt {a x^2 + 2 b x + c} } \rd x
| c... | Primitive of p x + q over Root of a x squared plus 2 b x plus c/Proof 1 | https://proofwiki.org/wiki/Primitive_of_p_x_+_q_over_Root_of_a_x_squared_plus_2_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_p_x_+_q_over_Root_of_a_x_squared_plus_2_b_x_plus_c/Proof_1 | [
"Primitive of p x + q over Root of a x squared plus 2 b x plus c",
"Primitives involving Root of a x squared plus b x plus c"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Power Rule for Derivatives",
"Square Root of Function under Derivative"
] |
proofwiki-20063 | All Normal Vectors of Simple Closed Contour Cannot Point into Interior | Let $C$ be a simple closed contour in the complex plane $\C$ with parameterization $\gamma: \closedint a b \to \C$.
Let $t \in \openint a b$ such that $\gamma$ is complex-differentiable at $t$.
Let $S \in \set {-1,1}$.
Let $\Int C$ denote the interior of $C$.
Then there exists $r_0 \in \R_{>0}$ such that:
:for all $\ep... | From Normal Vectors Form Space around Simple Complex Contour, it follows that there exists $\tilde r, \tilde R \in \R_{>0}$ such that:
:for all $s \in \openint { t - \tilde R }{ t + \tilde R }$ and for all $\epsilon \in \openint 0 {\tilde r}$: $\map \gamma s + \epsilon i \map {\gamma'} s \notin \Img C$, and $\map \gamm... | Let $C$ be a [[Definition:Simple Contour (Complex Plane)|simple]] [[Definition:Closed Contour (Complex Plane)|closed contour]] in the [[Definition:Complex Plane|complex plane]] $\C$ with [[Definition:Parameterization of Contour (Complex Plane)|parameterization]] $\gamma: \closedint a b \to \C$.
Let $t \in \openint a b... | From [[Normal Vectors Form Space around Simple Complex Contour]], it follows that there exists $\tilde r, \tilde R \in \R_{>0}$ such that:
:for all $s \in \openint { t - \tilde R }{ t + \tilde R }$ and for all $\epsilon \in \openint 0 {\tilde r}$: $\map \gamma s + \epsilon i \map {\gamma'} s \notin \Img C$, and $\map ... | All Normal Vectors of Simple Closed Contour Cannot Point into Interior | https://proofwiki.org/wiki/All_Normal_Vectors_of_Simple_Closed_Contour_Cannot_Point_into_Interior | https://proofwiki.org/wiki/All_Normal_Vectors_of_Simple_Closed_Contour_Cannot_Point_into_Interior | [
"Orientation of Complex Contour"
] | [
"Definition:Contour/Simple/Complex Plane",
"Definition:Contour/Closed/Complex Plane",
"Definition:Complex Number/Complex Plane",
"Definition:Contour/Parameterization/Complex Plane",
"Definition:Differentiable Mapping/Complex Function/Point",
"Definition:Interior of Simple Closed Contour"
] | [
"Normal Vectors Form Space around Simple Complex Contour",
"Definition:Contour/Image/Complex Plane",
"All Normal Vectors of Simple Closed Contour Cannot Point into Interior/Lemma 1",
"Definition:Open Set/Complex Analysis/Definition 1",
"Definition:Path (Topology)",
"Definition:Path (Topology)/Endpoint",
... |
proofwiki-20064 | Orientation of Simple Closed Contour is with Respect to Interior | Let $C$ be a simple closed contour in the complex plane $\C$.
Let $\Int C$ denote the interior of $C$.
$C$ is positively oriented, {{iff}} $C$ is positively oriented with respect to $\Int C$.
$C$ is negatively oriented, {{iff}} $C$ is negatively oriented with respect to $\Int C$. | === Sufficient condition ===
Suppose $C$ is positively oriented.
Let $\gamma : \closedint a b$ be a parameterization of $C$.
Set $K := \set { t \in \closedint a b : \textrm{ $\gamma$ is not differentiable at $t$ } }$.
By definition of positive orientation, it follows that for all $t \in \openint a b \setminus K$, there... | Let $C$ be a [[Definition:Simple Contour (Complex Plane)|simple]] [[Definition:Closed Contour (Complex Plane)|closed contour]] in the [[Definition:Complex Plane|complex plane]] $\C$.
Let $\Int C$ denote the [[Definition:Interior of Simple Closed Contour|interior]] of $C$.
$C$ is [[Definition:Positive Orientation of ... | === Sufficient condition ===
Suppose $C$ is [[Definition:Positive Orientation of Simple Closed Contour (Complex Plane)|positively oriented]].
Let $\gamma : \closedint a b$ be a [[Definition:Parameterization of Contour (Complex Plane)|parameterization]] of $C$.
Set $K := \set { t \in \closedint a b : \textrm{ $\gamma... | Orientation of Simple Closed Contour is with Respect to Interior | https://proofwiki.org/wiki/Orientation_of_Simple_Closed_Contour_is_with_Respect_to_Interior | https://proofwiki.org/wiki/Orientation_of_Simple_Closed_Contour_is_with_Respect_to_Interior | [
"Orientation of Complex Contour"
] | [
"Definition:Contour/Simple/Complex Plane",
"Definition:Contour/Closed/Complex Plane",
"Definition:Complex Number/Complex Plane",
"Definition:Interior of Simple Closed Contour",
"Definition:Orientation of Contour (Complex Plane)/Positive/Simple Closed",
"Definition:Orientation of Contour (Complex Plane)/Po... | [
"Definition:Orientation of Contour (Complex Plane)/Positive/Simple Closed",
"Definition:Contour/Parameterization/Complex Plane",
"Definition:Orientation of Contour (Complex Plane)/Positive/Simple Closed",
"Complex Plane is Homeomorphic to Real Plane",
"Definition:Complex Number/Complex Plane",
"Definition... |
proofwiki-20065 | Supremum Operator Norm of Cesàro Summation Operator | Let $A : \ell^\infty \to \ell^\infty$ be the Cesàro summation operator.
Let $\norm {\, \cdot \,}$ be the supremum operator norm.
Then $\norm A = 1$. | By Cesàro Summation Operator is Continuous Linear Transformation:
:$\norm {A \mathbf x}_\infty \le \norm {\mathbf x}_\infty$
where $\norm {\, \cdot \,}_\infty$ denotes the supremum norm.
By Universal Upper Bound greater than Supremum Operator Norm:
:$\norm A \le 1$
Let $\mathbf 1 := \tuple {1, 1, \ldots} \in \ell^\inft... | Let $A : \ell^\infty \to \ell^\infty$ be the [[Definition:Cesàro Summation Operator|Cesàro summation operator]].
Let $\norm {\, \cdot \,}$ be the [[Definition:Supremum Operator Norm|supremum operator norm]].
Then $\norm A = 1$. | By [[Cesàro Summation Operator is Continuous Linear Transformation]]:
:$\norm {A \mathbf x}_\infty \le \norm {\mathbf x}_\infty$
where $\norm {\, \cdot \,}_\infty$ denotes the supremum norm.
By [[Universal Upper Bound greater than Supremum Operator Norm]]:
:$\norm A \le 1$
Let $\mathbf 1 := \tuple {1, 1, \ldots} \... | Supremum Operator Norm of Cesàro Summation Operator | https://proofwiki.org/wiki/Supremum_Operator_Norm_of_Cesàro_Summation_Operator | https://proofwiki.org/wiki/Supremum_Operator_Norm_of_Cesàro_Summation_Operator | [
"Cesàro Summation Operator",
"Operator Theory",
"Linear Transformations",
"Continuous Transformations"
] | [
"Definition:Cesàro Summation Operator",
"Definition:Supremum Operator Norm"
] | [
"Cesàro Summation Operator is Continuous Linear Transformation",
"Universal Upper Bound greater than Supremum Operator Norm",
"Supremum Operator Norm as Universal Upper Bound"
] |
proofwiki-20066 | Integral of Positive Measurable Function with respect to Restricted Measure | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $\GG \subseteq \Sigma$ be a sub-$\sigma$-algebra of $\Sigma$.
Let $\mu \restriction_\GG$ be the restriction of $\mu$ to $\GG$.
Let $f : X \to \overline \R$ be a positive $\GG$-measurable function.
Then:
:$\ds \int f \rd \mu = \int f \rd \mu \restriction_\GG$ | We first prove the result for positive simple functions.
Let $f : X \to \R$ be a positive simple function that is $\GG$-measurable.
From Simple Function has Standard Representation there exists:
:a finite sequence $a_1, \ldots, a_n$ of real numbers
:a partition $E_1, \ldots, E_n$ of $\GG$-measurable sets
such that:
:... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $\GG \subseteq \Sigma$ be a [[Definition:Sub-Sigma-Algebra|sub-$\sigma$-algebra]] of $\Sigma$.
Let $\mu \restriction_\GG$ be the [[Definition:Restricted Measure|restriction]] of $\mu$ to $\GG$.
Let $f : X \to \overline \R$ be a [[De... | We first prove the result for [[Definition:Positive Simple Function|positive simple functions]].
Let $f : X \to \R$ be a [[Definition:Positive Simple Function|positive simple function]] that is [[Definition:Measurable Function|$\GG$-measurable]].
From [[Simple Function has Standard Representation]] there exists:
:a... | Integral of Positive Measurable Function with respect to Restricted Measure | https://proofwiki.org/wiki/Integral_of_Positive_Measurable_Function_with_respect_to_Restricted_Measure | https://proofwiki.org/wiki/Integral_of_Positive_Measurable_Function_with_respect_to_Restricted_Measure | [
"Restricted Measures",
"Integral of Positive Measurable Function"
] | [
"Definition:Measure Space",
"Definition:Sub-Sigma-Algebra",
"Definition:Restricted Measure",
"Definition:Measurable Function/Positive"
] | [
"Definition:Simple Function",
"Definition:Simple Function",
"Definition:Measurable Function",
"Measurable Function is Simple Function iff Finite Image Set/Corollary",
"Definition:Finite Sequence",
"Definition:Real Number",
"Definition:Set Partition",
"Definition:Measurable Set",
"Integral of Positiv... |
proofwiki-20067 | Existence and Essential Uniqueness of Conditional Expectation Conditioned on Sigma-Algebra | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be an integrable random variable on $\struct {\Omega, \Sigma, \Pr}$.
Let $\GG \subseteq \Sigma$ be a sub-$\sigma$-algebra of $\Sigma$.
Then there exists an integrable random variable $Z$ on $\struct {\Omega, \GG, \Pr}$ such that:
:$\ds \int_G Z \rd \P... | First take $X \ge 0$.
Define a function $\mu : \GG \to \R$ by:
:$\ds \map \mu A = \int_A X \rd \Pr$
for each $A \in \GG$.
From Integral of Integrable Function over Measurable Set is Well-Defined, this is well-defined.
From Measure with Density is Measure, $\mu$ is a measure.
Note that if $\map \Pr A = 0$ for $A \in... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ be an [[Definition:Integrable Random Variable|integrable random variable]] on $\struct {\Omega, \Sigma, \Pr}$.
Let $\GG \subseteq \Sigma$ be a [[Definition:Sub-Sigma-Algebra|sub-$\sigma$-algebra]] of $\Sigma$.
Then ... | First take $X \ge 0$.
Define a function $\mu : \GG \to \R$ by:
:$\ds \map \mu A = \int_A X \rd \Pr$
for each $A \in \GG$.
From [[Integral of Integrable Function over Measurable Set is Well-Defined]], this is well-defined.
From [[Measure with Density is Measure]], $\mu$ is a [[Definition:Measure (Measure Theory... | Existence and Essential Uniqueness of Conditional Expectation Conditioned on Sigma-Algebra/Proof 1 | https://proofwiki.org/wiki/Existence_and_Essential_Uniqueness_of_Conditional_Expectation_Conditioned_on_Sigma-Algebra | https://proofwiki.org/wiki/Existence_and_Essential_Uniqueness_of_Conditional_Expectation_Conditioned_on_Sigma-Algebra/Proof_1 | [
"Existence and Essential Uniqueness of Conditional Expectation Conditioned on Sigma-Algebra",
"Conditional Expectation"
] | [
"Definition:Probability Space",
"Definition:Integrable Random Variable",
"Definition:Sub-Sigma-Algebra",
"Definition:Integrable Random Variable",
"Definition:Integrable Random Variable",
"Definition:Almost Everywhere"
] | [
"Integral of Integrable Function over Measurable Set is Well-Defined",
"Measure with Density is Measure",
"Definition:Measure (Measure Theory)",
"Integral of Integrable Function over Null Set",
"Definition:Absolute Continuity/Measure",
"Radon-Nikodym Theorem",
"Definition:Integrable Function",
"Defini... |
proofwiki-20068 | Existence and Essential Uniqueness of Conditional Expectation Conditioned on Sigma-Algebra | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be an integrable random variable on $\struct {\Omega, \Sigma, \Pr}$.
Let $\GG \subseteq \Sigma$ be a sub-$\sigma$-algebra of $\Sigma$.
Then there exists an integrable random variable $Z$ on $\struct {\Omega, \GG, \Pr}$ such that:
:$\ds \int_G Z \rd \P... | Observe that:
:$\map {L^2} {\Omega, \GG, \Pr} \subseteq \map {L^2} {\Omega, \Sigma, \Pr}$
is a closed linear space.
Let:
:$P : \map {L^2} {\Omega, \Sigma, \Pr} \to \map {L^2} {\Omega, \GG, \Pr}$
be the orthogonal projection.
Observe that for all $f \in \map {L^2} {\Omega, \Sigma, \Pr}$ and $g \in \map {L^2} {\Omega, \G... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ be an [[Definition:Integrable Random Variable|integrable random variable]] on $\struct {\Omega, \Sigma, \Pr}$.
Let $\GG \subseteq \Sigma$ be a [[Definition:Sub-Sigma-Algebra|sub-$\sigma$-algebra]] of $\Sigma$.
Then ... | Observe that:
:$\map {L^2} {\Omega, \GG, \Pr} \subseteq \map {L^2} {\Omega, \Sigma, \Pr}$
is a [[Definition:Closed Linear Subspace|closed linear space]].
Let:
:$P : \map {L^2} {\Omega, \Sigma, \Pr} \to \map {L^2} {\Omega, \GG, \Pr}$
be the [[Definition:Orthogonal Projection|orthogonal projection]].
Observe that for a... | Existence and Essential Uniqueness of Conditional Expectation Conditioned on Sigma-Algebra/Proof 2 | https://proofwiki.org/wiki/Existence_and_Essential_Uniqueness_of_Conditional_Expectation_Conditioned_on_Sigma-Algebra | https://proofwiki.org/wiki/Existence_and_Essential_Uniqueness_of_Conditional_Expectation_Conditioned_on_Sigma-Algebra/Proof_2 | [
"Existence and Essential Uniqueness of Conditional Expectation Conditioned on Sigma-Algebra",
"Conditional Expectation"
] | [
"Definition:Probability Space",
"Definition:Integrable Random Variable",
"Definition:Sub-Sigma-Algebra",
"Definition:Integrable Random Variable",
"Definition:Integrable Random Variable",
"Definition:Almost Everywhere"
] | [
"Definition:Closed Linear Subspace",
"Definition:Orthogonal Projection",
"Cauchy-Bunyakovsky-Schwarz Inequality/Lebesgue 2-Space",
"Space of Simple P-Integrable Functions is Everywhere Dense in Lebesgue Space",
"Definition:Everywhere Dense",
"Definition:Vector Subspace/Hilbert Spaces",
"Definition:Integ... |
proofwiki-20069 | Inner Product with Vector is Linear Functional | Let $\GF$ be a subfield of $\C$.
Let $\struct{ V, \innerprod \cdot \cdot }$ be an inner product space over $\GF$.
Let $v_0 \in V$.
Then the mapping $L: V \to \GF$ defined by:
:$\map L v := \innerprod v {v_0}$
is a linear functional. | Let us directly check the definition of linear functional:
{{begin-eqn}}
{{eqn|l = \map L {\alpha v + \beta w}
|r = \innerprod { \alpha v + \beta w } {v_0}
}}
{{eqn|r = \innerprod {\alpha v} {v_0} + \innerprod {\beta w} {v_0}
}}
{{eqn|r = \alpha \innerprod v {v_0} + \beta \innerprod w {w_0}
}}
{{eqn|r = \alpha \ma... | Let $\GF$ be a [[Definition:Subfield|subfield]] of $\C$.
Let $\struct{ V, \innerprod \cdot \cdot }$ be an [[Definition:Inner Product Space|inner product space]] over $\GF$.
Let $v_0 \in V$.
Then the [[Definition:Mapping|mapping]] $L: V \to \GF$ defined by:
:$\map L v := \innerprod v {v_0}$
is a [[Definition:Linea... | Let us directly check the definition of [[Definition:Linear Functional|linear functional]]:
{{begin-eqn}}
{{eqn|l = \map L {\alpha v + \beta w}
|r = \innerprod { \alpha v + \beta w } {v_0}
}}
{{eqn|r = \innerprod {\alpha v} {v_0} + \innerprod {\beta w} {v_0}
}}
{{eqn|r = \alpha \innerprod v {v_0} + \beta \innerpro... | Inner Product with Vector is Linear Functional | https://proofwiki.org/wiki/Inner_Product_with_Vector_is_Linear_Functional | https://proofwiki.org/wiki/Inner_Product_with_Vector_is_Linear_Functional | [
"Inner Product Spaces",
"Linear Functionals"
] | [
"Definition:Subfield",
"Definition:Inner Product Space",
"Definition:Mapping",
"Definition:Linear Functional"
] | [
"Definition:Linear Functional"
] |
proofwiki-20070 | Inner Product with Vector is Bounded Linear Functional | Let $\GF$ be a subfield of $\C$.
Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space over $\GF$.
Let $v_0 \in V$.
Then the mapping $L: V \to \GF$ defined by:
:$\map L v := \innerprod v {v_0}$
is a bounded linear functional with norm equal to $\norm {v_0}$. | By Inner Product with Vector is Linear Functional, $L$ is a linear functional.
To check that $L$ is bounded:
{{begin-eqn}}
{{eqn | l = \cmod {\map L v}
| r = \cmod {\innerprod v {v_0} }
}}
{{eqn | o = \le
| r = \norm v \norm {v_0}
| c = Cauchy-Bunyakovsky-Schwarz Inequality for Inner Product Spaces
}}... | Let $\GF$ be a [[Definition:Subfield|subfield]] of $\C$.
Let $\struct {V, \innerprod \cdot \cdot}$ be an [[Definition:Inner Product Space|inner product space]] over $\GF$.
Let $v_0 \in V$.
Then the [[Definition:Mapping|mapping]] $L: V \to \GF$ defined by:
:$\map L v := \innerprod v {v_0}$
is a [[Definition:Bounde... | By [[Inner Product with Vector is Linear Functional]], $L$ is a [[Definition:Linear Functional|linear functional]].
To check that $L$ is [[Definition:Bounded Linear Functional|bounded]]:
{{begin-eqn}}
{{eqn | l = \cmod {\map L v}
| r = \cmod {\innerprod v {v_0} }
}}
{{eqn | o = \le
| r = \norm v \norm {v_... | Inner Product with Vector is Bounded Linear Functional | https://proofwiki.org/wiki/Inner_Product_with_Vector_is_Bounded_Linear_Functional | https://proofwiki.org/wiki/Inner_Product_with_Vector_is_Bounded_Linear_Functional | [
"Inner Product Spaces",
"Bounded Linear Functionals"
] | [
"Definition:Subfield",
"Definition:Inner Product Space",
"Definition:Mapping",
"Definition:Bounded Linear Functional",
"Definition:Norm/Bounded Linear Functional"
] | [
"Inner Product with Vector is Linear Functional",
"Definition:Linear Functional",
"Definition:Bounded Linear Functional",
"Cauchy-Bunyakovsky-Schwarz Inequality/Inner Product Spaces",
"Definition:Norm/Bounded Linear Functional"
] |
proofwiki-20071 | Kernel of Bounded Linear Transformation is Closed Linear Subspace | Let $V, W$ be normed vector spaces.
Let $f: V \to W$ be a bounded linear transformation.
Then $\ker f$, the kernel of $f$, is a closed linear subspace of $V$. | By Kernel of Linear Transformation is Linear Subspace, $\ker f$ is a subspace of $V$.
By Continuity of Linear Transformations, $f$ is continuous.
Since $\ker f = f^{-1} \sqbrk{ \set{ \mathbf 0_W } }$, it follows from Continuity Defined from Closed Sets that $\ker f$ is closed.
Hence the result by definition of closed l... | Let $V, W$ be [[Definition:Normed Vector Space|normed vector spaces]].
Let $f: V \to W$ be a [[Definition:Bounded Linear Transformation|bounded linear transformation]].
Then $\ker f$, the [[Definition:Kernel of Linear Transformation on Vector Space|kernel]] of $f$, is a [[Definition:Closed Linear Subspace|closed lin... | By [[Kernel of Linear Transformation is Linear Subspace]], $\ker f$ is a [[Definition:Vector Subspace|subspace]] of $V$.
By [[Continuity of Linear Transformations]], $f$ is [[Definition:Continuous Mapping (Topology)|continuous]].
Since $\ker f = f^{-1} \sqbrk{ \set{ \mathbf 0_W } }$, it follows from [[Continuity Defi... | Kernel of Bounded Linear Transformation is Closed Linear Subspace | https://proofwiki.org/wiki/Kernel_of_Bounded_Linear_Transformation_is_Closed_Linear_Subspace | https://proofwiki.org/wiki/Kernel_of_Bounded_Linear_Transformation_is_Closed_Linear_Subspace | [
"Bounded Linear Transformations"
] | [
"Definition:Normed Vector Space",
"Definition:Bounded Linear Transformation",
"Definition:Kernel of Linear Transformation/Vector Space",
"Definition:Closed Linear Subspace"
] | [
"Kernel of Linear Transformation is Linear Subspace",
"Definition:Vector Subspace",
"Continuity of Linear Transformations",
"Definition:Continuous Mapping (Topology)",
"Continuity Defined from Closed Sets",
"Definition:Closed Set/Topology",
"Definition:Closed Linear Subspace",
"Category:Bounded Linear... |
proofwiki-20072 | Sufficient Condition for Weak Convergence of Bounded Sequence in Hilbert Space in terms of Everywhere Dense Set | Let $\struct {\HH, \innerprod \cdot \cdot_\HH}$ be a Hilbert space.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a bounded sequence in $\HH$.
Let $D$ be an everywhere dense set in $\HH$ such that:
:$\innerprod {x_n} v \to \innerprod x v$
for all $v \in D$, for some $x \in \HH$.
Then:
:$x_n \weakconv x$
where $\weakcon... | We show the theorem in the case $x = 0$ first for neatness.
From Weak Convergence in Hilbert Space, we want to show that:
:$\innerprod {x_n} z \to \innerprod x z$
for all $z \in \HH$.
Let $\epsilon > 0$.
Since $D$ is everywhere dense in $\HH$, for each $n \in \N$ we can pick $z_n \in D$ such that:
:$\norm {z - z_n}... | Let $\struct {\HH, \innerprod \cdot \cdot_\HH}$ be a [[Definition:Hilbert Space|Hilbert space]].
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Bounded Sequence in Normed Vector Space|bounded sequence]] in $\HH$.
Let $D$ be an [[Definition:Everywhere Dense in Normed Vector Space|everywhere dense set]] in ... | We show the theorem in the case $x = 0$ first for neatness.
From [[Weak Convergence in Hilbert Space]], we want to show that:
:$\innerprod {x_n} z \to \innerprod x z$
for all $z \in \HH$.
Let $\epsilon > 0$.
Since $D$ is [[Definition:Everywhere Dense in Normed Vector Space|everywhere dense]] in $\HH$, for each ... | Sufficient Condition for Weak Convergence of Bounded Sequence in Hilbert Space in terms of Everywhere Dense Set | https://proofwiki.org/wiki/Sufficient_Condition_for_Weak_Convergence_of_Bounded_Sequence_in_Hilbert_Space_in_terms_of_Everywhere_Dense_Set | https://proofwiki.org/wiki/Sufficient_Condition_for_Weak_Convergence_of_Bounded_Sequence_in_Hilbert_Space_in_terms_of_Everywhere_Dense_Set | [
"Hilbert Spaces",
"Weak Convergence (Normed Vector Spaces)"
] | [
"Definition:Hilbert Space",
"Definition:Bounded Sequence/Normed Vector Space",
"Definition:Everywhere Dense/Normed Vector Space",
"Definition:Weak Convergence (Normed Vector Space)"
] | [
"Weak Convergence in Hilbert Space",
"Definition:Everywhere Dense/Normed Vector Space",
"Definition:Conjugate Linear Mapping",
"Definition:Inner Product",
"Triangle Inequality/Complex Numbers",
"Cauchy-Bunyakovsky-Schwarz Inequality/Inner Product Spaces",
"Definition:Linear Transformation",
"Definitio... |
proofwiki-20073 | Expectation of Conditional Expectation | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $\GG \subseteq \Sigma$ be a sub-$\sigma$-algebra.
Let $X$ be a integrable random variable.
Let $\expect {X \mid \GG}$ be a version of the conditional expectation of $X$ given $\GG$.
Then:
:$\expect {\expect {X \mid \GG} } = \expect X$ | We have:
{{begin-eqn}}
{{eqn | l = \expect {\expect {X \mid \GG} }
| r = \int_\Omega \expect {X \mid \GG} \rd \Pr
| c = {{Defof|Expectation|subdef = General Definition}}
}}
{{eqn | r = \int_\Omega X \rd \Pr
| c = {{Defof|Conditional Expectation on Sigma-Algebra}}
}}
{{eqn | r = \expect X
| c = {{Defof|Expectat... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $\GG \subseteq \Sigma$ be a [[Definition:Sub-Sigma-Algebra|sub-$\sigma$-algebra]].
Let $X$ be a [[Definition:Integrable Random Variable|integrable random variable]].
Let $\expect {X \mid \GG}$ be a version of the [[Defin... | We have:
{{begin-eqn}}
{{eqn | l = \expect {\expect {X \mid \GG} }
| r = \int_\Omega \expect {X \mid \GG} \rd \Pr
| c = {{Defof|Expectation|subdef = General Definition}}
}}
{{eqn | r = \int_\Omega X \rd \Pr
| c = {{Defof|Conditional Expectation on Sigma-Algebra}}
}}
{{eqn | r = \expect X
| c = {{Defof|Expecta... | Expectation of Conditional Expectation | https://proofwiki.org/wiki/Expectation_of_Conditional_Expectation | https://proofwiki.org/wiki/Expectation_of_Conditional_Expectation | [
"Conditional Expectation"
] | [
"Definition:Probability Space",
"Definition:Sub-Sigma-Algebra",
"Definition:Integrable Random Variable",
"Definition:Conditional Expectation/General Case/Sigma-Algebra"
] | [] |
proofwiki-20074 | Conditional Expectation of Measurable Random Variable | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $\GG \subseteq \Sigma$ be a sub-$\sigma$-algebra.
Let $X$ be a integrable random variable that is $\GG$-measurable.
Let $\expect {X \mid \GG}$ be a version of the conditional expectation of $X$ given $\GG$.
Then:
:$\expect {X \mid \GG} = X$ almost everywhe... | We show that $X$ is a version of $\expect {X \mid \GG}$.
Then since conditional expectation is essentially unique by Existence and Essential Uniqueness of Conditional Expectation Conditioned on Sigma-Algebra, we will obtain:
:$\expect {X \mid \GG} = X$ almost everywhere.
Note that $X$ is integrable and $\GG$-measurable... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $\GG \subseteq \Sigma$ be a [[Definition:Sub-Sigma-Algebra|sub-$\sigma$-algebra]].
Let $X$ be a [[Definition:Integrable Random Variable|integrable random variable]] that is [[Definition:Measurable Function|$\GG$-measurabl... | We show that $X$ is a version of $\expect {X \mid \GG}$.
Then since [[Definition:Conditional Expectation on Sigma-Algebra|conditional expectation]] is essentially unique by [[Existence and Essential Uniqueness of Conditional Expectation Conditioned on Sigma-Algebra]], we will obtain:
:$\expect {X \mid \GG} = X$ [[Def... | Conditional Expectation of Measurable Random Variable | https://proofwiki.org/wiki/Conditional_Expectation_of_Measurable_Random_Variable | https://proofwiki.org/wiki/Conditional_Expectation_of_Measurable_Random_Variable | [
"Conditional Expectation"
] | [
"Definition:Probability Space",
"Definition:Sub-Sigma-Algebra",
"Definition:Integrable Random Variable",
"Definition:Measurable Function",
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Definition:Almost Everywhere"
] | [
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Existence and Essential Uniqueness of Conditional Expectation Conditioned on Sigma-Algebra",
"Definition:Almost Everywhere",
"Definition:Integrable Random Variable",
"Definition:Measurable Function",
"Definition:Conditional Expectation/Gene... |
proofwiki-20075 | Tower Property of Conditional Expectation | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $\HH \subseteq \GG$ be sub-$\sigma$-algebras of $\Sigma$.
Let $X$ be a integrable random variable.
Let $\expect {X \mid \GG}$ be a version of the conditional expectation of $X$ given $\GG$.
Let $\expect {X \mid \HH}$ be a version of the conditional expect... | We show that $\expect {X \mid \HH}$ is a version of $\expect {\expect {X \mid \GG} \mid \HH}$.
Let $A \in \HH$.
Then:
{{begin-eqn}}
{{eqn | l = \int_A \expect {X \mid \HH} \rd \Pr
| r = \int_A X \rd \Pr
| c = {{Defof|Conditional Expectation on Sigma-Algebra}}
}}
{{eqn | r = \int_A \expect {X \mid \GG} \rd \Pr
| ... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $\HH \subseteq \GG$ be [[Definition:Sub-Sigma-Algebra|sub-$\sigma$-algebras]] of $\Sigma$.
Let $X$ be a [[Definition:Integrable Random Variable|integrable random variable]].
Let $\expect {X \mid \GG}$ be a version of th... | We show that $\expect {X \mid \HH}$ is a version of $\expect {\expect {X \mid \GG} \mid \HH}$.
Let $A \in \HH$.
Then:
{{begin-eqn}}
{{eqn | l = \int_A \expect {X \mid \HH} \rd \Pr
| r = \int_A X \rd \Pr
| c = {{Defof|Conditional Expectation on Sigma-Algebra}}
}}
{{eqn | r = \int_A \expect {X \mid \GG} \rd \Pr
... | Tower Property of Conditional Expectation | https://proofwiki.org/wiki/Tower_Property_of_Conditional_Expectation | https://proofwiki.org/wiki/Tower_Property_of_Conditional_Expectation | [
"Conditional Expectation"
] | [
"Definition:Probability Space",
"Definition:Sub-Sigma-Algebra",
"Definition:Integrable Random Variable",
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Def... | [
"Definition:Measurable Function",
"Existence and Essential Uniqueness of Conditional Expectation Conditioned on Sigma-Algebra",
"Definition:Almost Everywhere"
] |
proofwiki-20076 | Conditional Expectation is Linear | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $\GG \subseteq \Sigma$ be a sub-$\sigma$-algebra.
Let $X$ and $Y$ be integrable random variables.
Let $\alpha, \beta \in \R$.
Let $\expect {X \mid \GG}$ be a version of the conditional expectation of $X$ given $\GG$.
Let $\expect {Y \mid \GG}$ be a versio... | We show that $\alpha \expect {X \mid \GG} + \beta \expect {Y \mid \GG}$ is a version of the conditional expectation of $\alpha x + \beta Y$ given $\GG$.
From Linear Combination of Real-Valued Random Variables is Real-Valued Random Variable, we have that $\alpha \expect {X \mid \GG} + \beta \expect {Y \mid \GG}$ is a $\... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $\GG \subseteq \Sigma$ be a [[Definition:Sub-Sigma-Algebra|sub-$\sigma$-algebra]].
Let $X$ and $Y$ be [[Definition:Integrable Random Variable|integrable random variables]].
Let $\alpha, \beta \in \R$.
Let $\expect {X \... | We show that $\alpha \expect {X \mid \GG} + \beta \expect {Y \mid \GG}$ is a version of the [[Definition:Conditional Expectation on Sigma-Algebra|conditional expectation of $\alpha x + \beta Y$ given $\GG$]].
From [[Linear Combination of Real-Valued Random Variables is Real-Valued Random Variable]], we have that $\alp... | Conditional Expectation is Linear | https://proofwiki.org/wiki/Conditional_Expectation_is_Linear | https://proofwiki.org/wiki/Conditional_Expectation_is_Linear | [
"Conditional Expectation"
] | [
"Definition:Probability Space",
"Definition:Sub-Sigma-Algebra",
"Definition:Integrable Random Variable",
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Def... | [
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Linear Combination of Real-Valued Random Variables is Real-Valued Random Variable",
"Definition:Measurable Function",
"Definition:Random Variable",
"Integral of Integrable Function is Additive",
"Integral of Integrable Function is Homogeneo... |
proofwiki-20077 | Conditional Monotone Convergence Theorem | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be an non-negative integrable random variable.
Let $\sequence {X_n}_{n \in \N}$ be an sequence of non-negative integrable random variables converging almost surely to $X$, such that:
:$X_n \le X_{n + 1}$ almost everywhere for each $n \in \N$.
Let $\GG... | Note that for almost all $\omega \in \Omega$, $\sequence {\map {X_n} \omega}_{n \mathop \in \N}$ is an increasing real sequence with $\map {X_n} \omega \to \map X \omega$.
From Monotone Convergence Theorem (Real Analysis), we then have that:
:$\map {X_n} \omega \le \map X \omega$
for almost all $\omega \in \Omega$.
F... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ be an [[Definition:Positive Real Function|non-negative]] [[Definition:Integrable Random Variable|integrable random variable]].
Let $\sequence {X_n}_{n \in \N}$ be an [[Definition:Sequence|sequence]] of [[Definition:Po... | Note that for [[Definition:Almost All|almost all]] $\omega \in \Omega$, $\sequence {\map {X_n} \omega}_{n \mathop \in \N}$ is an [[Definition:Increasing Real Sequence|increasing real sequence]] with $\map {X_n} \omega \to \map X \omega$.
From [[Monotone Convergence Theorem (Real Analysis)]], we then have that:
:$\m... | Conditional Monotone Convergence Theorem | https://proofwiki.org/wiki/Conditional_Monotone_Convergence_Theorem | https://proofwiki.org/wiki/Conditional_Monotone_Convergence_Theorem | [
"Conditional Expectation"
] | [
"Definition:Probability Space",
"Definition:Positive Real Function",
"Definition:Integrable Random Variable",
"Definition:Sequence",
"Definition:Positive Real Function",
"Definition:Integrable Random Variable",
"Definition:Almost Sure Convergence",
"Definition:Almost Everywhere",
"Definition:Sub-Sig... | [
"Definition:Almost All",
"Definition:Increasing/Sequence/Real Sequence",
"Monotone Convergence Theorem (Real Analysis)",
"Definition:Almost All",
"Conditional Expectation is Monotone",
"Definition:Almost Everywhere",
"Conditional Expectation is Monotone",
"Definition:Almost Everywhere",
"Definition:... |
proofwiki-20078 | Tangent of Sum of Three Angles | :$\map \tan {A + B + C} = \dfrac {\tan A + \tan B + \tan C - \tan A \tan B \tan C} {1 - \tan B \tan C - \tan C \tan A - \tan A \tan B}$ | {{begin-eqn}}
{{eqn | l = \map \tan {A + B + C}
| r = \dfrac {\tan A + \map \tan {B + C} } {1 - \tan A \tan {B + C} }
| c = Tangent of Sum
}}
{{eqn | r = \dfrac {\tan A + \frac {\tan B + \tan C} {1 - \tan B \tan C} } {1 - \tan A \frac {\tan B + \tan C} {1 - \tan B \tan C} }
| c = Tangent of Sum
}}
{{e... | :$\map \tan {A + B + C} = \dfrac {\tan A + \tan B + \tan C - \tan A \tan B \tan C} {1 - \tan B \tan C - \tan C \tan A - \tan A \tan B}$ | {{begin-eqn}}
{{eqn | l = \map \tan {A + B + C}
| r = \dfrac {\tan A + \map \tan {B + C} } {1 - \tan A \tan {B + C} }
| c = [[Tangent of Sum]]
}}
{{eqn | r = \dfrac {\tan A + \frac {\tan B + \tan C} {1 - \tan B \tan C} } {1 - \tan A \frac {\tan B + \tan C} {1 - \tan B \tan C} }
| c = [[Tangent of Sum]... | Tangent of Sum of Three Angles/Proof 2 | https://proofwiki.org/wiki/Tangent_of_Sum_of_Three_Angles | https://proofwiki.org/wiki/Tangent_of_Sum_of_Three_Angles/Proof_2 | [
"Tangent of Sum of Three Angles",
"Tangent Function",
"Trigonometric Addition Formulas"
] | [] | [
"Tangent of Sum",
"Tangent of Sum",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-20079 | Tangent of Sum of Three Angles | :$\map \tan {A + B + C} = \dfrac {\tan A + \tan B + \tan C - \tan A \tan B \tan C} {1 - \tan B \tan C - \tan C \tan A - \tan A \tan B}$ | This is a special case of Tangent of Sum of Series of Angles, for $n = 3$.
{{qed}} | :$\map \tan {A + B + C} = \dfrac {\tan A + \tan B + \tan C - \tan A \tan B \tan C} {1 - \tan B \tan C - \tan C \tan A - \tan A \tan B}$ | This is a special case of [[Tangent of Sum of Series of Angles]], for $n = 3$.
{{qed}} | Tangent of Sum of Three Angles/Proof 3 | https://proofwiki.org/wiki/Tangent_of_Sum_of_Three_Angles | https://proofwiki.org/wiki/Tangent_of_Sum_of_Three_Angles/Proof_3 | [
"Tangent of Sum of Three Angles",
"Tangent Function",
"Trigonometric Addition Formulas"
] | [] | [
"Tangent of Sum of Series of Angles"
] |
proofwiki-20080 | Sine of Sum of Three Angles | :$\map \sin {A + B + C} = \sin A \cos B \cos C + \cos A \sin B \cos C + \cos A \cos B \sin C - \sin A \sin B \sin C$ | {{begin-eqn}}
{{eqn | l = \map \sin {A + B + C}
| r = \map \sin {A + B} \cos C + \map \cos {A + B} \sin C
| c = Sine of Sum
}}
{{eqn | r = \paren {\sin A \cos B + \cos A \sin B} \cos C + \paren {\cos A \cos B - \sin A \sin B} \sin C
| c = Sine of Sum, Cosine of Sum
}}
{{eqn | r = \sin A \cos B \cos C ... | :$\map \sin {A + B + C} = \sin A \cos B \cos C + \cos A \sin B \cos C + \cos A \cos B \sin C - \sin A \sin B \sin C$ | {{begin-eqn}}
{{eqn | l = \map \sin {A + B + C}
| r = \map \sin {A + B} \cos C + \map \cos {A + B} \sin C
| c = [[Sine of Sum]]
}}
{{eqn | r = \paren {\sin A \cos B + \cos A \sin B} \cos C + \paren {\cos A \cos B - \sin A \sin B} \sin C
| c = [[Sine of Sum]], [[Cosine of Sum]]
}}
{{eqn | r = \sin A \c... | Sine of Sum of Three Angles | https://proofwiki.org/wiki/Sine_of_Sum_of_Three_Angles | https://proofwiki.org/wiki/Sine_of_Sum_of_Three_Angles | [
"Sine Function"
] | [] | [
"Sine of Sum",
"Sine of Sum",
"Cosine of Sum",
"Category:Sine Function"
] |
proofwiki-20081 | Cosine of Sum of Three Angles | :$\map \cos {A + B + C} = \cos A \cos B \cos C - \sin A \sin B \cos C - \sin A \cos B \sin C - \cos A \sin B \sin C$ | {{begin-eqn}}
{{eqn | l = \map \cos {A + B + C}
| r = \map \cos {A + B} \cos C - \map \sin {A + B} \sin C
| c = Cosine of Sum
}}
{{eqn | r = \paren {\cos A \cos B - \sin A \sin B} \cos C - \paren {\sin A \cos B + \cos A \sin B} \sin C
| c = Cosine of Sum, Sine of Sum
}}
{{eqn | r = \cos A \cos B \cos ... | :$\map \cos {A + B + C} = \cos A \cos B \cos C - \sin A \sin B \cos C - \sin A \cos B \sin C - \cos A \sin B \sin C$ | {{begin-eqn}}
{{eqn | l = \map \cos {A + B + C}
| r = \map \cos {A + B} \cos C - \map \sin {A + B} \sin C
| c = [[Cosine of Sum]]
}}
{{eqn | r = \paren {\cos A \cos B - \sin A \sin B} \cos C - \paren {\sin A \cos B + \cos A \sin B} \sin C
| c = [[Cosine of Sum]], [[Sine of Sum]]
}}
{{eqn | r = \cos A ... | Cosine of Sum of Three Angles | https://proofwiki.org/wiki/Cosine_of_Sum_of_Three_Angles | https://proofwiki.org/wiki/Cosine_of_Sum_of_Three_Angles | [
"Cosine Function"
] | [] | [
"Cosine of Sum",
"Cosine of Sum",
"Sine of Sum",
"Category:Cosine Function"
] |
proofwiki-20082 | Tangent of Sum of Series of Angles | Let $\theta_1, \theta_2, \ldots, \theta_n$ be angles.
For all $k \in \set {1, 2, 3, \ldots, n}$, let $s_k$ be defined as the sum of the product of $\map \tan {\theta_1}, \map \tan {\theta_2}, \ldots, \map \tan {\theta_n}$ taken $k$ at a time:
{{begin-eqn}}
{{eqn | l = s_1
| r = \sum_{i \mathop = 1}^n \map \tan {\... | First we note:
{{begin-eqn}}
{{eqn | l = \cos \sum_j \theta_j + i \sin \sum_j \theta_j
| r = \prod_j \paren {\cos \theta_j + i \sin \theta_j}
| c = Product of Complex Numbers in Polar Form
}}
{{eqn | r = \prod_j \cos \theta_j \prod_j \paren {1 + i \tan \theta_j}
| c =
}}
{{eqn | r = \prod_j \cos \the... | Let $\theta_1, \theta_2, \ldots, \theta_n$ be [[Definition:Angle|angles]].
For all $k \in \set {1, 2, 3, \ldots, n}$, let $s_k$ be defined as the [[Definition:Real Addition|sum]] of the [[Definition:Real Multiplication|product]] of $\map \tan {\theta_1}, \map \tan {\theta_2}, \ldots, \map \tan {\theta_n}$ taken $k$ at... | First we note:
{{begin-eqn}}
{{eqn | l = \cos \sum_j \theta_j + i \sin \sum_j \theta_j
| r = \prod_j \paren {\cos \theta_j + i \sin \theta_j}
| c = [[Product of Complex Numbers in Polar Form]]
}}
{{eqn | r = \prod_j \cos \theta_j \prod_j \paren {1 + i \tan \theta_j}
| c =
}}
{{eqn | r = \prod_j \cos ... | Tangent of Sum of Series of Angles/Proof 1 | https://proofwiki.org/wiki/Tangent_of_Sum_of_Series_of_Angles | https://proofwiki.org/wiki/Tangent_of_Sum_of_Series_of_Angles/Proof_1 | [
"Tangent of Sum of Series of Angles",
"Tangent Function"
] | [
"Definition:Angle",
"Definition:Addition/Real Numbers",
"Definition:Multiplication/Real Numbers"
] | [
"Product of Complex Numbers in Polar Form",
"Product of Complex Numbers in Polar Form",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-20083 | Tangent of Sum of Series of Angles | Let $\theta_1, \theta_2, \ldots, \theta_n$ be angles.
For all $k \in \set {1, 2, 3, \ldots, n}$, let $s_k$ be defined as the sum of the product of $\map \tan {\theta_1}, \map \tan {\theta_2}, \ldots, \map \tan {\theta_n}$ taken $k$ at a time:
{{begin-eqn}}
{{eqn | l = s_1
| r = \sum_{i \mathop = 1}^n \map \tan {\... | First we note:
{{begin-eqn}}
{{eqn | l = \cos \sum_j \theta_j + i \sin \sum_j \theta_j
| r = \prod_j \paren {\cos \theta_j + i \sin \theta_j}
| c = Product of Complex Numbers in Polar Form
}}
{{eqn | r = \prod_j \cos \theta_j \prod_j \paren {1 + i \tan \theta_j}
| c =
}}
{{eqn | r = \prod_j \cos \the... | Let $\theta_1, \theta_2, \ldots, \theta_n$ be [[Definition:Angle|angles]].
For all $k \in \set {1, 2, 3, \ldots, n}$, let $s_k$ be defined as the [[Definition:Real Addition|sum]] of the [[Definition:Real Multiplication|product]] of $\map \tan {\theta_1}, \map \tan {\theta_2}, \ldots, \map \tan {\theta_n}$ taken $k$ at... | First we note:
{{begin-eqn}}
{{eqn | l = \cos \sum_j \theta_j + i \sin \sum_j \theta_j
| r = \prod_j \paren {\cos \theta_j + i \sin \theta_j}
| c = [[Product of Complex Numbers in Polar Form]]
}}
{{eqn | r = \prod_j \cos \theta_j \prod_j \paren {1 + i \tan \theta_j}
| c =
}}
{{eqn | r = \prod_j \cos ... | Tangent of Sum of Series of Angles/Proof 2 | https://proofwiki.org/wiki/Tangent_of_Sum_of_Series_of_Angles | https://proofwiki.org/wiki/Tangent_of_Sum_of_Series_of_Angles/Proof_2 | [
"Tangent of Sum of Series of Angles",
"Tangent Function"
] | [
"Definition:Angle",
"Definition:Addition/Real Numbers",
"Definition:Multiplication/Real Numbers"
] | [
"Product of Complex Numbers in Polar Form"
] |
proofwiki-20084 | Cotangent of Sum of Three Angles | :$\map \cot {A + B + C} = \dfrac {\cot A + \cot B + \cot C - \cot A \cot B \cot C} {1 - \cot B \cot C - \cot C \cot A - \cot A \cot B}$ | {{begin-eqn}}
{{eqn | l = \map \cos {A + B + C}
| r = \cos A \cos B \cos C - \sin A \sin B \cos C - \sin A \cos B \sin C - \cos A \sin B \sin C
| c = Cosine of Sum of Three Angles
}}
{{eqn | l = \map \sin {A + B + C}
| r = \sin A \cos B \cos C + \cos A \sin B \cos C + \cos A \cos B \sin C - \sin A \si... | :$\map \cot {A + B + C} = \dfrac {\cot A + \cot B + \cot C - \cot A \cot B \cot C} {1 - \cot B \cot C - \cot C \cot A - \cot A \cot B}$ | {{begin-eqn}}
{{eqn | l = \map \cos {A + B + C}
| r = \cos A \cos B \cos C - \sin A \sin B \cos C - \sin A \cos B \sin C - \cos A \sin B \sin C
| c = [[Cosine of Sum of Three Angles]]
}}
{{eqn | l = \map \sin {A + B + C}
| r = \sin A \cos B \cos C + \cos A \sin B \cos C + \cos A \cos B \sin C - \sin A... | Cotangent of Sum of Three Angles | https://proofwiki.org/wiki/Cotangent_of_Sum_of_Three_Angles | https://proofwiki.org/wiki/Cotangent_of_Sum_of_Three_Angles | [
"Cotangent of Sum",
"Cotangent Function",
"Trigonometric Addition Formulas"
] | [] | [
"Cosine of Sum of Three Angles",
"Sine of Sum of Three Angles",
"Cotangent is Cosine divided by Sine",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Cotangent is Cosine divided by Sine"
] |
proofwiki-20085 | Sum of Cotangents of Half Angles in Triangle | Let $\triangle ABC$ be a triangle.
Then:
:$\cot \dfrac A 2 + \cot \dfrac B 2 + \cot \dfrac C 2 = \cot \dfrac A 2 \cot \dfrac B 2 \cot \dfrac C 2$ | {{begin-eqn}}
{{eqn | l = \dfrac {\cot \frac A 2 + \cot \frac B 2 + \cot \frac C 2 - \cot \frac A 2 \cot \frac B 2 \cot \frac C 2} {1 - \cot \frac B 2 \cot \frac C 2 - \cot \frac C 2 \cot \frac A 2 - \cot \frac A 2 \cot \frac B 2}
| r = \map \cot {\frac A 2 + \frac B 2 + \frac C 2}
| c = Cotangent of Sum of... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Then:
:$\cot \dfrac A 2 + \cot \dfrac B 2 + \cot \dfrac C 2 = \cot \dfrac A 2 \cot \dfrac B 2 \cot \dfrac C 2$ | {{begin-eqn}}
{{eqn | l = \dfrac {\cot \frac A 2 + \cot \frac B 2 + \cot \frac C 2 - \cot \frac A 2 \cot \frac B 2 \cot \frac C 2} {1 - \cot \frac B 2 \cot \frac C 2 - \cot \frac C 2 \cot \frac A 2 - \cot \frac A 2 \cot \frac B 2}
| r = \map \cot {\frac A 2 + \frac B 2 + \frac C 2}
| c = [[Cotangent of Sum ... | Sum of Cotangents of Half Angles in Triangle/Proof | https://proofwiki.org/wiki/Sum_of_Cotangents_of_Half_Angles_in_Triangle | https://proofwiki.org/wiki/Sum_of_Cotangents_of_Half_Angles_in_Triangle/Proof | [
"Sum of Cotangents of Half Angles in Triangle",
"Triangles",
"Cotangent Function"
] | [
"Definition:Triangle (Geometry)"
] | [
"Cotangent of Sum of Three Angles",
"Sum of Angles of Triangle equals Two Right Angles",
"Cotangent of Right Angle"
] |
proofwiki-20086 | Sum of Sines of Angles in Triangle | Let $\triangle ABC$ be a triangle.
Then:
:$\sin A + \sin B + \sin C = 4 \cos \dfrac A 2 \cos \dfrac B 2 \cos \dfrac C 2$ | First we note that:
{{begin-eqn}}
{{eqn | l = A + B + C
| r = 180 \degrees
| c = Sum of Angles of Triangle equals Two Right Angles
}}
{{eqn | ll= \leadsto
| l = \dfrac A 2 + \dfrac {B + C} 2
| r = 90 \degrees
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \dfrac {B + C} 2
|... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Then:
:$\sin A + \sin B + \sin C = 4 \cos \dfrac A 2 \cos \dfrac B 2 \cos \dfrac C 2$ | First we note that:
{{begin-eqn}}
{{eqn | l = A + B + C
| r = 180 \degrees
| c = [[Sum of Angles of Triangle equals Two Right Angles]]
}}
{{eqn | ll= \leadsto
| l = \dfrac A 2 + \dfrac {B + C} 2
| r = 90 \degrees
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \dfrac {B + C} 2
... | Sum of Sines of Angles in Triangle/Proof | https://proofwiki.org/wiki/Sum_of_Sines_of_Angles_in_Triangle | https://proofwiki.org/wiki/Sum_of_Sines_of_Angles_in_Triangle/Proof | [
"Sum of Sines of Angles in Triangle",
"Triangles",
"Sine Function"
] | [
"Definition:Triangle (Geometry)"
] | [
"Sum of Angles of Triangle equals Two Right Angles",
"Prosthaphaeresis Formulas/Sine plus Sine",
"Double Angle Formulas/Sine",
"Sine of Complement equals Cosine",
"Werner Formulas/Cosine by Cosine",
"Cosine of Complement equals Sine"
] |
proofwiki-20087 | Sum of Cosines of Angles in Triangle | Let $\triangle ABC$ be a triangle.
Then:
:$\cos A + \cos B + \cos C = 1 - 4 \sin \dfrac A 2 \sin \dfrac B 2 \sin \dfrac C 2$ | First we note that:
{{begin-eqn}}
{{eqn | l = A + B + C
| r = 180 \degrees
| c = Sum of Angles of Triangle equals Two Right Angles
}}
{{eqn | ll= \leadsto
| l = \dfrac {A + B} 2 + \dfrac C 2
| r = 90 \degrees
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \dfrac {A + B} 2
|... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Then:
:$\cos A + \cos B + \cos C = 1 - 4 \sin \dfrac A 2 \sin \dfrac B 2 \sin \dfrac C 2$ | First we note that:
{{begin-eqn}}
{{eqn | l = A + B + C
| r = 180 \degrees
| c = [[Sum of Angles of Triangle equals Two Right Angles]]
}}
{{eqn | ll= \leadsto
| l = \dfrac {A + B} 2 + \dfrac C 2
| r = 90 \degrees
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \dfrac {A + B} 2
... | Sum of Cosines of Angles in Triangle/Proof | https://proofwiki.org/wiki/Sum_of_Cosines_of_Angles_in_Triangle | https://proofwiki.org/wiki/Sum_of_Cosines_of_Angles_in_Triangle/Proof | [
"Sum of Cosines of Angles in Triangle",
"Triangles",
"Cosine Function"
] | [
"Definition:Triangle (Geometry)"
] | [
"Sum of Angles of Triangle equals Two Right Angles",
"Prosthaphaeresis Formulas/Cosine plus Cosine",
"Cosine of Complement equals Sine",
"Sine of Complement equals Cosine",
"Prosthaphaeresis Formulas/Cosine minus Cosine"
] |
proofwiki-20088 | Solutions of sin x equals sin a | Let $\alpha \in \closedint {-1} 1$ be fixed.
Let:
:$(1): \quad \sin x = \sin \alpha$
The solution set of $(1)$ is:
:$\set {x \in \R: \forall n \in \Z: x = n \pi + \paren {-1}^n \alpha}$ | From Sine of Supplementary Angle:
:$\map \sin {\pi - x} = \sin x$
and so from Real Sine Function is Periodic:
{{begin-eqn}}
{{eqn | l = x
| r = 2 n \pi + a
| c =
}}
{{eqn | l = x
| r = \paren {2 n + 1} \pi - a
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = n \pi + \paren {-1}^n x
... | Let $\alpha \in \closedint {-1} 1$ be fixed.
Let:
:$(1): \quad \sin x = \sin \alpha$
The [[Definition:Solution Set|solution set]] of $(1)$ is:
:$\set {x \in \R: \forall n \in \Z: x = n \pi + \paren {-1}^n \alpha}$ | From [[Sine of Supplementary Angle]]:
:$\map \sin {\pi - x} = \sin x$
and so from [[Real Sine Function is Periodic]]:
{{begin-eqn}}
{{eqn | l = x
| r = 2 n \pi + a
| c =
}}
{{eqn | l = x
| r = \paren {2 n + 1} \pi - a
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = n \pi + \paren {-1... | Solutions of sin x equals sin a/Proof | https://proofwiki.org/wiki/Solutions_of_sin_x_equals_sin_a | https://proofwiki.org/wiki/Solutions_of_sin_x_equals_sin_a/Proof | [
"Solutions of sin x equals sin a",
"Sine Function"
] | [
"Definition:Fiber of Truth"
] | [
"Sine of Supplementary Angle",
"Sine and Cosine are Periodic on Reals/Sine"
] |
proofwiki-20089 | Solutions of cos x equals cos a | Let $\alpha \in \closedint {-1} 1$ be fixed.
Let:
:$(1): \quad \cos x = \cos \alpha$
The solution set of $(1)$ is:
:$\set {x \in \R: \forall n \in \Z: x = 2 n \pi \pm \alpha}$ | From Cosine of Supplementary Angle:
:$\map \cos {\pi - x} = -\cos x$
and so from Real Cosine Function is Periodic:
{{begin-eqn}}
{{eqn | l = x
| r = 2 n \pi \pm a
| c =
}}
{{end-eqn}}
{{qed}} | Let $\alpha \in \closedint {-1} 1$ be fixed.
Let:
:$(1): \quad \cos x = \cos \alpha$
The [[Definition:Solution Set|solution set]] of $(1)$ is:
:$\set {x \in \R: \forall n \in \Z: x = 2 n \pi \pm \alpha}$ | From [[Cosine of Supplementary Angle]]:
:$\map \cos {\pi - x} = -\cos x$
and so from [[Real Cosine Function is Periodic]]:
{{begin-eqn}}
{{eqn | l = x
| r = 2 n \pi \pm a
| c =
}}
{{end-eqn}}
{{qed}} | Solutions of cos x equals cos a/Proof | https://proofwiki.org/wiki/Solutions_of_cos_x_equals_cos_a | https://proofwiki.org/wiki/Solutions_of_cos_x_equals_cos_a/Proof | [
"Solutions of cos x equals cos a",
"Cosine Function"
] | [
"Definition:Fiber of Truth"
] | [
"Cosine of Supplementary Angle",
"Sine and Cosine are Periodic on Reals/Cosine"
] |
proofwiki-20090 | Solutions of tan x equals tan a | Let $\alpha \in \R$ be fixed.
Let:
:$(1): \quad \tan x = \tan \alpha$
The solution set of $(1)$ is:
:$\set {x \in \R: \forall n \in \Z: x = n \pi + \alpha}$ | From Tangent Function is Periodic on Reals:
:$\map \tan {\pi + x} = \tan x$
Hence:
{{begin-eqn}}
{{eqn | l = x
| r = n \pi + a
| c =
}}
{{end-eqn}}
{{qed}} | Let $\alpha \in \R$ be fixed.
Let:
:$(1): \quad \tan x = \tan \alpha$
The [[Definition:Solution Set|solution set]] of $(1)$ is:
:$\set {x \in \R: \forall n \in \Z: x = n \pi + \alpha}$ | From [[Tangent Function is Periodic on Reals]]:
:$\map \tan {\pi + x} = \tan x$
Hence:
{{begin-eqn}}
{{eqn | l = x
| r = n \pi + a
| c =
}}
{{end-eqn}}
{{qed}} | Solutions of tan x equals tan a/Proof | https://proofwiki.org/wiki/Solutions_of_tan_x_equals_tan_a | https://proofwiki.org/wiki/Solutions_of_tan_x_equals_tan_a/Proof | [
"Solutions of tan x equals tan a",
"Tangent Function"
] | [
"Definition:Fiber of Truth"
] | [
"Tangent Function is Periodic on Reals"
] |
proofwiki-20091 | Quotient Mapping is Linear Transformation | Let $K$ be a field.
Let $V$ be a vector space over $K$.
Let $M$ be a subspace of $V$.
Let $V / M$ be the quotient vector space.
Let $Q: V \to V / M$ be the quotient mapping.
Then $Q$ is a linear transformation. | Let $\lambda, \mu \in K$ and $x, y \in V$.
From Quotient Vector Space is Vector Space, it follows that $V / M$ is a vector space.
Using the definition of vector addition and scalar multiplication for $V/M$, we have:
{{begin-eqn}}
{{eqn | l = \map Q {\lambda x + \mu y}
| r = \paren {\lambda x + \mu y} + M
| c... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $V$ be a [[Definition:Vector Space|vector space]] over $K$.
Let $M$ be a [[Definition:Vector Subspace|subspace]] of $V$.
Let $V / M$ be the [[Definition:Quotient Vector Space|quotient vector space]].
Let $Q: V \to V / M$ be the [[Definition:Quotient Ma... | Let $\lambda, \mu \in K$ and $x, y \in V$.
From [[Quotient Vector Space is Vector Space]], it follows that $V / M$ is a [[Definition:Vector Space|vector space]].
Using the definition of [[Definition:Vector Addition on Vector Space|vector addition]] and [[Definition:Scalar Multiplication on Vector Space|scalar multipl... | Quotient Mapping is Linear Transformation | https://proofwiki.org/wiki/Quotient_Mapping_is_Linear_Transformation | https://proofwiki.org/wiki/Quotient_Mapping_is_Linear_Transformation | [
"Linear Transformations",
"Quotient Vector Spaces"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Vector Subspace",
"Definition:Quotient Vector Space",
"Definition:Quotient Mapping",
"Definition:Linear Transformation"
] | [
"Quotient Vector Space is Vector Space",
"Definition:Vector Space",
"Definition:Vector Addition/Vector Space",
"Definition:Scalar Multiplication/Vector Space",
"Definition:Quotient Mapping",
"Definition:Quotient Mapping"
] |
proofwiki-20092 | Kernel of Quotient Mapping | Let $V$ be a vector space.
Let $M$ be a subspace of $V$.
Let $Q: V \to V / M$ be the quotient mapping.
Then:
:$\ker Q = M$
where $\ker Q$ is the kernel of $Q$. | For $v \in V$, we have that:
:$v \in \ker Q$
{{iff}}
:$v + M = 0 + M$
That is, {{iff}}:
:$v \in M$
Hence:
:$\ker Q = M$
{{qed}} | Let $V$ be a [[Definition:Vector Space|vector space]].
Let $M$ be a [[Definition:Vector Subspace|subspace]] of $V$.
Let $Q: V \to V / M$ be the [[Definition:Quotient Mapping|quotient mapping]].
Then:
:$\ker Q = M$
where $\ker Q$ is the [[Definition:Kernel of Linear Transformation on Vector Space|kernel]] of $Q$. | For $v \in V$, we have that:
:$v \in \ker Q$
{{iff}}
:$v + M = 0 + M$
That is, {{iff}}:
:$v \in M$
Hence:
:$\ker Q = M$
{{qed}} | Kernel of Quotient Mapping | https://proofwiki.org/wiki/Kernel_of_Quotient_Mapping | https://proofwiki.org/wiki/Kernel_of_Quotient_Mapping | [
"Quotient Vector Spaces",
"Quotient Mappings"
] | [
"Definition:Vector Space",
"Definition:Vector Subspace",
"Definition:Quotient Mapping",
"Definition:Kernel of Linear Transformation/Vector Space"
] | [] |
proofwiki-20093 | Cosine of Half Angle in Triangle | Let $\triangle ABC$ be a triangle whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.
Let $s$ denote the semiperimeter of $\triangle ABC$.
Then:
:$\cos \dfrac C 2 = \sqrt {\dfrac {s \paren {s - c} } {a b} }$ | {{begin-eqn}}
{{eqn | l = \cos C
| r = \dfrac {a^2 + b^2 - c^2} {2 a b}
| c = Law of Cosines
}}
{{eqn | ll= \leadsto
| l = 2 \cos^2 \dfrac C 2 - 1
| r = \dfrac {a^2 + b^2 - c^2} {2 a b}
| c = {{Corollary|Double Angle Formula for Cosine|1}}
}}
{{eqn | ll= \leadsto
| l = 2 \cos^2 \dfra... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.
Let $s$ denote the [[Definition:Semiperimeter|semiperimeter]] of $\triangle ABC$.
Then:
:$\cos \dfrac C 2 = \sqrt {\dfrac {s \paren {s - c} } {a b... | {{begin-eqn}}
{{eqn | l = \cos C
| r = \dfrac {a^2 + b^2 - c^2} {2 a b}
| c = [[Law of Cosines]]
}}
{{eqn | ll= \leadsto
| l = 2 \cos^2 \dfrac C 2 - 1
| r = \dfrac {a^2 + b^2 - c^2} {2 a b}
| c = {{Corollary|Double Angle Formula for Cosine|1}}
}}
{{eqn | ll= \leadsto
| l = 2 \cos^2 \... | Cosine of Half Angle in Triangle/Proof | https://proofwiki.org/wiki/Cosine_of_Half_Angle_in_Triangle | https://proofwiki.org/wiki/Cosine_of_Half_Angle_in_Triangle/Proof | [
"Cosine of Half Angle in Triangle",
"Triangles",
"Cosine Function"
] | [
"Definition:Triangle (Geometry)",
"Definition:Semiperimeter"
] | [
"Law of Cosines",
"Definition:Common Denominator",
"Square of Sum",
"Difference of Two Squares",
"Definition:Square Root"
] |
proofwiki-20094 | Sine of Half Angle in Triangle | Let $\triangle ABC$ be a triangle whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.
Let $s$ denote the semiperimeter of $\triangle ABC$.
Then:
:$\sin \dfrac C 2 = \sqrt {\dfrac {\paren {s - a} \paren {s - b} } {a b} }$ | {{begin-eqn}}
{{eqn | l = \cos C
| r = \dfrac {a^2 + b^2 - c^2} {2 a b}
| c = Law of Cosines
}}
{{eqn | ll= \leadsto
| l = 1 - 2 \sin^2 \dfrac C 2
| r = \dfrac {a^2 + b^2 - c^2} {2 a b}
| c = {{Corollary|Double Angle Formula for Cosine|2}}
}}
{{eqn | ll= \leadsto
| l = 2 \sin^2 \dfra... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.
Let $s$ denote the [[Definition:Semiperimeter|semiperimeter]] of $\triangle ABC$.
Then:
:$\sin \dfrac C 2 = \sqrt {\dfrac {\paren {s - a} \paren {... | {{begin-eqn}}
{{eqn | l = \cos C
| r = \dfrac {a^2 + b^2 - c^2} {2 a b}
| c = [[Law of Cosines]]
}}
{{eqn | ll= \leadsto
| l = 1 - 2 \sin^2 \dfrac C 2
| r = \dfrac {a^2 + b^2 - c^2} {2 a b}
| c = {{Corollary|Double Angle Formula for Cosine|2}}
}}
{{eqn | ll= \leadsto
| l = 2 \sin^2 \... | Sine of Half Angle in Triangle/Proof | https://proofwiki.org/wiki/Sine_of_Half_Angle_in_Triangle | https://proofwiki.org/wiki/Sine_of_Half_Angle_in_Triangle/Proof | [
"Sine of Half Angle in Triangle",
"Triangles",
"Sine Function"
] | [
"Definition:Triangle (Geometry)",
"Definition:Semiperimeter"
] | [
"Law of Cosines",
"Definition:Common Denominator",
"Square of Sum",
"Difference of Two Squares",
"Definition:Square Root"
] |
proofwiki-20095 | Tangent of Half Angle in Triangle | Let $\triangle ABC$ be a triangle whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.
Let $s$ denote the semiperimeter of $\triangle ABC$:
:$s = \dfrac {a + b + c} 2$
Then:
:$\tan \dfrac C 2 = \sqrt {\dfrac {\paren {s - a} \paren {s - b} } {s \paren {s - c} } }$ | {{begin-eqn}}
{{eqn | l = \tan \dfrac C 2
| r = \dfrac {\sin \dfrac C 2} {\cos \dfrac C 2}
| c =
}}
{{eqn | r = \dfrac {\sqrt {\dfrac {\paren {s - a} \paren {s - b} } {a b} } } {\sqrt {\dfrac {s \paren {s - c} } {a b} } }
| c = Sine of Half Angle in Triangle, Cosine of Half Angle in Triangle
}}
{{eqn... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.
Let $s$ denote the [[Definition:Semiperimeter|semiperimeter]] of $\triangle ABC$:
:$s = \dfrac {a + b + c} 2$
Then:
:$\tan \dfrac C 2 = \sqrt {\dfr... | {{begin-eqn}}
{{eqn | l = \tan \dfrac C 2
| r = \dfrac {\sin \dfrac C 2} {\cos \dfrac C 2}
| c =
}}
{{eqn | r = \dfrac {\sqrt {\dfrac {\paren {s - a} \paren {s - b} } {a b} } } {\sqrt {\dfrac {s \paren {s - c} } {a b} } }
| c = [[Sine of Half Angle in Triangle]], [[Cosine of Half Angle in Triangle]]
... | Tangent of Half Angle in Triangle/Proof | https://proofwiki.org/wiki/Tangent_of_Half_Angle_in_Triangle | https://proofwiki.org/wiki/Tangent_of_Half_Angle_in_Triangle/Proof | [
"Tangent of Half Angle in Triangle",
"Triangles",
"Tangent Function"
] | [
"Definition:Triangle (Geometry)",
"Definition:Semiperimeter"
] | [
"Sine of Half Angle in Triangle",
"Cosine of Half Angle in Triangle"
] |
proofwiki-20096 | Condition for Mapping from Quotient Vector Space to be Well-Defined | Let $V, W$ be vector spaces.
Let $T: V \to W$ be a linear transformation.
Let $M$ be a subspace of $V$.
Let $V / M$ be the quotient vector space of $V$ by $M$.
Let $Q_M: V \to V / M$ be the associated quotient mapping.
Then:
:there exists a linear transformation $L: V / M \to W$ such that $L \circ Q_M = T$
{{iff}}:
:$M... | By Condition for Mapping from Quotient Set to be Well-Defined, it follows that:
:$L: V / M \to W$ exists
{{iff}}
:$\forall v, v' \in V: v + M = v' + M \implies T v = T v'$
Now $v + M = v' + M$ {{iff}} $v - v' \in M$.
We are given that $T$ is a linear transformation, so:
:$T v = T v' \iff T \paren{ v - v' } = 0$
In part... | Let $V, W$ be [[Definition:Vector Space|vector spaces]].
Let $T: V \to W$ be a [[Definition:Linear Transformation on Vector Space|linear transformation]].
Let $M$ be a [[Definition:Vector Subspace|subspace]] of $V$.
Let $V / M$ be the [[Definition:Quotient Vector Space|quotient vector space]] of $V$ by $M$.
Let $Q... | By [[Condition for Mapping from Quotient Set to be Well-Defined]], it follows that:
:$L: V / M \to W$ exists
{{iff}}
:$\forall v, v' \in V: v + M = v' + M \implies T v = T v'$
Now $v + M = v' + M$ {{iff}} $v - v' \in M$.
We are [[Definition:Given|given]] that $T$ is a [[Definition:Linear Transformation on Vector ... | Condition for Mapping from Quotient Vector Space to be Well-Defined | https://proofwiki.org/wiki/Condition_for_Mapping_from_Quotient_Vector_Space_to_be_Well-Defined | https://proofwiki.org/wiki/Condition_for_Mapping_from_Quotient_Vector_Space_to_be_Well-Defined | [
"Quotient Vector Spaces"
] | [
"Definition:Vector Space",
"Definition:Linear Transformation/Vector Space",
"Definition:Vector Subspace",
"Definition:Quotient Vector Space",
"Definition:Quotient Mapping",
"Definition:Linear Transformation"
] | [
"Condition for Mapping from Quotient Set to be Well-Defined",
"Definition:Given",
"Definition:Linear Transformation/Vector Space"
] |
proofwiki-20097 | Distance from Vertex of Triangle to Orthocenter | Let $\triangle ABC$ be a triangle with sides $a$, $b$ and $c$ opposite vertices $A$, $B$ and $C$ respectively.
Let $H$ be the orthocenter of $\triangle ABC$.
Then:
:$AH = 2 R \size {\cos A}$
where $R$ denotes the circumradius of $\triangle ABC$. | :400px
We construct the circumcircle of $\triangle ABC$, whose circumcenter is $K$ and whose circumradius is $R$.
We construct the orthocenter $H$ of $\triangle ABC$ as the intersection of the altitudes $AD$ and $BE$.
First we note the following:
{{begin-eqn}}
{{eqn | l = \angle ABH
| r = \angle ABE
| c =
... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] with [[Definition:Side of Polygon|sides]] $a$, $b$ and $c$ [[Definition:Opposite (in Triangle)|opposite]] [[Definition:Vertex of Polygon|vertices]] $A$, $B$ and $C$ respectively.
Let $H$ be the [[Definition:Orthocenter|orthocenter]] of $\triangle ABC... | :[[File:Vertex-to-Orthocenter.png|400px]]
We construct the [[Definition:Circumcircle of Triangle|circumcircle]] of $\triangle ABC$, whose [[Definition:Circumcenter of Triangle|circumcenter]] is $K$ and whose [[Definition:Circumradius of Triangle|circumradius]] is $R$.
We construct the [[Definition:Orthocenter|orthoce... | Distance from Vertex of Triangle to Orthocenter/Proof | https://proofwiki.org/wiki/Distance_from_Vertex_of_Triangle_to_Orthocenter | https://proofwiki.org/wiki/Distance_from_Vertex_of_Triangle_to_Orthocenter/Proof | [
"Distance from Vertex of Triangle to Orthocenter",
"Orthocenters of Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)/Opposite",
"Definition:Polygon/Vertex",
"Definition:Orthocenter",
"Definition:Circumcircle of Triangle/Circumradius"
] | [
"File:Vertex-to-Orthocenter.png",
"Definition:Circumcircle of Triangle",
"Definition:Circumcircle of Triangle/Circumcenter",
"Definition:Circumcircle of Triangle/Circumradius",
"Definition:Orthocenter",
"Definition:Intersection (Geometry)",
"Definition:Altitude of Triangle",
"Sum of Angles of Triangle... |
proofwiki-20098 | Gauss Lemma for Riemannian Manifolds | Let $\struct {M, g}$ be a Riemannian manifold.
Let $U = \map {\exp_p} {\map {B_\epsilon} 0 }$ be a geodesic ball centered at $p \in M$.
Let $\partial_r$ be the radial vector field on $U \setminus \set p$, where $\setminus$ denotes the set difference.
Then $\partial_r$ is a unit vector field orthogonal to the geodesic s... | {{ProofWanted}}
{{Namedfor|Carl Friedrich Gauss|cat = Gauss}} | Let $\struct {M, g}$ be a [[Definition:Riemannian Manifold|Riemannian manifold]].
Let $U = \map {\exp_p} {\map {B_\epsilon} 0 }$ be a [[Definition:Open Geodesic Ball in Riemannian Manifold|geodesic ball]] centered at $p \in M$.
Let $\partial_r$ be the [[Definition:Radial Vector Field|radial vector field]] on $U \setm... | {{ProofWanted}}
{{Namedfor|Carl Friedrich Gauss|cat = Gauss}} | Gauss Lemma for Riemannian Manifolds | https://proofwiki.org/wiki/Gauss_Lemma_for_Riemannian_Manifolds | https://proofwiki.org/wiki/Gauss_Lemma_for_Riemannian_Manifolds | [
"Riemannian Manifolds"
] | [
"Definition:Riemannian Manifold",
"Definition:Open Geodesic Ball in Riemannian Manifold",
"Definition:Radial Vector Field",
"Definition:Set Difference",
"Definition:Unit Vector",
"Definition:Vector Field",
"Definition:Orthogonal to Surface",
"Definition:Geodesic Sphere in Riemannian Manifold"
] | [] |
proofwiki-20099 | Angles of Orthic Triangle of Acute Triangle | Let $\triangle ABC$ be an acute triangle with sides $a$, $b$ and $c$ opposite vertices $A$, $B$ and $C$ respectively.
Let $\triangle DEF$ be the orthic triangle of $\triangle ABC$.
Then the angles of $\triangle DEF$ are $180 \degrees - 2 A$, $180 \degrees - 2 B$ and $180 \degrees - 2 C$. | :420px
Let $H$ be the orthocenter of $\triangle ABC$.
The quadrilateral $\Box FHDB$ is cyclic.
That is, $\Box FHDB$ can be circumscribed.
Hence:
{{begin-eqn}}
{{eqn | l = \angle HDF
| r = \angle HBF
| c = Angles in Same Segment of Circle are Equal
}}
{{eqn | n = 1
| r = 90 \degrees - A
| c = as ... | Let $\triangle ABC$ be an [[Definition:Acute Triangle|acute triangle]] with [[Definition:Side of Polygon|sides]] $a$, $b$ and $c$ [[Definition:Opposite (in Triangle)|opposite]] [[Definition:Vertex of Polygon|vertices]] $A$, $B$ and $C$ respectively.
Let $\triangle DEF$ be the [[Definition:Orthic Triangle|orthic triang... | :[[File:Orthic-Triangle.png|420px]]
Let $H$ be the [[Definition:Orthocenter|orthocenter]] of $\triangle ABC$.
The [[Definition:Quadrilateral|quadrilateral]] $\Box FHDB$ is [[Definition:Cyclic Quadrilateral|cyclic]].
That is, $\Box FHDB$ can be [[Definition:Circle Circumscribed around Polygon|circumscribed]].
Hence... | Angles of Orthic Triangle of Acute Triangle/Proof | https://proofwiki.org/wiki/Angles_of_Orthic_Triangle_of_Acute_Triangle | https://proofwiki.org/wiki/Angles_of_Orthic_Triangle_of_Acute_Triangle/Proof | [
"Angles of Orthic Triangle of Acute Triangle",
"Orthic Triangles",
"Acute Triangles"
] | [
"Definition:Triangle (Geometry)/Acute",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)/Opposite",
"Definition:Polygon/Vertex",
"Definition:Orthic Triangle",
"Definition:Polygon/Internal Angle"
] | [
"File:Orthic-Triangle.png",
"Definition:Orthocenter",
"Definition:Quadrilateral",
"Definition:Cyclic Quadrilateral",
"Definition:Circumscribe/Circle around Polygon",
"Angles in Same Segment of Circle are Equal",
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Cyclic Quadrilateral",
"Angle... |
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