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-17900 | Combination Theorem for Continuous Mappings/Metric Space/Product Rule | :$f g$ is continuous on $M$. | By definition of continuous:
:$\forall a \in M: \ds \lim_{x \mathop \to a} \map f x = \map f a$
:$\forall a \in M: \ds \lim_{x \mathop \to a} \map g x = \map g a$
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{x \mathop \to a} \map f x = l$
:$\ds \lim_{x \mathop \to a} \map g x = m$
From the Product Rule for... | :$f g$ is [[Definition:Continuous Mapping (Metric Space)|continuous]] on $M$. | By definition of [[Definition:Continuous Mapping (Metric Space)|continuous]]:
:$\forall a \in M: \ds \lim_{x \mathop \to a} \map f x = \map f a$
:$\forall a \in M: \ds \lim_{x \mathop \to a} \map g x = \map g a$
Let $f$ and $g$ tend to the following [[Definition:Limit of Real Function|limits]]:
:$\ds \lim_{x \mathop ... | Combination Theorem for Continuous Mappings/Metric Space/Product Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Metric_Space/Product_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Metric_Space/Product_Rule | [
"Combination Theorem for Continuous Mappings"
] | [
"Definition:Continuous Mapping (Metric Space)"
] | [
"Definition:Continuous Mapping (Metric Space)",
"Definition:Limit of Real Function",
"Combination Theorem for Limits of Functions/Real/Product Rule",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Continuous Mapping (Metric Space)"
] |
proofwiki-17901 | Combination Theorem for Continuous Mappings/Metric Space/Quotient Rule | :$\dfrac f g$ is continuous on $M \setminus \set {x \in A: \map g x = 0}$.
that is, on all the points $x$ of $A$ where $\map g x \ne 0$. | By definition of continuous:
:$\forall a \in M: \ds \lim_{x \mathop \to a} \map f x = \map f a$
:$\forall a \in M: \ds \lim_{x \mathop \to a} \map g x = \map g a$
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{x \mathop \to a} \map f x = l$
:$\ds \lim_{x \mathop \to a} \map g x = m$
From the Product Rule for... | :$\dfrac f g$ is [[Definition:Continuous Mapping (Metric Space)|continuous]] on $M \setminus \set {x \in A: \map g x = 0}$.
that is, on all the points $x$ of $A$ where $\map g x \ne 0$. | By definition of [[Definition:Continuous Mapping (Metric Space)|continuous]]:
:$\forall a \in M: \ds \lim_{x \mathop \to a} \map f x = \map f a$
:$\forall a \in M: \ds \lim_{x \mathop \to a} \map g x = \map g a$
Let $f$ and $g$ tend to the following [[Definition:Limit of Real Function|limits]]:
:$\ds \lim_{x \mathop ... | Combination Theorem for Continuous Mappings/Metric Space/Quotient Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Metric_Space/Quotient_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Metric_Space/Quotient_Rule | [
"Combination Theorem for Continuous Mappings"
] | [
"Definition:Continuous Mapping (Metric Space)"
] | [
"Definition:Continuous Mapping (Metric Space)",
"Definition:Limit of Real Function",
"Combination Theorem for Limits of Functions/Real/Product Rule",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Continuous Mapping (Metric Space)"
] |
proofwiki-17902 | Projection from Metric Space Product with P-Product Metric is Continuous | Let $M_1 = \struct {A_1, d}$ and $M_2 = \struct {A_2, d'}$ be metric spaces.
Let $\AA := A_1 \times A_2$ be the cartesian product of $A_1$ and $A_2$.
Let $\MM = \struct {\AA, d_p}$ denote the metric space on $\AA$ where $d_p: \AA \to \R$ is the $p$-product metric on $\AA$:
:$\map {d_p} {x, y} := \paren {\paren {\map d ... | We want to show that, for all $a \in \AA$:
:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall z \in \AA: \map {d_p} {z, a} < \delta \implies \map d {\map {\pr_1} z, \map {\pr_1} a} < \epsilon$
and:
:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall z \in \AA: \map {d_p} {z, a} < \delta... | Let $M_1 = \struct {A_1, d}$ and $M_2 = \struct {A_2, d'}$ be [[Definition:Metric Space|metric spaces]].
Let $\AA := A_1 \times A_2$ be the [[Definition:Cartesian Product|cartesian product]] of $A_1$ and $A_2$.
Let $\MM = \struct {\AA, d_p}$ denote the [[Definition:Metric Space|metric space]] on $\AA$ where $d_p: \A... | We want to show that, for all $a \in \AA$:
:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall z \in \AA: \map {d_p} {z, a} < \delta \implies \map d {\map {\pr_1} z, \map {\pr_1} a} < \epsilon$
and:
:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall z \in \AA: \map {d_p} {z, a} < \delt... | Projection from Metric Space Product with P-Product Metric is Continuous | https://proofwiki.org/wiki/Projection_from_Metric_Space_Product_with_P-Product_Metric_is_Continuous | https://proofwiki.org/wiki/Projection_from_Metric_Space_Product_with_P-Product_Metric_is_Continuous | [
"Projections",
"P-Product Metrics",
"Continuous Mappings on Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Cartesian Product",
"Definition:Metric Space",
"Definition:P-Product Metric",
"Definition:Projection (Mapping Theory)/First Projection",
"Definition:Projection (Mapping Theory)/Second Projection",
"Definition:Continuous Mapping (Metric Space)"
] | [
"Definition:Continuous Mapping (Metric Space)",
"Category:Projections",
"Category:P-Product Metrics",
"Category:Continuous Mappings on Metric Spaces"
] |
proofwiki-17903 | Canonical Injection into Metric Space Product with P-Product Metric is Continuous | Let $M_1 = \struct {A_1, d}$ and $M_2 = \struct {A_2, d'}$ be metric spaces.
Let $\AA := A_1 \times A_2$ be the cartesian product of $A_1$ and $A_2$.
Let $\MM = \struct {\AA, d_p}$ denote the metric space on $\AA$ where $d_p: \AA \to \R$ is one of the $p$-product metrics on $\AA$:
:$\map {d_p} {x, y} := \begin {cases}
... | Let $\pr_1: \MM \to M_1$ and $\pr_2: \MM \to T_2$ be the first and second projections from $\MM$ onto its factors.
From Projection from Metric Space Product with P-Product Metric is Continuous, both $\pr_1$ and $\pr_2$ are continuous
The result follows from Continuous Mapping to Product Space.
{{qed}} | Let $M_1 = \struct {A_1, d}$ and $M_2 = \struct {A_2, d'}$ be [[Definition:Metric Space|metric spaces]].
Let $\AA := A_1 \times A_2$ be the [[Definition:Cartesian Product|cartesian product]] of $A_1$ and $A_2$.
Let $\MM = \struct {\AA, d_p}$ denote the [[Definition:Metric Space|metric space]] on $\AA$ where $d_p: \A... | Let $\pr_1: \MM \to M_1$ and $\pr_2: \MM \to T_2$ be the [[Definition:Projection (Mapping Theory)|first and second projections]] from $\MM$ onto its [[Definition:Factor Space|factors]].
From [[Projection from Metric Space Product with P-Product Metric is Continuous]], both $\pr_1$ and $\pr_2$ are [[Definition:Continuo... | Canonical Injection into Metric Space Product with P-Product Metric is Continuous/Proof 1 | https://proofwiki.org/wiki/Canonical_Injection_into_Metric_Space_Product_with_P-Product_Metric_is_Continuous | https://proofwiki.org/wiki/Canonical_Injection_into_Metric_Space_Product_with_P-Product_Metric_is_Continuous/Proof_1 | [
"P-Product Metrics",
"Canonical Injections",
"Continuous Mappings on Metric Spaces",
"Canonical Injection into Metric Space Product with P-Product Metric is Continuous"
] | [
"Definition:Metric Space",
"Definition:Cartesian Product",
"Definition:Metric Space",
"Definition:P-Product Metric",
"Definition:Mapping",
"Definition:Mapping",
"Definition:Continuous Mapping (Metric Space)"
] | [
"Definition:Projection (Mapping Theory)",
"Definition:Product Topology/Factor Space",
"Projection from Metric Space Product with P-Product Metric is Continuous",
"Definition:Continuous Mapping (Metric Space)",
"Continuous Mapping to Product Space"
] |
proofwiki-17904 | Metric is Continous Mapping | Let $M = \struct {A, d}$ be a metric space.
Consider the distance function:
:$d: A \times A \to \R$
Then $\R$ is a continuous function. | Let $\epsilon > 0$.
Let $\tuple {x_1, x_2} \in A \times A$.
Let $\delta = \dfrac \epsilon 2$.
Then $U = \map {B_\delta} {x_1} \times \map {B_\delta} {x_2}$ is a neighborhood of $\tuple {x_1, x_2}$ in $X \times X$ such that:
:$d \sqbrk U \subseteq \openint {\map d {x_1, x_2} - \epsilon} {\map d {x_1, x_2} + \epsilon}$
T... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Consider the [[Definition:Distance Function|distance function]]:
:$d: A \times A \to \R$
Then $\R$ is a [[Definition:Continuous Real-Valued Function|continuous function]]. | Let $\epsilon > 0$.
Let $\tuple {x_1, x_2} \in A \times A$.
Let $\delta = \dfrac \epsilon 2$.
Then $U = \map {B_\delta} {x_1} \times \map {B_\delta} {x_2}$ is a [[Definition:Neighborhood|neighborhood]] of $\tuple {x_1, x_2}$ in $X \times X$ such that:
:$d \sqbrk U \subseteq \openint {\map d {x_1, x_2} - \epsilon} {\... | Metric is Continous Mapping | https://proofwiki.org/wiki/Metric_is_Continous_Mapping | https://proofwiki.org/wiki/Metric_is_Continous_Mapping | [
"Metric Spaces",
"Continuous Mappings"
] | [
"Definition:Metric Space",
"Definition:Distance Function",
"Definition:Continuous Real-Valued Vector Function"
] | [
"Definition:Neighborhood",
"Definition:Continuous Real-Valued Vector Function",
"Category:Metric Spaces",
"Category:Continuous Mappings"
] |
proofwiki-17905 | Taxicab Metric on Metric Space Product is Continuous | Let $M = \struct {A, d}$ be a metric space.
Let $\AA$ be the Cartesian product $A \times A$.
Let $d_1$ be the taxicab metric on $\AA$:
:$\ds \map {d_1} {x, y} = \sum_{i \mathop = 1}^n \map {d_{i'} } {x_{i'}, y_{i'} }$
for $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \AA$.
Then $d_1: \AA \... | Recall the definition of continuous mapping in this context.
Given metric spaces $M_X = \struct {X, d_X}$ and $M_Y = \struct {Y, d_Y}$, and a mapping $f : X \to Y$, we say that $f$ is $\struct {X, d_X} \to \struct {Y, d_Y}$-continuous {{iff}}:
:$\forall x_0 \in X: \forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $\AA$ be the [[Definition:Finite Cartesian Product|Cartesian product]] $A \times A$.
Let $d_1$ be the [[Definition:Taxicab Metric|taxicab metric]] on $\AA$:
:$\ds \map {d_1} {x, y} = \sum_{i \mathop = 1}^n \map {d_{i'} } {x_{i'}, y_{i'} }$
fo... | Recall the definition of [[Definition:Continuous Mapping (Metric Space)|continuous mapping]] in this context.
Given [[Definition:Metric Space|metric spaces]] $M_X = \struct {X, d_X}$ and $M_Y = \struct {Y, d_Y}$, and a [[Definition:Mapping|mapping]] $f : X \to Y$, we say that $f$ is [[Definition:Continuous Mapping (Me... | Taxicab Metric on Metric Space Product is Continuous | https://proofwiki.org/wiki/Taxicab_Metric_on_Metric_Space_Product_is_Continuous | https://proofwiki.org/wiki/Taxicab_Metric_on_Metric_Space_Product_is_Continuous | [
"Taxicab Metric",
"Continuous Mappings on Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Cartesian Product/Finite",
"Definition:Taxicab Metric",
"Definition:Continuous Real-Valued Vector Function"
] | [
"Definition:Continuous Mapping (Metric Space)",
"Definition:Metric Space",
"Definition:Mapping",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Continuous Mapping (Metric Space)"
] |
proofwiki-17906 | Space of Continuous on Closed Interval Real-Valued Functions with Supremum Norm forms Normed Vector Space | Let $I := \closedint a b$ be a closed real interval.
The space of continuous real-valued functions on $I$ with supremum norm forms a normed vector space. | We have that:
:Space of Continuous on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space
:Supremum Norm is Norm/Continuous on Closed Interval Real-Valued Function
By definition, $\struct {\map C I, \norm {\, \cdot \,}_\infty}$ is a normed vector space.
{... | Let $I := \closedint a b$ be a [[Definition:Closed Real Interval|closed real interval]].
The [[Definition:Space of Real-Valued Functions Continuous on Closed Interval|space of continuous real-valued functions on $I$]] with [[Definition:Supremum Norm|supremum norm]] forms a [[Definition:Normed Vector Space|normed vecto... | We have that:
:[[Space of Continuous on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space]]
:[[Supremum Norm is Norm/Continuous on Closed Interval Real-Valued Function]]
By definition, $\struct {\map C I, \norm {\, \cdot \,}_\infty}$ is a [[Definitio... | Space of Continuous on Closed Interval Real-Valued Functions with Supremum Norm forms Normed Vector Space | https://proofwiki.org/wiki/Space_of_Continuous_on_Closed_Interval_Real-Valued_Functions_with_Supremum_Norm_forms_Normed_Vector_Space | https://proofwiki.org/wiki/Space_of_Continuous_on_Closed_Interval_Real-Valued_Functions_with_Supremum_Norm_forms_Normed_Vector_Space | [
"Examples of Normed Vector Spaces"
] | [
"Definition:Real Interval/Closed",
"Definition:Space of Real-Valued Functions Continuous on Closed Interval",
"Definition:Supremum Norm",
"Definition:Normed Vector Space"
] | [
"Space of Continuous on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space",
"Supremum Norm is Norm/Continuous on Closed Interval Real-Valued Function",
"Definition:Normed Vector Space"
] |
proofwiki-17907 | Neighborhood Filter of Point is Filter | Let $T = \struct {S, \tau}$ be a topological space.
Let $x \in S$ be a point of $T$.
Let $\NN_x$ denote the neighborhood filter of $x$ in $T$.
Then $\NN_x$ is a filter on $S$. | By definition, $\NN_x$ is the set of all neighborhoods of $x$ in $T$.
It is to be demonstrated that all the conditions are satisfied for $\NN_x$ to be a filter. | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x \in S$ be a [[Definition:Point|point]] of $T$.
Let $\NN_x$ denote the [[Definition:Neighborhood Filter of Point|neighborhood filter]] of $x$ in $T$.
Then $\NN_x$ is a [[Definition:Filter on Set|filter]] on $S$. | By definition, $\NN_x$ is the [[Definition:Set|set]] of all [[Definition:Neighborhood of Point|neighborhoods]] of $x$ in $T$.
It is to be demonstrated that all the conditions are satisfied for $\NN_x$ to be a [[Definition:Filter on Set|filter]]. | Neighborhood Filter of Point is Filter | https://proofwiki.org/wiki/Neighborhood_Filter_of_Point_is_Filter | https://proofwiki.org/wiki/Neighborhood_Filter_of_Point_is_Filter | [
"Neighborhoods",
"Filter Theory"
] | [
"Definition:Topological Space",
"Definition:Point",
"Definition:Neighborhood Filter/Point",
"Definition:Filter on Set"
] | [
"Definition:Set",
"Definition:Neighborhood (Topology)/Point",
"Definition:Filter on Set",
"Definition:Neighborhood (Topology)/Point"
] |
proofwiki-17908 | Intersection of Compact and Closed Subsets of Normed Finite-Dimensional Real Vector Space with Euclidean Norm is Compact | Let $\struct {\R^d, \norm {\, \cdot \,}_2}$ be the normed finite-dimensional real vector space with Euclidean norm.
Let $K$ be a compact subset of $\struct {\R^d, \norm {\, \cdot \,}_2}$.
Let $F$ be a closed subset of $\struct {\R^d, \norm {\, \cdot \,}_2}$.
Then $F \cap K$ is compact in $\struct {\R^d, \norm {\, \cdot... | By assumption, $K$ is compact.
We have that a compact subset of normed vector space is closed and bounded.
Hence, $K$ is closed and bounded.
Since $K$ is bounded:
:$\exists C \in \R_{> 0} : \forall \mathbf x \in K : \norm {\mathbf x}_2 \le C$.
Then:
:$\forall \mathbf x \in K \cap F : \norm {\mathbf x}_2 \le C$
Hence, $... | Let $\struct {\R^d, \norm {\, \cdot \,}_2}$ be the [[Finite Dimensional Real Vector Space with Euclidean Norm form Normed Vector Space|normed finite-dimensional real vector space with Euclidean norm]].
Let $K$ be a [[Definition:Compact Subset of Normed Vector Space|compact]] [[Definition:Subset|subset]] of $\struct {\... | By [[Definition:Assumption|assumption]], $K$ is [[Definition:Compact Subset of Normed Vector Space|compact]].
We have that a [[Compact Subset of Normed Vector Space is Closed and Bounded|compact subset of normed vector space is closed and bounded]].
Hence, $K$ is [[Definition:Closed Set of Normed Vector Space|closed]... | Intersection of Compact and Closed Subsets of Normed Finite-Dimensional Real Vector Space with Euclidean Norm is Compact | https://proofwiki.org/wiki/Intersection_of_Compact_and_Closed_Subsets_of_Normed_Finite-Dimensional_Real_Vector_Space_with_Euclidean_Norm_is_Compact | https://proofwiki.org/wiki/Intersection_of_Compact_and_Closed_Subsets_of_Normed_Finite-Dimensional_Real_Vector_Space_with_Euclidean_Norm_is_Compact | [
"Closed Sets",
"Compact Normed Vector Spaces"
] | [
"Finite Dimensional Real Vector Space with Euclidean Norm form Normed Vector Space",
"Definition:Compact Space/Normed Vector Space",
"Definition:Subset",
"Definition:Closed Set/Normed Vector Space",
"Definition:Subset",
"Definition:Compact Space/Normed Vector Space"
] | [
"Definition:Assumption",
"Definition:Compact Space/Normed Vector Space",
"Compact Subset of Normed Vector Space is Closed and Bounded",
"Definition:Closed Set/Normed Vector Space",
"Definition:Bounded Subset of Normed Vector Space",
"Definition:Bounded Subset of Normed Vector Space",
"Definition:Bounded... |
proofwiki-17909 | Eigenvectors of Symmetric Matrix are Orthogonal | Let $K$ be a ring.
Let $A$ be a symmetric matrix over $K$.
Let $\lambda_1, \lambda_2$ be distinct eigenvalues of $A$.
Let $\mathbf v_1, \mathbf v_2$ be eigenvectors of $A$ corresponding to the eigenvalues $\lambda_1$ and $\lambda_2$ respectively.
Then $\mathbf v_1$ and $\mathbf v_2$ are orthogonal. | We have:
:$A \mathbf v_1 = \lambda_1 \mathbf v_1$
and:
:$A \mathbf v_2 = \lambda_2 \mathbf v_2$
We also have:
{{begin-eqn}}
{{eqn | l = \mathbf v_1 \cdot \paren {A \mathbf v_2}
| r = \mathbf v_1 \cdot \paren {\lambda_2 \mathbf v_2}
}}
{{eqn | r = \lambda_2 \mathbf v_1 \cdot \mathbf v_2
| c = Dot Product Ope... | Let $K$ be a [[Definition:Ring|ring]].
Let $A$ be a [[Definition:Symmetric Matrix|symmetric matrix]] over $K$.
Let $\lambda_1, \lambda_2$ be distinct [[Definition:Eigenvalue of Square Matrix|eigenvalues]] of $A$.
Let $\mathbf v_1, \mathbf v_2$ be [[Definition:Eigenvector of Square Matrix|eigenvectors]] of $A$ corre... | We have:
:$A \mathbf v_1 = \lambda_1 \mathbf v_1$
and:
:$A \mathbf v_2 = \lambda_2 \mathbf v_2$
We also have:
{{begin-eqn}}
{{eqn | l = \mathbf v_1 \cdot \paren {A \mathbf v_2}
| r = \mathbf v_1 \cdot \paren {\lambda_2 \mathbf v_2}
}}
{{eqn | r = \lambda_2 \mathbf v_1 \cdot \mathbf v_2
| c = [[Dot Product... | Eigenvectors of Symmetric Matrix are Orthogonal | https://proofwiki.org/wiki/Eigenvectors_of_Symmetric_Matrix_are_Orthogonal | https://proofwiki.org/wiki/Eigenvectors_of_Symmetric_Matrix_are_Orthogonal | [
"Eigenvectors of Square Matrices",
"Symmetric Matrices"
] | [
"Definition:Ring",
"Definition:Symmetric Matrix",
"Definition:Eigenvalue/Square Matrix",
"Definition:Eigenvector/Square Matrix",
"Definition:Eigenvalue/Square Matrix",
"Definition:Orthogonal (Linear Algebra)"
] | [
"Dot Product Operator is Bilinear",
"Transpose of Matrix Product",
"Dot Product Operator is Bilinear",
"Definition:Orthogonal",
"Category:Eigenvectors of Square Matrices",
"Category:Symmetric Matrices"
] |
proofwiki-17910 | Product of Diagonal Matrices is Diagonal | Let $A$ and $B$ be $n \times n$ diagonal matrices.
Then the matrix product $A B$ is an $n \times n$ diagonal matrix.
Further:
:$\paren {A B}_{i j} = \begin {cases} \paren A_{i i} \paren B_{i i} & i = j \\ 0 & i \ne j \end {cases}$ | We have:
:$\ds \paren {A B}_{ij} = \sum_{k \mathop = 1}^n \paren A_{i k} \paren B_{k j}$
Since $A$ and $B$ are diagonal:
:$\paren A_{i k} = 0$ for $i \ne k$,
and:
:$\paren B_{k j} = 0$ for $k \ne j$.
If $i \ne j$, for each $k$ we either have $i \ne k$ or $k \ne j$, so:
:$\paren A_{i k} \paren B_{k j} = 0$ for each $1 ... | Let $A$ and $B$ be [[Definition:Diagonal Matrix|$n \times n$ diagonal matrices]].
Then the [[Definition:Matrix Product|matrix product]] $A B$ is an $n \times n$ [[Definition:Diagonal Matrix|diagonal matrix]].
Further:
:$\paren {A B}_{i j} = \begin {cases} \paren A_{i i} \paren B_{i i} & i = j \\ 0 & i \ne j \end {... | We have:
:$\ds \paren {A B}_{ij} = \sum_{k \mathop = 1}^n \paren A_{i k} \paren B_{k j}$
Since $A$ and $B$ are [[Definition:Diagonal Matrix|diagonal]]:
:$\paren A_{i k} = 0$ for $i \ne k$,
and:
:$\paren B_{k j} = 0$ for $k \ne j$.
If $i \ne j$, for each $k$ we either have $i \ne k$ or $k \ne j$, so:
:$\paren A_... | Product of Diagonal Matrices is Diagonal | https://proofwiki.org/wiki/Product_of_Diagonal_Matrices_is_Diagonal | https://proofwiki.org/wiki/Product_of_Diagonal_Matrices_is_Diagonal | [
"Diagonal Matrices",
"Conventional Matrix Multiplication"
] | [
"Definition:Diagonal Matrix",
"Definition:Matrix Product",
"Definition:Diagonal Matrix"
] | [
"Definition:Diagonal Matrix",
"Definition:Diagonal Matrix",
"Category:Diagonal Matrices",
"Category:Conventional Matrix Multiplication"
] |
proofwiki-17911 | Open Real Interval is Homeomorphic to Real Number Line | Let $\R$ be the real number line with the Euclidean topology.
Let $I := \openint a b$ be a non-empty open real interval.
Then $I$ and $\R$ are homeomorphic. | By definition of open real interval, for $I$ to be non-empty it must be the case that $a < b$.
In particular it is noted that $a \ne b$.
Thus $a - b \ne 0$.
Let $I' := \openint {-1} 1$ denote the open real interval from $-1$ to $1$.
From Open Real Intervals are Homeomorphic, $I$ and $I'$ are homeomorphic.
Consider the ... | Let $\R$ be the [[Definition:Real Number Line|real number line]] with the [[Definition:Real Number Line with Euclidean Topology|Euclidean topology]].
Let $I := \openint a b$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Open Real Interval|open real interval]].
Then $I$ and $\R$ are [[Definition:Homeomorph... | By definition of [[Definition:Open Real Interval|open real interval]], for $I$ to be [[Definition:Non-Empty Set|non-empty]] it must be the case that $a < b$.
In particular it is noted that $a \ne b$.
Thus $a - b \ne 0$.
Let $I' := \openint {-1} 1$ denote the [[Definition:Open Real Interval|open real interval]] from ... | Open Real Interval is Homeomorphic to Real Number Line/Proof 1 | https://proofwiki.org/wiki/Open_Real_Interval_is_Homeomorphic_to_Real_Number_Line | https://proofwiki.org/wiki/Open_Real_Interval_is_Homeomorphic_to_Real_Number_Line/Proof_1 | [
"Examples of Homeomorphisms",
"Open Real Interval is Homeomorphic to Real Number Line"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Non-Empty Set",
"Definition:Real Interval/Open",
"Definition:Homeomorphism/Topological Spaces"
] | [
"Definition:Real Interval/Open",
"Definition:Non-Empty Set",
"Definition:Real Interval/Open",
"Open Real Intervals are Homeomorphic",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Real Function",
"Combination Theorem for Continuous Functions/Real",
"Definition:Continuous Real Function",
... |
proofwiki-17912 | Open Real Interval is Homeomorphic to Real Number Line | Let $\R$ be the real number line with the Euclidean topology.
Let $I := \openint a b$ be a non-empty open real interval.
Then $I$ and $\R$ are homeomorphic. | Let $I := \openint {-\dfrac \pi 2} {\dfrac \pi 2}$ denote the open real interval from $-\dfrac \pi 2$ to $\dfrac \pi 2$.
Consider the real function $f: I \to \R$ defined as:
:$\forall x \in I: \map f x = \tan x$
Then we have:
:$\forall x \in \R: \map {f^{-1} } x = \arctan x$
From Homeomorphism Relation is Equivalence i... | Let $\R$ be the [[Definition:Real Number Line|real number line]] with the [[Definition:Real Number Line with Euclidean Topology|Euclidean topology]].
Let $I := \openint a b$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Open Real Interval|open real interval]].
Then $I$ and $\R$ are [[Definition:Homeomorph... | Let $I := \openint {-\dfrac \pi 2} {\dfrac \pi 2}$ denote the [[Definition:Open Real Interval|open real interval]] from $-\dfrac \pi 2$ to $\dfrac \pi 2$.
Consider the [[Definition:Real Function|real function]] $f: I \to \R$ defined as:
:$\forall x \in I: \map f x = \tan x$
Then we have:
:$\forall x \in \R: \map {f^{... | Open Real Interval is Homeomorphic to Real Number Line/Proof 2 | https://proofwiki.org/wiki/Open_Real_Interval_is_Homeomorphic_to_Real_Number_Line | https://proofwiki.org/wiki/Open_Real_Interval_is_Homeomorphic_to_Real_Number_Line/Proof_2 | [
"Examples of Homeomorphisms",
"Open Real Interval is Homeomorphic to Real Number Line"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Non-Empty Set",
"Definition:Real Interval/Open",
"Definition:Homeomorphism/Topological Spaces"
] | [
"Definition:Real Interval/Open",
"Definition:Real Function",
"Homeomorphism Relation is Equivalence",
"Definition:Homeomorphism/Topological Spaces",
"Open Real Intervals are Homeomorphic",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Real Interval/Open"
] |
proofwiki-17913 | Closed Unit Interval is Homeomorphic to Letter L | Let $\R$ be the real number line under the Euclidean metric.
Let $\Bbb I := \closedint 0 1$ be the closed unit interval.
Let $\mathsf L \subseteq \R^2$ denote the letter $L$:
:$\mathsf L := \closedint 0 1 \times \set 0 \cup \set 0 \times \closedint 0 1$
Then $\Bbb I$ and $\mathsf L$ are homeomorphic. | :thumbright
Consider the mapping $f: \Bbb I \to \mathsf L$ defined as:
:$\forall x \in \Bbb I: \map f x = \begin {cases} \tuple {0, 1 - 2 x} & : x \in \closedint 0 {\dfrac 1 2} \\ \tuple {2 x - 1, 0} & : x \in \closedint {\dfrac 1 2} 1 \end {cases}$
It is seen that:
:$f \closedint 0 {\dfrac 1 2} = \set 0 \times \closed... | Let $\R$ be the [[Definition:Real Number Line|real number line]] under the [[Definition:Euclidean Metric on Real Number Line|Euclidean metric]].
Let $\Bbb I := \closedint 0 1$ be the [[Definition:Closed Unit Interval|closed unit interval]].
Let $\mathsf L \subseteq \R^2$ denote the [[Definition:Letter L|letter $L$]]:... | :[[File:Letter-L.png|thumb|right]]
Consider the [[Definition:Mapping|mapping]] $f: \Bbb I \to \mathsf L$ defined as:
:$\forall x \in \Bbb I: \map f x = \begin {cases} \tuple {0, 1 - 2 x} & : x \in \closedint 0 {\dfrac 1 2} \\ \tuple {2 x - 1, 0} & : x \in \closedint {\dfrac 1 2} 1 \end {cases}$
It is seen that:
:$f ... | Closed Unit Interval is Homeomorphic to Letter L | https://proofwiki.org/wiki/Closed_Unit_Interval_is_Homeomorphic_to_Letter_L | https://proofwiki.org/wiki/Closed_Unit_Interval_is_Homeomorphic_to_Letter_L | [
"Examples of Homeomorphisms",
"Letter L"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Metric/Real Number Line",
"Definition:Real Interval/Unit Interval/Closed",
"Definition:Letter L",
"Definition:Homeomorphism/Topological Spaces"
] | [
"File:Letter-L.png",
"Definition:Mapping",
"Definition:Bijection",
"Definition:Continuous Real Function",
"Combination Theorem for Continuous Functions/Real"
] |
proofwiki-17914 | Letter L and Letter T are not Homeomorphic | Let $\R^2$ denote the real number plane under the Euclidean topology.
Let $\mathsf L \subseteq \R^2$ denote the letter $L$:
:$\mathsf L := \closedint 0 1 \times \set 0 \cup \set 0 \times \closedint 0 1$
Let $\mathsf T \subseteq \R^2$ denote the letter $T$:
:$\mathsf T := \closedint {-1} 1 \times \set 0 \cup \set 0 \tim... | {{AimForCont}} $f: \mathsf T \to \mathsf L$ is a homeomorphism.
Let $g$ be the restriction of $f$ to $\mathsf T \setminus \set \bszero$, where $\bszero := \tuple {0, 0}$ denotes the origin of $\R^2$.
Then from Restriction of Homeomorphism is Homeomorphism, $g$ is also a homeomorphism.
But $\bszero$ is the junction poin... | Let $\R^2$ denote the [[Definition:Real Number Plane|real number plane]] under the [[Definition:Real Number Plane with Euclidean Topology|Euclidean topology]].
Let $\mathsf L \subseteq \R^2$ denote the [[Definition:Letter L|letter $L$]]:
:$\mathsf L := \closedint 0 1 \times \set 0 \cup \set 0 \times \closedint 0 1$
L... | {{AimForCont}} $f: \mathsf T \to \mathsf L$ is a [[Definition:Homeomorphism (Topological Spaces)|homeomorphism]].
Let $g$ be the [[Definition:Restriction of Mapping|restriction]] of $f$ to $\mathsf T \setminus \set \bszero$, where $\bszero := \tuple {0, 0}$ denotes the [[Definition:Origin|origin]] of $\R^2$.
Then fro... | Letter L and Letter T are not Homeomorphic | https://proofwiki.org/wiki/Letter_L_and_Letter_T_are_not_Homeomorphic | https://proofwiki.org/wiki/Letter_L_and_Letter_T_are_not_Homeomorphic | [
"Letter L",
"Letter T"
] | [
"Definition:Real Number Plane",
"Definition:Euclidean Space/Euclidean Topology/Real Number Plane",
"Definition:Letter L",
"Definition:Letter T",
"Definition:Homeomorphism/Topological Spaces"
] | [
"Definition:Homeomorphism/Topological Spaces",
"Definition:Restriction/Mapping",
"Definition:Coordinate System/Origin",
"Restriction of Homeomorphism is Homeomorphism",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Disjoint Sets",
"Definition:Real Interval/Half-Open",
"Definition:Set",
... |
proofwiki-17915 | Real Symmetric Positive Definite Matrix has Positive Eigenvalues | Let $A$ be a symmetric positive definite matrix over $\mathbb R$.
Let $\lambda$ be an eigenvalue of $A$.
Then $\lambda$ is real with $\lambda > 0$. | Let $\lambda$ be an eigenvalue of $A$ and let $\mathbf v$ be a corresponding eigenvector.
From Eigenvalues of Symmetric Matrix are Real, $\lambda$ is real.
From the definition of a positive definite matrix, we have:
:$\mathbf v^\intercal A \mathbf v > 0$
That is:
{{begin-eqn}}
{{eqn | l = 0
| o = <
| r = ... | Let $A$ be a [[Definition:Symmetric Matrix|symmetric]] [[Definition:Positive Definite Matrix|positive definite matrix]] over $\mathbb R$.
Let $\lambda$ be an [[Definition:Eigenvalue of Real Square Matrix|eigenvalue]] of $A$.
Then $\lambda$ is [[Definition:Real Number|real]] with $\lambda > 0$. | Let $\lambda$ be an [[Definition:Eigenvalue of Real Square Matrix|eigenvalue]] of $A$ and let $\mathbf v$ be a corresponding [[Definition:Eigenvector of Real Square Matrix|eigenvector]].
From [[Eigenvalues of Symmetric Matrix are Real]], $\lambda$ is [[Definition:Real Number|real]].
From the definition of a [[Definit... | Real Symmetric Positive Definite Matrix has Positive Eigenvalues | https://proofwiki.org/wiki/Real_Symmetric_Positive_Definite_Matrix_has_Positive_Eigenvalues | https://proofwiki.org/wiki/Real_Symmetric_Positive_Definite_Matrix_has_Positive_Eigenvalues | [
"Symmetric Matrices",
"Positive Definite Matrices"
] | [
"Definition:Symmetric Matrix",
"Definition:Positive Definite Matrix",
"Definition:Eigenvalue/Real Square Matrix",
"Definition:Real Number"
] | [
"Definition:Eigenvalue/Real Square Matrix",
"Definition:Eigenvector/Real Square Matrix",
"Eigenvalues of Symmetric Matrix are Real",
"Definition:Real Number",
"Definition:Positive Definite Matrix",
"Dot Product of Vector with Itself",
"Euclidean Space is Normed Vector Space",
"Category:Symmetric Matri... |
proofwiki-17916 | Characterisation of Real Symmetric Positive Definite Matrix/Necessary Condition | Let $A$ be an $n \times n$ positive definite symmetric matrix over $\R$.
Then:
:there exists a nonsingular matrix $C$ such that $A = C^\intercal C$. | Let $A$ be positive definite.
From Spectral Theorem for Real Symmetric Matrices:
:there exists an orthogonal matrix $P$ and diagonal matrix $D$ such that $A = P^\intercal D P$.
Further, from Characterization of Diagonalizable Matrices:
:the diagonal entries of $D$ are the eigenvalues of $A$.
From Real Symmetric Posit... | Let $A$ be an $n \times n$ [[Definition:Positive Definite Matrix|positive definite]] [[Definition:Symmetric Matrix|symmetric matrix]] over $\R$.
Then:
:there exists a [[Definition:Nonsingular Matrix|nonsingular matrix]] $C$ such that $A = C^\intercal C$. | Let $A$ be [[Definition:Positive Definite Matrix|positive definite]].
From [[Spectral Theorem for Real Symmetric Matrices]]:
:there exists an [[Definition:Orthogonal Matrix|orthogonal matrix]] $P$ and [[Definition:Diagonal Matrix|diagonal matrix]] $D$ such that $A = P^\intercal D P$.
Further, from [[Characterizatio... | Characterisation of Real Symmetric Positive Definite Matrix/Necessary Condition | https://proofwiki.org/wiki/Characterisation_of_Real_Symmetric_Positive_Definite_Matrix/Necessary_Condition | https://proofwiki.org/wiki/Characterisation_of_Real_Symmetric_Positive_Definite_Matrix/Necessary_Condition | [
"Characterisation of Real Symmetric Positive Definite Matrix",
"Positive Definite Matrices",
"Symmetric Matrices"
] | [
"Definition:Positive Definite Matrix",
"Definition:Symmetric Matrix",
"Definition:Nonsingular Matrix"
] | [
"Definition:Positive Definite Matrix",
"Spectral Theorem for Real Symmetric Matrices",
"Definition:Orthogonal Matrix",
"Definition:Diagonal Matrix",
"Characterization of Diagonalizable Matrices",
"Definition:Matrix/Diagonal/Main",
"Definition:Matrix/Element",
"Definition:Eigenvalue",
"Real Symmetric... |
proofwiki-17917 | Characterisation of Real Symmetric Positive Definite Matrix/Sufficient Condition | Let $A$ be an $n \times n$ symmetric matrix over $\mathbb R$ such that:
:there exists a nonsingular matrix $C$ such that $A = C^\intercal C$.
Then $A$ is positive definite. | Let $A$ be a symmetric matrix such that:
:there exists an nonsingular matrix $C$ such that $A = C^\intercal C$.
Let $\mathbf v$ be a non-zero vector.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf v^\intercal A \mathbf v
| r = \mathbf v^\intercal C^\intercal C \mathbf v
}}
{{eqn | r = \paren {C \mathbf v}^\intercal C ... | Let $A$ be an [[Definition:Symmetric Matrix|$n \times n$ symmetric matrix]] over $\mathbb R$ such that:
:there exists a [[Definition:Nonsingular Matrix|nonsingular matrix]] $C$ such that $A = C^\intercal C$.
Then $A$ is [[Definition:Positive Definite Matrix|positive definite]]. | Let $A$ be a [[Definition:Symmetric Matrix|symmetric matrix]] such that:
:there exists an [[Definition:Nonsingular Matrix|nonsingular matrix]] $C$ such that $A = C^\intercal C$.
Let $\mathbf v$ be a non-[[Definition:Zero Vector|zero vector]].
Then:
{{begin-eqn}}
{{eqn | l = \mathbf v^\intercal A \mathbf v
|... | Characterisation of Real Symmetric Positive Definite Matrix/Sufficient Condition | https://proofwiki.org/wiki/Characterisation_of_Real_Symmetric_Positive_Definite_Matrix/Sufficient_Condition | https://proofwiki.org/wiki/Characterisation_of_Real_Symmetric_Positive_Definite_Matrix/Sufficient_Condition | [
"Characterisation of Real Symmetric Positive Definite Matrix",
"Positive Definite Matrices",
"Symmetric Matrices"
] | [
"Definition:Symmetric Matrix",
"Definition:Nonsingular Matrix",
"Definition:Positive Definite Matrix"
] | [
"Definition:Symmetric Matrix",
"Definition:Nonsingular Matrix",
"Definition:Zero Vector",
"Transpose of Matrix Product",
"Dot Product of Vector with Itself",
"Euclidean Space is Normed Vector Space",
"Definition:Positive Definite Matrix"
] |
proofwiki-17918 | Characterisation of Real Symmetric Positive Definite Matrix | Let $A$ be an $n \times n$ symmetric matrix over $\mathbb R$.
Then $A$ is positive definite {{iff}}:
:there exists a nonsingular matrix $C$ such that $A = C^\intercal C$. | === Necessary Condition ===
{{:Characterisation of Real Symmetric Positive Definite Matrix/Necessary Condition}}{{qed|lemma}} | Let $A$ be an [[Definition:Symmetric Matrix|$n \times n$ symmetric matrix]] over $\mathbb R$.
Then $A$ is [[Definition:Positive Definite|positive definite]] {{iff}}:
:there exists a [[Definition:Nonsingular Matrix|nonsingular matrix]] $C$ such that $A = C^\intercal C$. | === [[Characterisation of Real Symmetric Positive Definite Matrix/Necessary Condition|Necessary Condition]] ===
{{:Characterisation of Real Symmetric Positive Definite Matrix/Necessary Condition}}{{qed|lemma}} | Characterisation of Real Symmetric Positive Definite Matrix | https://proofwiki.org/wiki/Characterisation_of_Real_Symmetric_Positive_Definite_Matrix | https://proofwiki.org/wiki/Characterisation_of_Real_Symmetric_Positive_Definite_Matrix | [
"Characterisation of Real Symmetric Positive Definite Matrix",
"Symmetric Matrices",
"Positive Definite Matrices"
] | [
"Definition:Symmetric Matrix",
"Definition:Positive Definite",
"Definition:Nonsingular Matrix"
] | [
"Characterisation of Real Symmetric Positive Definite Matrix/Necessary Condition"
] |
proofwiki-17919 | Coffee Mug and Doughnut are Homeomorphic | A {{WP|Doughnut|doughnut}} and a {{WP|Coffee mug|coffee mug}} are homeomorphic. | They are both solid objects with one hole.
A classical {{WP|Doughnut|doughnut}} is in the shape of a torus.
That is, it is a solid figure with one hole.
The classic coffee-cup shape is a receptacle with a closed loop of china forming a handle.
The hole through which you put your fingers is homeomorphic to the hole in t... | A {{WP|Doughnut|doughnut}} and a {{WP|Coffee mug|coffee mug}} are [[Definition:Homeomorphic Topological Spaces|homeomorphic]]. | They are both solid objects with one hole.
A classical {{WP|Doughnut|doughnut}} is in the shape of a [[Definition:Torus (Geometry)|torus]].
That is, it is a [[Definition:Solid Figure|solid figure]] with one hole.
The classic coffee-cup shape is a receptacle with a closed loop of china forming a handle.
The hole thr... | Coffee Mug and Doughnut are Homeomorphic | https://proofwiki.org/wiki/Coffee_Mug_and_Doughnut_are_Homeomorphic | https://proofwiki.org/wiki/Coffee_Mug_and_Doughnut_are_Homeomorphic | [
"Examples of Homeomorphisms"
] | [
"Definition:Homeomorphism/Topological Spaces"
] | [
"Definition:Torus (Geometry)",
"Definition:Geometric Figure/Three-Dimensional Figure"
] |
proofwiki-17920 | Trefoil Knot is Homeomorphic to Circle | The trefoil knot is homeomorphic to the circle. | Despite the fact that you cannot actually rearrange a trefoil knot actually into a circle in the usual $\R^3$ space, you can set up a mappping from one to the other.
{{finish}} | The [[Definition:Trefoil Knot|trefoil knot]] is [[Definition:Homeomorphic Topological Spaces|homeomorphic]] to the [[Definition:Circle|circle]]. | Despite the fact that you cannot actually rearrange a [[Definition:Trefoil Knot|trefoil knot]] actually into a [[Definition:Circle|circle]] in the [[Definition:Cartesian Space|usual $\R^3$ space]], you can set up a [[Definition:Mapping|mappping]] from one to the other.
{{finish}} | Trefoil Knot is Homeomorphic to Circle | https://proofwiki.org/wiki/Trefoil_Knot_is_Homeomorphic_to_Circle | https://proofwiki.org/wiki/Trefoil_Knot_is_Homeomorphic_to_Circle | [
"Examples of Homeomorphisms"
] | [
"Definition:Trefoil Knot",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Circle"
] | [
"Definition:Trefoil Knot",
"Definition:Circle",
"Definition:Cartesian Product/Cartesian Space",
"Definition:Mapping"
] |
proofwiki-17921 | Homeomorphism between Topological Spaces may not be Unique | Let $T_1$ and $T_2$ be topological spaces.
Let $f$ be a homeomorphism from $T_1$ to $T_2$.
Then $f$ may not necessarily be unique. | Let $\R$ be the real number line with the Euclidean topology.
Let $I_1 := \openint a b$ and $I_2 := \openint c d$ be non-empty open real intervals.
From Open Real Intervals are Homeomorphic, $I_1$ and $I_2$ are homeomorphic.
The example given of a homeomorphism was the real function $f: I_1 \to I_2$ defined as:
:$\fora... | Let $T_1$ and $T_2$ be [[Definition:Topological Space|topological spaces]].
Let $f$ be a [[Definition:Homeomorphism (Topological Spaces)|homeomorphism]] from $T_1$ to $T_2$.
Then $f$ may not necessarily be [[Definition:Unique|unique]]. | Let $\R$ be the [[Definition:Real Number Line|real number line]] with the [[Definition:Real Number Line with Euclidean Topology|Euclidean topology]].
Let $I_1 := \openint a b$ and $I_2 := \openint c d$ be [[Definition:Non-Empty Set|non-empty]] [[Definition:Open Real Interval|open real intervals]].
From [[Open Real In... | Homeomorphism between Topological Spaces may not be Unique | https://proofwiki.org/wiki/Homeomorphism_between_Topological_Spaces_may_not_be_Unique | https://proofwiki.org/wiki/Homeomorphism_between_Topological_Spaces_may_not_be_Unique | [
"Homeomorphisms (Topological Spaces)"
] | [
"Definition:Topological Space",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Unique"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Non-Empty Set",
"Definition:Real Interval/Open",
"Open Real Intervals are Homeomorphic",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Homeomorphism/Topological Spaces",
... |
proofwiki-17922 | Cardinality of Set is Topological Property | Let $T = \struct {S, \tau}$ be a topological space.
The cardinality of $S$ is a topological property of $T$. | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.
Let $T_1$ and $T_2$ be homeomorphic.
Then by definition there exists a homeomorphism $f: T_1 \to T_2$.
Hence by definition $f$ is a bijection.
Hence by definition $S$ and $T$ are equivalent.
That is, they have the same cardinalit... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
The [[Definition:Cardinality|cardinality]] of $S$ is a [[Definition:Topological Property|topological property]] of $T$. | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $T_1$ and $T_2$ be [[Definition:Homeomorphic Topological Spaces|homeomorphic]].
Then by definition there exists a [[Definition:Homeomorphism (Topological Spaces)|homeomorphism]] $f: T_1 \to ... | Cardinality of Set is Topological Property | https://proofwiki.org/wiki/Cardinality_of_Set_is_Topological_Property | https://proofwiki.org/wiki/Cardinality_of_Set_is_Topological_Property | [
"Cardinality of Set is Topological Property",
"Cardinality",
"Examples of Topological Properties"
] | [
"Definition:Topological Space",
"Definition:Cardinality",
"Definition:Topological Property"
] | [
"Definition:Topological Space",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Bijection",
"Definition:Set Equivalence",
"Definition:Cardinality",
"Definition:Cardinality",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Topologic... |
proofwiki-17923 | Boundedness is not Topological Property | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $M_1$ and $M_2$ be homeomorphic.
Then it is not necessarily the case that:
:$M_1$ is bounded {{iff}} $M_2$ is bounded.
That is, boundedness is not a topological property. | ;Proof by Counterexample
Let the metric space $M_1 = \struct {S_1, d}$ such that:
:$S_1 = \openint 0 1$ is the open unit interval
:$d$ is the usual (Euclidean) metric on $S_1$.
Let the metric space $M_2 = \struct {\R, d}$ such that:
:$\R$ is the set of real numbers
:$d$ is again the usual (Euclidean) metric on $\R$.
Th... | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be [[Definition:Metric Space|metric spaces]].
Let $M_1$ and $M_2$ be [[Definition:Homeomorphic Metric Spaces|homeomorphic]].
Then it is not necessarily the case that:
:$M_1$ is [[Definition:Bounded Metric Space|bounded]] {{iff}} $M_2$ is [[Definition:Boun... | ;[[Proof by Counterexample]]
Let the [[Definition:Metric Space|metric space]] $M_1 = \struct {S_1, d}$ such that:
:$S_1 = \openint 0 1$ is the [[Definition:Open Unit Interval|open unit interval]]
:$d$ is the [[Definition:Euclidean Metric|usual (Euclidean) metric]] on $S_1$.
Let the [[Definition:Metric Space|metric sp... | Boundedness is not Topological Property | https://proofwiki.org/wiki/Boundedness_is_not_Topological_Property | https://proofwiki.org/wiki/Boundedness_is_not_Topological_Property | [
"Examples of Topological Properties"
] | [
"Definition:Metric Space",
"Definition:Homeomorphism/Metric Spaces",
"Definition:Bounded Metric Space",
"Definition:Bounded Metric Space",
"Definition:Bounded Metric Space",
"Definition:Topological Property"
] | [
"Proof by Counterexample",
"Definition:Metric Space",
"Definition:Real Interval/Unit Interval/Open",
"Definition:Euclidean Metric",
"Definition:Metric Space",
"Definition:Real Number",
"Definition:Euclidean Metric",
"Definition:Bounded Metric Space",
"Definition:Bounded Metric Space",
"Open Real I... |
proofwiki-17924 | Minimum of Exponential Random Variables has Exponential Distribution | Let $\beta_1, \beta_2, \ldots, \beta_n$ be positive real numbers.
Let $X_1, X_2, \ldots, X_n$ be independent random variables.
For each $i$, let $X_i \sim \Exponential {\beta_i}$, where $\Exponential {\beta_i}$ is the exponential distribution with parameter $\beta_i$.
Let:
:$\ds M = \map {\min_{1 \mathop \le i \matho... | We aim to show that:
:$\ds \map \Pr {M \le m} = 1 - \map \exp {-m \sum_{i \mathop = 1}^n \frac 1 {\beta_i} }$
for each $m > 0$.
Note that:
:$\ds M > m$
{{iff}}:
:$\ds X_i > m$
for each $i$.
We therefore have:
{{begin-eqn}}
{{eqn | l = \map \Pr {M > m}
| r = \map \Pr {\bigcap_{i \mathop = 1}^n \set {X_i > m} }
}}
{... | Let $\beta_1, \beta_2, \ldots, \beta_n$ be [[Definition:Positive Real Number|positive real numbers]].
Let $X_1, X_2, \ldots, X_n$ be [[Definition:Independent Random Variables|independent random variables]].
For each $i$, let $X_i \sim \Exponential {\beta_i}$, where $\Exponential {\beta_i}$ is the [[Definition:Exponen... | We aim to show that:
:$\ds \map \Pr {M \le m} = 1 - \map \exp {-m \sum_{i \mathop = 1}^n \frac 1 {\beta_i} }$
for each $m > 0$.
Note that:
:$\ds M > m$
{{iff}}:
:$\ds X_i > m$
for each $i$.
We therefore have:
{{begin-eqn}}
{{eqn | l = \map \Pr {M > m}
| r = \map \Pr {\bigcap_{i \mathop = 1}^n \set {X_i > ... | Minimum of Exponential Random Variables has Exponential Distribution | https://proofwiki.org/wiki/Minimum_of_Exponential_Random_Variables_has_Exponential_Distribution | https://proofwiki.org/wiki/Minimum_of_Exponential_Random_Variables_has_Exponential_Distribution | [
"Exponential Distribution"
] | [
"Definition:Positive/Real Number",
"Definition:Independent Random Variables",
"Definition:Exponential Distribution"
] | [
"Exponential of Sum",
"Category:Exponential Distribution"
] |
proofwiki-17925 | Multiple of Exponential Random Variable has Exponential Distribution | Let $\beta, k$ be real numbers with $\beta > 0$.
Let $X$ be a random variable.
Let $X \sim \Exponential \beta$, where $\Exponential \beta$ is the exponential distribution with parameter $\beta$.
Then:
:$k X \sim \Exponential {k \beta}$ | Let:
:$Y \sim k X$
We aim to show that:
:$\ds \map \Pr {Y \le y} = 1 - \map \exp {-\frac y {k \beta} }$
for each $y > 0$.
We have:
{{begin-eqn}}
{{eqn | l = \map \Pr {Y \le y}
| r = \map \Pr {k X \le y}
}}
{{eqn | r = \map \Pr {X \le \frac y k}
}}
{{eqn | r = 1 - \map \exp {-\frac y {k \beta} }
| c = {{Defof|Expon... | Let $\beta, k$ be [[Definition:Real Number|real numbers]] with $\beta > 0$.
Let $X$ be a [[Definition:Random Variable|random variable]].
Let $X \sim \Exponential \beta$, where $\Exponential \beta$ is the [[Definition:Exponential Distribution|exponential distribution]] with parameter $\beta$.
Then:
:$k X \sim \Exp... | Let:
:$Y \sim k X$
We aim to show that:
:$\ds \map \Pr {Y \le y} = 1 - \map \exp {-\frac y {k \beta} }$
for each $y > 0$.
We have:
{{begin-eqn}}
{{eqn | l = \map \Pr {Y \le y}
| r = \map \Pr {k X \le y}
}}
{{eqn | r = \map \Pr {X \le \frac y k}
}}
{{eqn | r = 1 - \map \exp {-\frac y {k \beta} }
| c = {{Defof... | Multiple of Exponential Random Variable has Exponential Distribution | https://proofwiki.org/wiki/Multiple_of_Exponential_Random_Variable_has_Exponential_Distribution | https://proofwiki.org/wiki/Multiple_of_Exponential_Random_Variable_has_Exponential_Distribution | [
"Exponential Distribution"
] | [
"Definition:Real Number",
"Definition:Random Variable",
"Definition:Exponential Distribution"
] | [
"Category:Exponential Distribution"
] |
proofwiki-17926 | Exponential of Negative of Exponential Random Variable has Beta Distribution | Let $\beta$ be a positive real number.
Let $X \sim \Exponential \beta$ where $\Exponential \beta$ is the exponential distribution with parameter $\beta$.
Then:
:$e^{-X} \sim \BetaDist {\dfrac 1 \beta} 1$ | Note that if:
:$Y \sim \BetaDist {\dfrac 1 \beta} 1$
then the probability density function of $Y$, $f_Y$ is given by:
{{begin-eqn}}
{{eqn | l = \map {f_Y} y
| r = \frac {y^{\frac 1 \beta - 1} \paren {1 - y}^{1 - 1} } {\map \Beta {\frac 1 \beta, 1} }
| c = {{Defof|Beta Distribution}}
}}
{{eqn | r = \frac {y^{\frac 1 \... | Let $\beta$ be a [[Definition:Positive Real Number|positive real number]].
Let $X \sim \Exponential \beta$ where $\Exponential \beta$ is the [[Definition:Exponential Distribution|exponential distribution]] with parameter $\beta$.
Then:
:$e^{-X} \sim \BetaDist {\dfrac 1 \beta} 1$ | Note that if:
:$Y \sim \BetaDist {\dfrac 1 \beta} 1$
then the [[Definition:Probability Density Function|probability density function]] of $Y$, $f_Y$ is given by:
{{begin-eqn}}
{{eqn | l = \map {f_Y} y
| r = \frac {y^{\frac 1 \beta - 1} \paren {1 - y}^{1 - 1} } {\map \Beta {\frac 1 \beta, 1} }
| c = {{Defof|Beta Di... | Exponential of Negative of Exponential Random Variable has Beta Distribution | https://proofwiki.org/wiki/Exponential_of_Negative_of_Exponential_Random_Variable_has_Beta_Distribution | https://proofwiki.org/wiki/Exponential_of_Negative_of_Exponential_Random_Variable_has_Beta_Distribution | [
"Exponential Distribution",
"Beta Distribution"
] | [
"Definition:Positive/Real Number",
"Definition:Exponential Distribution"
] | [
"Definition:Probability Density Function",
"Gamma Difference Equation",
"Definition:Probability Density Function",
"Power Rule for Derivatives",
"Definition:Probability Density Function",
"Category:Exponential Distribution",
"Category:Beta Distribution"
] |
proofwiki-17927 | Normed Vector Space is Finite Dimensional iff Unit Sphere is Compact/Necessary Condition | Let $X$ be a finite dimensional normed vector space.
Let $\Bbb S = \map {\Bbb S_1} 0$ be the unit sphere centred at $0$ in $X$.
Then $\Bbb S$ is compact. | Let $X$ be a finite dimensional vector space $\R^d$.
We have that all norms on finite-dimensional vector space are equivalent.
Choose Euclidean norm $\norm {\, \cdot \,}_2$.
Let $\struct {\R^d, \norm {\, \cdot \,}_2}$ be the normed finite-dimensional real vector space with Euclidean norm.
Let $\map {\Bbb S^{d - 1}_1} 0... | Let $X$ be a [[Definition:Finite Dimensional Vector Space|finite dimensional]] [[Definition:Normed Vector Space|normed vector space]].
Let $\Bbb S = \map {\Bbb S_1} 0$ be the [[Definition:Sphere in Normed Vector Space|unit sphere]] [[Definition:Sphere/Normed Vector Space/Center|centred]] at $0$ in $X$.
Then $\Bbb S$... | Let $X$ be a [[Definition:Finite Dimensional Vector Space|finite dimensional]] [[Real Vector Space is Vector Space|vector space $\R^d$]].
We have that [[Norms on Finite-Dimensional Real Vector Space are Equivalent|all norms on finite-dimensional vector space are equivalent]].
Choose [[Definition:Euclidean Norm|Euclid... | Normed Vector Space is Finite Dimensional iff Unit Sphere is Compact/Necessary Condition | https://proofwiki.org/wiki/Normed_Vector_Space_is_Finite_Dimensional_iff_Unit_Sphere_is_Compact/Necessary_Condition | https://proofwiki.org/wiki/Normed_Vector_Space_is_Finite_Dimensional_iff_Unit_Sphere_is_Compact/Necessary_Condition | [
"Normed Vector Space is Finite Dimensional iff Unit Sphere is Compact"
] | [
"Definition:Dimension of Vector Space/Finite",
"Definition:Normed Vector Space",
"Definition:Sphere/Normed Vector Space",
"Definition:Sphere/Normed Vector Space/Center",
"Definition:Compact Space/Normed Vector Space"
] | [
"Definition:Dimension of Vector Space/Finite",
"Real Vector Space is Vector Space",
"Norms on Finite-Dimensional Real Vector Space are Equivalent",
"Definition:Euclidean Norm",
"Finite Dimensional Real Vector Space with Euclidean Norm form Normed Vector Space",
"Definition:Unit Sphere/Normed Vector Space"... |
proofwiki-17928 | Limit Point of Set may or may not be Element of Set | Let $S$ be a set.
Let $H \subseteq S$ be a subset of $S$.
Let $T = \struct {H, \tau}$ be a topological space on the underlying set $H$.
Let $a \in S$ be a limit point of $T$.
Then $a$ may or may not be an element of $H$.
Whether it is or not depends upon the nature of both $a$ and $T$. | Consider:
:the open real interval $\openint a b$
:the closed real interval $\closedint a b$.
Both of these are subsets of the set of real numbers $\R$.
From Limit Point Examples: End Points of Real Interval, $a$ is a limit point of both $\openint a b$ and $\closedint a b$.
But $a \in \closedint a b$ while $a \notin \op... | Let $S$ be a [[Definition:Set|set]].
Let $H \subseteq S$ be a [[Definition:Subset|subset]] of $S$.
Let $T = \struct {H, \tau}$ be a [[Definition:Topological Space|topological space]] on the [[Definition:Underlying Set of Topological Space|underlying set]] $H$.
Let $a \in S$ be a [[Definition:Limit Point (Topology)|l... | Consider:
:the [[Definition:Open Real Interval|open real interval]] $\openint a b$
:the [[Definition:Closed Real Interval|closed real interval]] $\closedint a b$.
Both of these are [[Definition:Subset|subsets]] of the [[Definition:Real Numbers|set of real numbers]] $\R$.
From [[Limit Point/Examples/End Points of Rea... | Limit Point of Set may or may not be Element of Set | https://proofwiki.org/wiki/Limit_Point_of_Set_may_or_may_not_be_Element_of_Set | https://proofwiki.org/wiki/Limit_Point_of_Set_may_or_may_not_be_Element_of_Set | [
"Limit Points"
] | [
"Definition:Set",
"Definition:Subset",
"Definition:Topological Space",
"Definition:Underlying Set/Topological Space",
"Definition:Limit Point/Topology",
"Definition:Element"
] | [
"Definition:Real Interval/Open",
"Definition:Real Interval/Closed",
"Definition:Subset",
"Definition:Real Number",
"Limit Point/Examples/End Points of Real Interval",
"Definition:Limit Point/Topology"
] |
proofwiki-17929 | Element of Topological Space may or may not be Limit Point | Let $S$ be a set.
Let $H \subseteq S$ be a subset of $S$.
Let $T = \struct {H, \tau}$ be a topological space on the underlying set $H$.
Let $a \in H$.
Then $a$ may or may not be a limit point of $T$.
Whether it is or not depends upon the nature of both $a$ and $T$. | Let $\R$ be the set of real numbers.
Let $H \subseteq \R$ be the subset of $\R$ defined as:
:$H = \set 0 \cup \openint 1 2$
From Limit Point Examples: Union of Singleton with Open Real Interval, $0$ is not a limit point of $H$, although $0 \in H$.
From Limit Point Examples: End Points of Real Interval, $a$ is a limit p... | Let $S$ be a [[Definition:Set|set]].
Let $H \subseteq S$ be a [[Definition:Subset|subset]] of $S$.
Let $T = \struct {H, \tau}$ be a [[Definition:Topological Space|topological space]] on the [[Definition:Underlying Set of Topological Space|underlying set]] $H$.
Let $a \in H$.
Then $a$ may or may not be a [[Definiti... | Let $\R$ be the [[Definition:Real Number|set of real numbers]].
Let $H \subseteq \R$ be the [[Definition:Subset|subset]] of $\R$ defined as:
:$H = \set 0 \cup \openint 1 2$
From [[Limit Point/Examples/Union of Singleton with Open Real Interval|Limit Point Examples: Union of Singleton with Open Real Interval]], $0$ is... | Element of Topological Space may or may not be Limit Point | https://proofwiki.org/wiki/Element_of_Topological_Space_may_or_may_not_be_Limit_Point | https://proofwiki.org/wiki/Element_of_Topological_Space_may_or_may_not_be_Limit_Point | [
"Limit Points"
] | [
"Definition:Set",
"Definition:Subset",
"Definition:Topological Space",
"Definition:Underlying Set/Topological Space",
"Definition:Limit Point/Topology"
] | [
"Definition:Real Number",
"Definition:Subset",
"Limit Point/Examples/Union of Singleton with Open Real Interval",
"Definition:Limit Point/Topology",
"Limit Point/Examples/End Points of Real Interval",
"Definition:Limit Point/Topology"
] |
proofwiki-17930 | Countable Set may have Uncountable Limit Points | Let $S$ be an uncountable set.
Let $H \subseteq S$ be a countable subset of $S$.
Let $T = \struct {H, \tau}$ be a topological space on the underlying set $H$.
Then despite the fact that $H$ is countable, the set of limit points of $T$ may be uncountable. | Let $\R$ be the set of real numbers.
Let $\Q$ be the set of rational numbers.
Let $x \in \R$.
Then from Real Number is Limit Point of Rational Numbers in Real Numbers, $x$ is a limit point of $T$.
As $x$ is arbitrary, it follows that every element of $\R$ is a limit point of $T$.
From Rational Numbers are Countably Inf... | Let $S$ be an [[Definition:Uncountable Set|uncountable set]].
Let $H \subseteq S$ be a [[Definition:Countable Set|countable]] [[Definition:Subset|subset]] of $S$.
Let $T = \struct {H, \tau}$ be a [[Definition:Topological Space|topological space]] on the [[Definition:Underlying Set of Topological Space|underlying set]... | Let $\R$ be the [[Definition:Real Number|set of real numbers]].
Let $\Q$ be the [[Definition:Rational Number|set of rational numbers]].
Let $x \in \R$.
Then from [[Real Number is Limit Point of Rational Numbers in Real Numbers]], $x$ is a [[Definition:Limit Point (Topology)|limit point]] of $T$.
As $x$ is arbitrary... | Countable Set may have Uncountable Limit Points | https://proofwiki.org/wiki/Countable_Set_may_have_Uncountable_Limit_Points | https://proofwiki.org/wiki/Countable_Set_may_have_Uncountable_Limit_Points | [
"Limit Points"
] | [
"Definition:Uncountable/Set",
"Definition:Countable Set",
"Definition:Subset",
"Definition:Topological Space",
"Definition:Underlying Set/Topological Space",
"Definition:Countable Set",
"Definition:Set",
"Definition:Limit Point/Topology",
"Definition:Uncountable/Set"
] | [
"Definition:Real Number",
"Definition:Rational Number",
"Real Number is Limit Point of Rational Numbers in Real Numbers",
"Definition:Limit Point/Topology",
"Definition:Element",
"Definition:Limit Point/Topology",
"Rational Numbers are Countably Infinite",
"Definition:Countable Set",
"Real Numbers a... |
proofwiki-17931 | Condition for Point being in Closure/Metric Space | Let $M = \struct {S, d}$ be a metric space.
Let $H \subseteq S$.
Let $\map \cl H$ denote the closure of $H$ in $M$.
Let $x \in S$.
Then $x \in \map \cl H$ {{iff}}:
:$\forall \epsilon \in \R_{>0}: \map {B_\epsilon} x \cap H \ne \O$
where $\map {B_\epsilon} x$ denotes the open $\epsilon$-ball of $x$. | By definition of closure of $H$ in $M$:
:$\map \cl H = H^i \cup H'$
where:
:$H^i$ denotes the set of isolated points of $H$
:$H'$ denotes the set of limit points of $H$. | Let $M = \struct {S, d}$ be a [[Definition:Metric Space|metric space]].
Let $H \subseteq S$.
Let $\map \cl H$ denote the [[Definition:Closure (Metric Space)|closure]] of $H$ in $M$.
Let $x \in S$.
Then $x \in \map \cl H$ {{iff}}:
:$\forall \epsilon \in \R_{>0}: \map {B_\epsilon} x \cap H \ne \O$
where $\map {B_\ep... | By definition of [[Definition:Closure (Metric Space)|closure]] of $H$ in $M$:
:$\map \cl H = H^i \cup H'$
where:
:$H^i$ denotes the [[Definition:Set|set]] of [[Definition:Isolated Point of Subset of Metric Space|isolated points]] of $H$
:$H'$ denotes the [[Definition:Set|set]] of [[Definition:Limit Point (Metric Space... | Condition for Point being in Closure/Metric Space | https://proofwiki.org/wiki/Condition_for_Point_being_in_Closure/Metric_Space | https://proofwiki.org/wiki/Condition_for_Point_being_in_Closure/Metric_Space | [
"Set Closures",
"Condition for Point being in Closure"
] | [
"Definition:Metric Space",
"Definition:Closure (Topology)/Metric Space",
"Definition:Open Ball"
] | [
"Definition:Closure (Topology)/Metric Space",
"Definition:Set",
"Definition:Isolated Point (Metric Space)/Subset",
"Definition:Set",
"Definition:Limit Point/Metric Space",
"Definition:Isolated Point (Metric Space)/Subset",
"Definition:Isolated Point (Metric Space)/Subset",
"Definition:Limit Point/Metr... |
proofwiki-17932 | Equivalence of Definitions of Everywhere Dense | {{TFAE|def = Everywhere Dense}}
Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$ be a subset. | === $(1)$ implies $(2)$ ===
Let $H$ be a subset of $S$ which is everywhere dense in $T$ by definition $1$.
Then by definition:
:$H^- = S$
where $H^-$ is the closure of $H$.
{{AimForCont}} there exists $U \in \tau$ such that $U \cap H = \O$.
Let $x \in S$ such that $x \in U$.
Thus $U$ is an open set of $T$ which does no... | {{TFAE|def = Everywhere Dense}}
Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $H \subseteq S$ be a [[Definition:Subset|subset]]. | === $(1)$ implies $(2)$ ===
Let $H$ be a [[Definition:Subset|subset]] of $S$ which is [[Definition:Everywhere Dense/Definition 1|everywhere dense in $T$ by definition $1$]].
Then by definition:
:$H^- = S$
where $H^-$ is the [[Definition:Closure (Topology)|closure]] of $H$.
{{AimForCont}} there exists $U \in \tau$ s... | Equivalence of Definitions of Everywhere Dense | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Everywhere_Dense | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Everywhere_Dense | [
"Everywhere Dense"
] | [
"Definition:Topological Space",
"Definition:Subset"
] | [
"Definition:Subset",
"Definition:Everywhere Dense/Definition 1",
"Definition:Closure (Topology)",
"Definition:Open Set/Topology",
"Definition:Element",
"Definition:Distinct/Plural",
"Definition:Open Set/Topology",
"Definition:Element",
"Definition:Limit Point/Topology/Set",
"Definition:Closure (To... |
proofwiki-17933 | Closure of Open Ball may not equal Closed Ball of Same Radius | Let $M = \struct {A, d}$ be a metric space.
Let $x \in A$.
Let $\map {B_\epsilon} x$ be the open $\epsilon$-ball of $x$ of radius $\epsilon$ for some $\epsilon \in \R_{>0}$.
Let $\map { {B_\epsilon}^-} x$ be the closed $\epsilon$-ball of $x$ of radius $\epsilon$.
Then it is not necessarily the case that:
:$\map \cl {\m... | Proof by Counterexample:
Let $M = \struct {A, d}$ be the standard discrete metric space on a set $A$.
From Closure of Open $1$-Ball in Standard Discrete Metric Space we have that:
:$\map \cl {\map {B_1} x} = \set x$
but:
{{begin-eqn}}
{{eqn | l = \map { {B_1}^-} x
| r = \set {y \in A: \map d {x, y} \le 1}
|... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $x \in A$.
Let $\map {B_\epsilon} x$ be the [[Definition:Open Ball of Metric Space|open $\epsilon$-ball of $x$]] of [[Definition:Radius of Open Ball|radius]] $\epsilon$ for some $\epsilon \in \R_{>0}$.
Let $\map { {B_\epsilon}^-} x$ be the [... | [[Proof by Counterexample]]:
Let $M = \struct {A, d}$ be the [[Definition:Standard Discrete Metric|standard discrete metric space]] on a [[Definition:Set|set]] $A$.
From [[Closure of Open 1-Ball in Standard Discrete Metric Space|Closure of Open $1$-Ball in Standard Discrete Metric Space]] we have that:
:$\map \cl {\m... | Closure of Open Ball may not equal Closed Ball of Same Radius | https://proofwiki.org/wiki/Closure_of_Open_Ball_may_not_equal_Closed_Ball_of_Same_Radius | https://proofwiki.org/wiki/Closure_of_Open_Ball_may_not_equal_Closed_Ball_of_Same_Radius | [
"Open Balls"
] | [
"Definition:Metric Space",
"Definition:Open Ball",
"Definition:Open Ball/Radius",
"Definition:Closed Ball/Metric Space",
"Definition:Open Ball/Radius",
"Definition:Closure (Topology)/Metric Space"
] | [
"Proof by Counterexample",
"Definition:Standard Discrete Metric",
"Definition:Set",
"Closure of Open 1-Ball in Standard Discrete Metric Space"
] |
proofwiki-17934 | Set in Standard Discrete Metric Space has no Limit Points | Let $M = \struct {S, d}$ be the standard discrete metric space on a set $A$.
Let $H \subseteq S$ be a subset of $S$.
Then $H$ has no limit points. | By definition of the standard discrete metric:
:$\map d {x, y} = \begin {cases} 0 & : x = y \\ 1 & : x \ne y \end {cases}$
Let $\alpha \in S$.
By definition, $\alpha$ is a limit point of $H$ {{iff}} ''every'' deleted $\epsilon$-neighborhood $\map {B_\epsilon} \alpha \setminus \set \alpha$ of $\alpha$ contains a point i... | Let $M = \struct {S, d}$ be the [[Definition:Standard Discrete Metric|standard discrete metric space]] on a [[Definition:Set|set]] $A$.
Let $H \subseteq S$ be a [[Definition:Subset|subset]] of $S$.
Then $H$ has no [[Definition:Limit Point (Metric Space)|limit points]]. | By definition of the [[Definition:Standard Discrete Metric|standard discrete metric]]:
:$\map d {x, y} = \begin {cases} 0 & : x = y \\ 1 & : x \ne y \end {cases}$
Let $\alpha \in S$.
By definition, $\alpha$ is a [[Definition:Limit Point (Metric Space)|limit point]] of $H$ {{iff}} ''every'' [[Definition:Deleted Neig... | Set in Standard Discrete Metric Space has no Limit Points | https://proofwiki.org/wiki/Set_in_Standard_Discrete_Metric_Space_has_no_Limit_Points | https://proofwiki.org/wiki/Set_in_Standard_Discrete_Metric_Space_has_no_Limit_Points | [
"Standard Discrete Metric",
"Examples of Limit Points"
] | [
"Definition:Standard Discrete Metric",
"Definition:Set",
"Definition:Subset",
"Definition:Limit Point/Metric Space"
] | [
"Definition:Standard Discrete Metric",
"Definition:Limit Point/Metric Space",
"Definition:Deleted Neighborhood/Metric Space",
"Definition:Open Ball",
"Intersection with Empty Set",
"Definition:Deleted Neighborhood/Metric Space",
"Definition:Limit Point/Metric Space",
"Category:Standard Discrete Metric... |
proofwiki-17935 | Closure of Open 1-Ball in Standard Discrete Metric Space | Let $M = \struct {A, d}$ be the standard discrete metric space on a set $A$.
Let $x \in A$.
Let $\map {B_1} x$ be the open $1$-ball of $x$ in $M$.
Then:
:$\map \cl {\map {B_1} x} = \set x$
while:
:$\set {y \in A: \map d {x, y} \le 1} = A$ | By definition of the standard discrete metric:
:$\map d {x, y} = \begin {cases}
0 & : x = y \\
1 & : x \ne y
\end {cases}$
That is:
:$\forall \tuple {x, y} \in A: \map d {x, y} \le 1$
Thus:
:$\set {y \in A: \map d {x, y} \le 1} = A$
From Open Ball in Standard Discrete Metric Space:
:$\map {B_1} x = \set x$
Let $y \in... | Let $M = \struct {A, d}$ be the [[Definition:Standard Discrete Metric|standard discrete metric space]] on a [[Definition:Set|set]] $A$.
Let $x \in A$.
Let $\map {B_1} x$ be the [[Definition:Open Ball of Metric Space|open $1$-ball of $x$]] in $M$.
Then:
:$\map \cl {\map {B_1} x} = \set x$
while:
:$\set {y \in A: \... | By definition of the [[Definition:Standard Discrete Metric|standard discrete metric]]:
:$\map d {x, y} = \begin {cases}
0 & : x = y \\
1 & : x \ne y
\end {cases}$
That is:
:$\forall \tuple {x, y} \in A: \map d {x, y} \le 1$
Thus:
:$\set {y \in A: \map d {x, y} \le 1} = A$
From [[Open Ball in Standard Discrete Me... | Closure of Open 1-Ball in Standard Discrete Metric Space | https://proofwiki.org/wiki/Closure_of_Open_1-Ball_in_Standard_Discrete_Metric_Space | https://proofwiki.org/wiki/Closure_of_Open_1-Ball_in_Standard_Discrete_Metric_Space | [
"Open Balls"
] | [
"Definition:Standard Discrete Metric",
"Definition:Set",
"Definition:Open Ball"
] | [
"Definition:Standard Discrete Metric",
"Open Ball in Standard Discrete Metric Space",
"Definition:Closure (Topology)/Metric Space",
"Definition:Set",
"Definition:Isolated Point (Metric Space)/Subset",
"Definition:Set",
"Definition:Limit Point/Metric Space",
"Point in Standard Discrete Metric Space is ... |
proofwiki-17936 | Point in Standard Discrete Metric Space is Isolated | Let $M = \struct {S, d}$ be the standard discrete metric space on a set $A$.
Let $H \subseteq S$ be a subset of $S$.
Let $\alpha \in H$.
The $\alpha$ is an isolated point of $H$. | By definition of the standard discrete metric:
:$\map d {x, y} = \begin {cases} 0 & : x = y \\ 1 & : x \ne y \end {cases}$
Let $\alpha \in H$.
Let $\map {B_1} \alpha$ be the open $1$-ball of $\alpha$ in $M$.
Thus:
{{begin-eqn}}
{{eqn | l = \map {B_1} \alpha \cap H
| r = \set {y \in H: \map d {\alpha, y} < 1}
... | Let $M = \struct {S, d}$ be the [[Definition:Standard Discrete Metric|standard discrete metric space]] on a [[Definition:Set|set]] $A$.
Let $H \subseteq S$ be a [[Definition:Subset|subset]] of $S$.
Let $\alpha \in H$.
The $\alpha$ is an [[Definition:Isolated Point of Subset of Metric Space|isolated point]] of $H$. | By definition of the [[Definition:Standard Discrete Metric|standard discrete metric]]:
:$\map d {x, y} = \begin {cases} 0 & : x = y \\ 1 & : x \ne y \end {cases}$
Let $\alpha \in H$.
Let $\map {B_1} \alpha$ be the [[Definition:Open Ball of Metric Space|open $1$-ball of $\alpha$]] in $M$.
Thus:
{{begin-eqn}}
{{eqn... | Point in Standard Discrete Metric Space is Isolated | https://proofwiki.org/wiki/Point_in_Standard_Discrete_Metric_Space_is_Isolated | https://proofwiki.org/wiki/Point_in_Standard_Discrete_Metric_Space_is_Isolated | [
"Standard Discrete Metric",
"Isolated Points"
] | [
"Definition:Standard Discrete Metric",
"Definition:Set",
"Definition:Subset",
"Definition:Isolated Point (Metric Space)/Subset"
] | [
"Definition:Standard Discrete Metric",
"Definition:Open Ball",
"Definition:Isolated Point (Metric Space)/Subset",
"Category:Standard Discrete Metric",
"Category:Isolated Points"
] |
proofwiki-17937 | Interior of Set of Rational Numbers in Real Numbers is Empty | Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Let $\Q$ be the subspace of rational numbers.
Then the interior of $\Q$ in $\R$ is the empty set $\O$. | Consider the set of set of irrational numbers $\R \setminus \Q$.
By definition:
:$\R \setminus \Q = \relcomp \R \Q$
where $\relcomp \R \Q$ denotes the relative complement of $\Q$ in $\R$.
We have that Irrationals are Everywhere Dense in Reals.
Hence by definition of everywhere dense, the closure of $\R \setminus \Q$ in... | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Let $\Q$ be the [[Definition:Topological Subspace|subspace]] of [[Definition:Rational Number|rational numbers]].
Then the [[Definition:Interior (Topology)|interior]] of $\Q... | Consider the [[Definition:Set|set]] of [[Definition:Irrational Number|set of irrational numbers]] $\R \setminus \Q$.
By definition:
:$\R \setminus \Q = \relcomp \R \Q$
where $\relcomp \R \Q$ denotes the [[Definition:Relative Complement|relative complement]] of $\Q$ in $\R$.
We have that [[Irrationals are Everywhere D... | Interior of Set of Rational Numbers in Real Numbers is Empty | https://proofwiki.org/wiki/Interior_of_Set_of_Rational_Numbers_in_Real_Numbers_is_Empty | https://proofwiki.org/wiki/Interior_of_Set_of_Rational_Numbers_in_Real_Numbers_is_Empty | [
"Examples of Set Interiors",
"Rational Number Space"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Topological Subspace",
"Definition:Rational Number",
"Definition:Interior (Topology)",
"Definition:Empty Set"
] | [
"Definition:Set",
"Definition:Irrational Number",
"Definition:Relative Complement",
"Irrationals are Everywhere Dense in Reals",
"Definition:Everywhere Dense",
"Definition:Closure (Topology)",
"Relative Complement with Self is Empty Set",
"Interior equals Complement of Closure of Complement"
] |
proofwiki-17938 | Equivalence of Definitions of Nowhere Dense | Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$.
{{TFAE|def = Nowhere Dense}} | === $(1)$ implies $(2)$ ===
Let $H$ be nowhere dense in $T$ by definition $1$.
Then by definition:
:$\paren {H^-}^\circ = \O$
Hence by definition of interior:
:the union of all subsets of $H$ which are open in $T$.
But this union is empty.
Hence all subsets of $H$ which are open in $T$ must themselves be empty.
Thus $H... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $H \subseteq S$.
{{TFAE|def = Nowhere Dense}} | === $(1)$ implies $(2)$ ===
Let $H$ be [[Definition:Nowhere Dense/Definition 1|nowhere dense in $T$ by definition $1$]].
Then by definition:
:$\paren {H^-}^\circ = \O$
Hence by definition of [[Definition:Interior (Topology)|interior]]:
:the [[Definition:Set Union|union]] of all [[Definition:Subset|subsets]] of $H$ ... | Equivalence of Definitions of Nowhere Dense | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Nowhere_Dense | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Nowhere_Dense | [
"Nowhere Dense"
] | [
"Definition:Topological Space"
] | [
"Definition:Nowhere Dense/Definition 1",
"Definition:Interior (Topology)",
"Definition:Set Union",
"Definition:Subset",
"Definition:Open Set/Topology",
"Definition:Set Union",
"Definition:Empty Set",
"Definition:Subset",
"Definition:Open Set/Topology",
"Definition:Empty Set",
"Definition:Nowhere... |
proofwiki-17939 | Exponential Distribution in terms of Beta Distribution | Let $\sequence {X_n}$ be a sequence of independent random variables with:
:$X_n \sim \BetaDist 1 n$
for each natural number $n$, where $\BetaDist 1 n$ denotes the beta distribution with parameters $1$ and $n$.
Then:
:$n X_n \xrightarrow d X$
with:
:$X \sim \Exponential 1$
where:
:$\Exponential 1$ denotes the exponen... | We aim to show that for each real $x > 0$, we have:
:$\ds \lim_{n \mathop \to \infty} \map \Pr {X_n \le x} = \map \Pr {X \le x}$
From the definition of the exponential distribution, we have:
:$\map \Pr {X \le x} = 1 - e^{-x}$
Note that, from the definition of the beta distribution:
:$0 \le X_n \le 1$
So, if $n \le x$... | Let $\sequence {X_n}$ be a sequence of [[Definition:Independent Random Variables|independent]] [[Definition:Random Variable|random variables]] with:
:$X_n \sim \BetaDist 1 n$
for each [[Definition:Natural Number|natural number]] $n$, where $\BetaDist 1 n$ denotes the [[Definition:Beta Distribution|beta distribution]]... | We aim to show that for each [[Definition:Real Number|real]] $x > 0$, we have:
:$\ds \lim_{n \mathop \to \infty} \map \Pr {X_n \le x} = \map \Pr {X \le x}$
From the definition of the [[Definition:Exponential Distribution|exponential distribution]], we have:
:$\map \Pr {X \le x} = 1 - e^{-x}$
Note that, from the d... | Exponential Distribution in terms of Beta Distribution | https://proofwiki.org/wiki/Exponential_Distribution_in_terms_of_Beta_Distribution | https://proofwiki.org/wiki/Exponential_Distribution_in_terms_of_Beta_Distribution | [
"Exponential Distribution",
"Beta Distribution"
] | [
"Definition:Independent Random Variables",
"Definition:Random Variable",
"Definition:Natural Numbers",
"Definition:Beta Distribution",
"Definition:Exponential Distribution",
"Definition:Convergence in Distribution"
] | [
"Definition:Real Number",
"Definition:Exponential Distribution",
"Definition:Beta Distribution",
"Gamma Difference Equation",
"Primitive of Power",
"Definition:Exponential Function/Real/Limit of Sequence",
"Category:Exponential Distribution",
"Category:Beta Distribution"
] |
proofwiki-17940 | Topologies on Doubleton | Let $S = \set {a, b}$ be a doubleton.
Then there exist $4$ possible different topologies on $S$:
{{begin-eqn}}
{{eqn | l = \tau_a
| r = \set {\O, \set {a, b} }
| c = Indiscrete topology on doubleton
}}
{{eqn | l = \tau_b
| r = \set {\O, \set a, \set {a, b} }
| c = Sierpiński topology
}}
{{eqn | ... | The power set of $S$ is the set:
:$\powerset S = \set {\O, \set a, \set b, \set {a, b} }$
Because all topologies on $S$ are subsets of $\powerset S$, one of the following must hold:
{{begin-eqn}}
{{eqn | l = \tau_1
| r = \O
| c =
}}
{{eqn | l = \tau_2
| r = \set \O
| c =
}}
{{eqn | l = \tau_3
... | Let $S = \set {a, b}$ be a [[Definition:Doubleton|doubleton]].
Then there exist $4$ possible different [[Definition:Topology|topologies]] on $S$:
{{begin-eqn}}
{{eqn | l = \tau_a
| r = \set {\O, \set {a, b} }
| c = [[Definition:Indiscrete Topology on Doubleton|Indiscrete topology on doubleton]]
}}
{{eqn |... | The [[Definition:Power Set|power set]] of $S$ is the set:
:$\powerset S = \set {\O, \set a, \set b, \set {a, b} }$
Because all [[Definition:Topology|topologies]] on $S$ are [[Definition:Subset|subsets]] of $\powerset S$, one of the following must hold:
{{begin-eqn}}
{{eqn | l = \tau_1
| r = \O
| c =
}}
... | Topologies on Doubleton | https://proofwiki.org/wiki/Topologies_on_Doubleton | https://proofwiki.org/wiki/Topologies_on_Doubleton | [
"Doubletons",
"Examples of Topologies"
] | [
"Definition:Doubleton",
"Definition:Topology",
"Definition:Indiscrete Topology on Two-Point Set",
"Definition:Sierpiński Space",
"Definition:Sierpiński Space",
"Definition:Discrete Topology on Two-Point Set"
] | [
"Definition:Power Set",
"Definition:Topology",
"Definition:Subset",
"Definition:Topology",
"Definition:Element",
"Definition:Topology",
"Definition:Topology",
"Empty Set is Element of Topology",
"Definition:Topology",
"Definition:Topology",
"Indiscrete Topology is Topology",
"Definition:Topolo... |
proofwiki-17941 | Topologies on Set with 3 Elements | Let $S = \set {a, b, c}$ be a set with $3$ elements.
Then there exist $29$ possible different topologies on $S$. | The power set of $S$ is the set:
:$\powerset S = \set {\O, \set a, \set b, \set c, \set {a, b}, \set {a, c}, \set {b, c}, \set {a, b, c} }$
A topology on $S$ is a subset of $\powerset S$.
Thus the set of all topologies on $S$ is a subset of the power set of $\powerset S$.
From Cardinality of Power Set of Finite Set:
:$... | Let $S = \set {a, b, c}$ be a [[Definition:Set|set]] with $3$ [[Definition:Element|elements]].
Then there exist $29$ possible different [[Definition:Topology|topologies]] on $S$. | The [[Definition:Power Set|power set]] of $S$ is the set:
:$\powerset S = \set {\O, \set a, \set b, \set c, \set {a, b}, \set {a, c}, \set {b, c}, \set {a, b, c} }$
A [[Definition:Topology|topology]] on $S$ is a [[Definition:Subset|subset]] of $\powerset S$.
Thus the [[Definition:Set|set]] of all [[Definition:Topolo... | Topologies on Set with 3 Elements | https://proofwiki.org/wiki/Topologies_on_Set_with_3_Elements | https://proofwiki.org/wiki/Topologies_on_Set_with_3_Elements | [
"Topologies on Set with 3 Elements",
"Examples of Topologies",
"3"
] | [
"Definition:Set",
"Definition:Element",
"Definition:Topology"
] | [
"Definition:Power Set",
"Definition:Topology",
"Definition:Subset",
"Definition:Set",
"Definition:Topology",
"Definition:Subset",
"Definition:Power Set",
"Cardinality of Power Set of Finite Set",
"Definition:Subset",
"Definition:Topology",
"Definition:Subset",
"Definition:Topology",
"Definit... |
proofwiki-17942 | Set of 2-Dimensional Real Orthogonal Matrices is Compact in Normed Real Square Matrix Vector Space | Let $\struct {\R^{2 \times2}, \norm {\, \cdot \,}_\infty}$ be the normed 2-dimensional real square matrix vector space.
Let $\map O 2 := \set {\mathbf R \in \R^{2 \times2}: \mathbf R^\intercal \mathbf R = \mathbf I_2}$ be the orthogonal group of degree $2$ over real numbers.
Then $\map O 2$ is a compact set in $\struct... | {{tidy|eqn template could (and probably should) be used on this page to good effect}}
Let $\sequence {\mathbf R_n}_{n \mathop \in \N}$ be a sequence in $\map O 2$.
Let:
:<nowiki>$\begin {bmatrix}
a_n & b_n \\
c_n & d_n \\
\end {bmatrix} := \mathbf R_n$</nowiki>
where $\sequence {a_n}_{n \mathop \in \N}$, $\sequence {b_... | Let $\struct {\R^{2 \times2}, \norm {\, \cdot \,}_\infty}$ be the [[Definition:Normed 2-Dimensional Real Square Matrix Vector Space|normed 2-dimensional real square matrix vector space]].
Let $\map O 2 := \set {\mathbf R \in \R^{2 \times2}: \mathbf R^\intercal \mathbf R = \mathbf I_2}$ be the [[Definition:Orthogonal G... | {{tidy|eqn template could (and probably should) be used on this page to good effect}}
Let $\sequence {\mathbf R_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $\map O 2$.
Let:
:<nowiki>$\begin {bmatrix}
a_n & b_n \\
c_n & d_n \\
\end {bmatrix} := \mathbf R_n$</nowiki>
where $\sequence {a_n}_{n \mat... | Set of 2-Dimensional Real Orthogonal Matrices is Compact in Normed Real Square Matrix Vector Space | https://proofwiki.org/wiki/Set_of_2-Dimensional_Real_Orthogonal_Matrices_is_Compact_in_Normed_Real_Square_Matrix_Vector_Space | https://proofwiki.org/wiki/Set_of_2-Dimensional_Real_Orthogonal_Matrices_is_Compact_in_Normed_Real_Square_Matrix_Vector_Space | [
"Matrix Groups",
"Compact Normed Vector Spaces"
] | [
"Definition:Normed 2-Dimensional Real Square Matrix Vector Space",
"Definition:Orthogonal Group",
"Definition:Real Number",
"Definition:Compact Space/Normed Vector Space"
] | [
"Definition:Sequence",
"Definition:Real Sequence",
"Bolzano-Weierstrass Theorem",
"Definition:Set",
"Definition:Finite Subset",
"Definition:Set",
"Definition:Real Number",
"Real Numbers form Totally Ordered Field",
"Finite Non-Empty Subset of Ordered Set has Maximal and Minimal Elements",
"Definit... |
proofwiki-17943 | Mapping from Unit Circle defines Periodic Function | Let $\SS$ denote the unit circle whose center is at the origin of the Cartesian plane $\R^2$.
Let $p: \R \to \SS$ be the mapping defined as:
:$\forall x \in \R: \map p x = \tuple {\cos x, \sin x}$
Let $f': \SS \to \R$ be a real-valued function.
Then the composition $f' \circ p$ defines a periodic real function whose pe... | Let $f := f' \circ p$ denote the composition of $f$ with $p$.
We have:
{{begin-eqn}}
{{eqn | q = \forall x \in \R
| l = \map f {x + 2 \pi}
| r = \map {f'} {\map p {x + 2 \pi} }
| c = {{Defof|Composition of Mappings}}
}}
{{eqn | r = \map {f'} {\map \cos {x + 2 \pi}, \map \sin {x + 2 \pi} }
| c = ... | Let $\SS$ denote the [[Definition:Unit Circle|unit circle]] whose [[Definition:Center of Circle|center]] is at the [[Definition:Origin|origin]] of the [[Definition:Cartesian Plane|Cartesian plane]] $\R^2$.
Let $p: \R \to \SS$ be the [[Definition:Mapping|mapping]] defined as:
:$\forall x \in \R: \map p x = \tuple {\cos... | Let $f := f' \circ p$ denote the [[Definition:Composition of Mappings|composition]] of $f$ with $p$.
We have:
{{begin-eqn}}
{{eqn | q = \forall x \in \R
| l = \map f {x + 2 \pi}
| r = \map {f'} {\map p {x + 2 \pi} }
| c = {{Defof|Composition of Mappings}}
}}
{{eqn | r = \map {f'} {\map \cos {x + 2 \... | Mapping from Unit Circle defines Periodic Function | https://proofwiki.org/wiki/Mapping_from_Unit_Circle_defines_Periodic_Function | https://proofwiki.org/wiki/Mapping_from_Unit_Circle_defines_Periodic_Function | [
"Periodic Functions"
] | [
"Definition:Unit Circle",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Plane",
"Definition:Mapping",
"Definition:Real-Valued Function",
"Definition:Composition of Mappings",
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period"
] | [
"Definition:Composition of Mappings",
"Cosine of Angle plus Full Angle",
"Sine of Angle plus Full Angle"
] |
proofwiki-17944 | Periodic Function as Mapping from Unit Circle | Let $\SS$ denote the unit circle whose center is at the origin of the Cartesian plane $\R^2$.
Let $p: \R \to \SS$ be the mapping defined as:
:$\forall x \in \R: \map p x = \tuple {\cos x, \sin x}$
Let $f: \R \to \R$ be a periodic real function whose period is $2 \pi$.
Then there exists a well-defined real-valued functi... | Let $f': \SS \to \R$ be defined as:
:$\forall \tuple {x, y} \in \SS: \map {f'} {x, y} = \map f x$
Consider the inverse $p^{-1}: \SS \to \R$ of $p$:
:$\forall \tuple {x', y'} \in \SS: p^{-1} \sqbrk {x', y'} = \set {x \in \R: \cos x = x', \sin x = y'}$
Let $\RR$ be the equivalence relation on $\R$ induced by $p$:
:$\fora... | Let $\SS$ denote the [[Definition:Unit Circle|unit circle]] whose [[Definition:Center of Circle|center]] is at the [[Definition:Origin|origin]] of the [[Definition:Cartesian Plane|Cartesian plane]] $\R^2$.
Let $p: \R \to \SS$ be the [[Definition:Mapping|mapping]] defined as:
:$\forall x \in \R: \map p x = \tuple {\cos... | Let $f': \SS \to \R$ be defined as:
:$\forall \tuple {x, y} \in \SS: \map {f'} {x, y} = \map f x$
Consider the [[Definition:Inverse of Mapping|inverse]] $p^{-1}: \SS \to \R$ of $p$:
:$\forall \tuple {x', y'} \in \SS: p^{-1} \sqbrk {x', y'} = \set {x \in \R: \cos x = x', \sin x = y'}$
Let $\RR$ be the [[Definition:E... | Periodic Function as Mapping from Unit Circle | https://proofwiki.org/wiki/Periodic_Function_as_Mapping_from_Unit_Circle | https://proofwiki.org/wiki/Periodic_Function_as_Mapping_from_Unit_Circle | [
"Periodic Functions"
] | [
"Definition:Unit Circle",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Plane",
"Definition:Mapping",
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period",
"Definition:Well-Defined/Mapping",
"Definition:Real-Valued Function",
"D... | [
"Definition:Inverse of Mapping",
"Definition:Equivalence Relation Induced by Mapping",
"Quotient Theorem for Sets",
"Definition:Well-Defined/Mapping",
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period",
"Conditions for Commutative Diagram on Quotient Mappings between Mapping... |
proofwiki-17945 | Periodic Function is Continuous iff Mapping from Unit Circle is Continuous | Let $f: \R \to \R$ be a periodic real function whose period is $2 \pi$.
Let $\SS$ denote the unit circle whose center is at the origin of the Cartesian plane $\R^2$.
Let $p: \R \to \SS$ be the mapping defined as:
:$\forall x \in \R: \map p x = \tuple {\cos x, \sin x}$
Let $f': \SS \to \R$ be the well-defined real-value... | The existence and well-definedness of $f'$ are demonstrated in Periodic Function as Mapping from Unit Circle. | Let $f: \R \to \R$ be a [[Definition:Periodic Real Function|periodic real function]] whose [[Definition:Period of Periodic Real Function|period]] is $2 \pi$.
Let $\SS$ denote the [[Definition:Unit Circle|unit circle]] whose [[Definition:Center of Circle|center]] is at the [[Definition:Origin|origin]] of the [[Definiti... | The existence and [[Definition:Well-Defined Mapping|well-definedness]] of $f'$ are demonstrated in [[Periodic Function as Mapping from Unit Circle]]. | Periodic Function is Continuous iff Mapping from Unit Circle is Continuous | https://proofwiki.org/wiki/Periodic_Function_is_Continuous_iff_Mapping_from_Unit_Circle_is_Continuous | https://proofwiki.org/wiki/Periodic_Function_is_Continuous_iff_Mapping_from_Unit_Circle_is_Continuous | [
"Periodic Functions"
] | [
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period",
"Definition:Unit Circle",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Plane",
"Definition:Mapping",
"Definition:Well-Defined/Mapping",
"Definition:Real-Valued Function",
"D... | [
"Definition:Well-Defined/Mapping",
"Periodic Function as Mapping from Unit Circle"
] |
proofwiki-17946 | Set of 2-Dimensional Indefinite Real Orthogonal Matrices is not Compact in Normed Real Square Matrix Vector Space | Let $\struct {\R^{2 \times2}, \norm {\, \cdot \,}_\infty}$ be the normed real matrix vector space.
Let $\map O {1, 1} := \set {\mathbf R \in \R^{2 \times2} : \mathbf R^\intercal \mathbf J_{1,1} \mathbf R = \mathbf J_{1,1}}$ be the indefinite orthogonal group of degree $\paren {1, 1}$ over real numbers where:
:<nowiki>$... | Let:
:<nowiki>$\begin{bmatrix}
\map \cosh t& \map \sinh t\\
\map \sinh t & \map \cosh t \\
\end{bmatrix} := \map {\mathbf R} t$</nowiki>
We have that:
{{begin-eqn}}
{{eqn | l = \map {\mathbf R^\intercal} t \mathbf J_{1,1} \map {\mathbf R} t
| r = <nowiki>\begin{bmatrix}
\map \cosh t & \map \sinh t \\
\map \sinh t... | Let $\struct {\R^{2 \times2}, \norm {\, \cdot \,}_\infty}$ be the [[Definition:Normed Real Matrix Vector Space|normed real matrix vector space]].
Let $\map O {1, 1} := \set {\mathbf R \in \R^{2 \times2} : \mathbf R^\intercal \mathbf J_{1,1} \mathbf R = \mathbf J_{1,1}}$ be the [[Definition:Indefinite Orthogonal Group|... | Let:
:<nowiki>$\begin{bmatrix}
\map \cosh t& \map \sinh t\\
\map \sinh t & \map \cosh t \\
\end{bmatrix} := \map {\mathbf R} t$</nowiki>
We have that:
{{begin-eqn}}
{{eqn | l = \map {\mathbf R^\intercal} t \mathbf J_{1,1} \map {\mathbf R} t
| r = <nowiki>\begin{bmatrix}
\map \cosh t & \map \sinh t \\
\map \sin... | Set of 2-Dimensional Indefinite Real Orthogonal Matrices is not Compact in Normed Real Square Matrix Vector Space | https://proofwiki.org/wiki/Set_of_2-Dimensional_Indefinite_Real_Orthogonal_Matrices_is_not_Compact_in_Normed_Real_Square_Matrix_Vector_Space | https://proofwiki.org/wiki/Set_of_2-Dimensional_Indefinite_Real_Orthogonal_Matrices_is_not_Compact_in_Normed_Real_Square_Matrix_Vector_Space | [
"Matrix Groups",
"Compact Normed Vector Spaces"
] | [
"Definition:Normed Real Matrix Vector Space",
"Definition:Indefinite Orthogonal Group",
"Definition:Real Number",
"Definition:Compact Space/Normed Vector Space"
] | [
"Difference of Squares of Hyperbolic Cosine and Sine",
"Difference of Squares of Hyperbolic Cosine and Sine",
"Definition:Set",
"Definition:Matrix/Element",
"Definition:Finite Subset",
"Definition:Set",
"Definition:Real Number",
"Real Numbers form Totally Ordered Field",
"Finite Non-Empty Subset of ... |
proofwiki-17947 | Real-Valued Mapping is Continuous if Inverse Images of Unbounded Open Intervals are Open | Let $T = \struct {S, \tau}$ be a topological space.
Let the real number line $\R$ be considered as a topology under the usual (Euclidean) topology.
Let $f: T \to \R$ be a real-valued function on $T$.
Then:
:$f$ is continuous
{{iff}}:
:for all $a \in \R$: $f^{-1} \openint \gets a$ and $f^{-1} \openint a \to$ are open in... | === Sufficient Condition ===
Let $f$ be a continuous mapping.
From {{Corollary|Open Real Interval is Open Set}}, both $\openint \gets a$ and $\openint a \to$ are open in $\R$.
Then by definition of continuous mapping, $f^{-1} \openint \gets a$ and $f^{-1} \openint a \to$ are both open in $T$.
{{qed|lemma}}
=== Necessar... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let the [[Definition:Real Number Line|real number line]] $\R$ be considered as a [[Definition:Topology|topology]] under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]].
Let $f: T \to \R$ be a [[De... | === Sufficient Condition ===
Let $f$ be a [[Definition:Continuous Mapping (Topology)|continuous mapping]].
From {{Corollary|Open Real Interval is Open Set}}, both $\openint \gets a$ and $\openint a \to$ are [[Definition:Open Set (Topology)|open]] in $\R$.
Then by definition of [[Definition:Continuous Mapping (Topolo... | Real-Valued Mapping is Continuous if Inverse Images of Unbounded Open Intervals are Open/Proof 1 | https://proofwiki.org/wiki/Real-Valued_Mapping_is_Continuous_if_Inverse_Images_of_Unbounded_Open_Intervals_are_Open | https://proofwiki.org/wiki/Real-Valued_Mapping_is_Continuous_if_Inverse_Images_of_Unbounded_Open_Intervals_are_Open/Proof_1 | [
"Real-Valued Mapping is Continuous if Inverse Images of Unbounded Open Intervals are Open",
"Continuous Real-Valued Functions"
] | [
"Definition:Topological Space",
"Definition:Real Number/Real Number Line",
"Definition:Topology",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Real-Valued Function",
"Definition:Continuous Mapping (Topology)",
"Definition:Open Set/Topology"
] | [
"Definition:Continuous Mapping (Topology)",
"Definition:Open Set/Topology",
"Definition:Continuous Mapping (Topology)",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Sub-Basis for Real Number Line",
"Definition:Sub-Basis",
"Definition:Open Set/Topology",
"Preimage of Intersection ... |
proofwiki-17948 | Real-Valued Mapping is Continuous if Inverse Images of Unbounded Open Intervals are Open | Let $T = \struct {S, \tau}$ be a topological space.
Let the real number line $\R$ be considered as a topology under the usual (Euclidean) topology.
Let $f: T \to \R$ be a real-valued function on $T$.
Then:
:$f$ is continuous
{{iff}}:
:for all $a \in \R$: $f^{-1} \openint \gets a$ and $f^{-1} \openint a \to$ are open in... | From Sub-Basis for Real Number Line:
:$\set {\openint \gets a, \openint b \to: a, b \in \R}$ is a sub-basis for $\R$.
The result follows from Continuity Test using Sub-Basis.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let the [[Definition:Real Number Line|real number line]] $\R$ be considered as a [[Definition:Topology|topology]] under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]].
Let $f: T \to \R$ be a [[De... | From [[Sub-Basis for Real Number Line]]:
:$\set {\openint \gets a, \openint b \to: a, b \in \R}$ is a [[Definition:Sub-Basis|sub-basis]] for $\R$.
The result follows from [[Continuity Test using Sub-Basis]].
{{qed}} | Real-Valued Mapping is Continuous if Inverse Images of Unbounded Open Intervals are Open/Proof 2 | https://proofwiki.org/wiki/Real-Valued_Mapping_is_Continuous_if_Inverse_Images_of_Unbounded_Open_Intervals_are_Open | https://proofwiki.org/wiki/Real-Valued_Mapping_is_Continuous_if_Inverse_Images_of_Unbounded_Open_Intervals_are_Open/Proof_2 | [
"Real-Valued Mapping is Continuous if Inverse Images of Unbounded Open Intervals are Open",
"Continuous Real-Valued Functions"
] | [
"Definition:Topological Space",
"Definition:Real Number/Real Number Line",
"Definition:Topology",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Real-Valued Function",
"Definition:Continuous Mapping (Topology)",
"Definition:Open Set/Topology"
] | [
"Sub-Basis for Real Number Line",
"Definition:Sub-Basis",
"Continuity Test using Sub-Basis"
] |
proofwiki-17949 | Open Balls on Rational Centers form Basis for Usual Topology on Plane | Let $\R^2$ be the real number plane with the usual (Euclidean) topology.
Let $S$ be the set defined as:
:$S = \set {\tuple {x, y} \in \R^2: x, y \in \Q}$
That is, let $S$ be the set of all points in $\R^2$ whose coordinates are rational numbers.
Let $\BB$ denote the set defined as:
:$\BB = \set {\map {B_q} s: s \in S, ... | Let $d: \R^2 \times \R^2 \to \R$ be the usual (Euclidean) metric on $\R^2$.
Let $U$ be an open set of $\R^2$.
Let $z = \tuple {x, y} \in U$.
Then by definition of open set:
:$\exists \epsilon \in \R_{>0}: \map {B_\epsilon} z \subseteq U$
By Rationals are Everywhere Dense in Topological Space of Reals:
:$\exists \tuple ... | Let $\R^2$ be the [[Definition:Real Number Plane with Euclidean Topology|real number plane with the usual (Euclidean) topology]].
Let $S$ be the [[Definition:Set|set]] defined as:
:$S = \set {\tuple {x, y} \in \R^2: x, y \in \Q}$
That is, let $S$ be the [[Definition:Set|set]] of all [[Definition:Point|points]] in $\... | Let $d: \R^2 \times \R^2 \to \R$ be the [[Definition:Euclidean Metric|usual (Euclidean) metric]] on $\R^2$.
Let $U$ be an [[Definition:Open Set (Topology)|open set]] of $\R^2$.
Let $z = \tuple {x, y} \in U$.
Then by definition of [[Definition:Open Set (Topology)|open set]]:
:$\exists \epsilon \in \R_{>0}: \map {B_\e... | Open Balls on Rational Centers form Basis for Usual Topology on Plane | https://proofwiki.org/wiki/Open_Balls_on_Rational_Centers_form_Basis_for_Usual_Topology_on_Plane | https://proofwiki.org/wiki/Open_Balls_on_Rational_Centers_form_Basis_for_Usual_Topology_on_Plane | [
"Topological Bases"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Plane",
"Definition:Set",
"Definition:Set",
"Definition:Point",
"Definition:Cartesian Coordinate System",
"Definition:Rational Number",
"Definition:Set",
"Definition:Set",
"Definition:Open Ball",
"Definition:Open Ball/Center",
"Definitio... | [
"Definition:Euclidean Metric",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Rational Numbers are Everywhere Dense in Set of Real Numbers/Topology",
"Definition:Set Union",
"Definition:Open Ball",
"Definition:Open Ball/Center",
"Definition:Rational Number",
"Definition:Open Ball/R... |
proofwiki-17950 | Openness Relation on Topological Spaces is Transitive | Let $T_1 = \struct {S_1, \tau_1}$ be a topological space
Let $S_2 \subseteq S_1$ be a subset of $S_1$.
Let $S_3 \subseteq S_2$ be a subset of $S_2$.
Let $T_2 = \struct {S_2, \tau_2}$ be the topological subspace of $T_1$ such that $\tau_2$ is the subspace topology induced by $\tau_1$.
Let $T_3 = \struct {S_3, \tau_3}$ b... | We have by definition of subspace topology that:
:$\tau_2 = \set {U \cap S_2: U \in \tau_1}$
Then we have {{hypothesis}} that:
:$S_3 \in \tau_2$
and so:
:$S_3 \in \set {U \cap S_2: U \in \tau_1}$
That is, $S_3$ is the intersection of $U$ and $S_2$, both of which are open sets of $T_1$.
Hence by {{Open-set-axiom|2}}, $S... | Let $T_1 = \struct {S_1, \tau_1}$ be a [[Definition:Topological Space|topological space]]
Let $S_2 \subseteq S_1$ be a [[Definition:Subset|subset]] of $S_1$.
Let $S_3 \subseteq S_2$ be a [[Definition:Subset|subset]] of $S_2$.
Let $T_2 = \struct {S_2, \tau_2}$ be the [[Definition:Topological Subspace|topological subs... | We have by definition of [[Definition:Subspace Topology|subspace topology]] that:
:$\tau_2 = \set {U \cap S_2: U \in \tau_1}$
Then we have {{hypothesis}} that:
:$S_3 \in \tau_2$
and so:
:$S_3 \in \set {U \cap S_2: U \in \tau_1}$
That is, $S_3$ is the [[Definition:Set Intersection|intersection]] of $U$ and $S_2$, bot... | Openness Relation on Topological Spaces is Transitive | https://proofwiki.org/wiki/Openness_Relation_on_Topological_Spaces_is_Transitive | https://proofwiki.org/wiki/Openness_Relation_on_Topological_Spaces_is_Transitive | [
"Topological Subspaces"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Subset",
"Definition:Topological Subspace",
"Definition:Topological Subspace",
"Definition:Topological Subspace",
"Definition:Topological Subspace",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Open Set... | [
"Definition:Topological Subspace",
"Definition:Set Intersection",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology"
] |
proofwiki-17951 | Induced Topology on Subspace of Subspace Coincides with Induced Topology on Subspace | Let $T_1 = \struct {S_1, \tau_1}$ be a topological space
Let $S_2 \subseteq S_1$ be a subset of $S_1$.
Let $S_3 \subseteq S_2$ be a subset of $S_2$.
Let $T_2 = \struct {S_2, \tau_2}$ be the topological subspace of $T_1$ such that $\tau_2$ is the subspace topology on $T_2$ induced by $\tau_1$.
Let $T_3 = \struct {S_3, \... | Let $\tau_P$ denote the subspace topology on $T_3$ induced by $\tau_1$.
Let $\tau_Q$ denote the subspace topology on $T_3$ induced by $\tau_2$.
The object of this exercise is to demonstrate that $\tau_P = \tau_Q$.
This will be done by showing that an arbitrary set $V$ is in $\tau_P$ {{iff}} $V$ is in $\tau_Q$.
We have ... | Let $T_1 = \struct {S_1, \tau_1}$ be a [[Definition:Topological Space|topological space]]
Let $S_2 \subseteq S_1$ be a [[Definition:Subset|subset]] of $S_1$.
Let $S_3 \subseteq S_2$ be a [[Definition:Subset|subset]] of $S_2$.
Let $T_2 = \struct {S_2, \tau_2}$ be the [[Definition:Topological Subspace|topological sub... | Let $\tau_P$ denote the [[Definition:Subspace Topology|subspace topology]] on $T_3$ induced by $\tau_1$.
Let $\tau_Q$ denote the [[Definition:Subspace Topology|subspace topology]] on $T_3$ induced by $\tau_2$.
The object of this exercise is to demonstrate that $\tau_P = \tau_Q$.
This will be done by showing that an ... | Induced Topology on Subspace of Subspace Coincides with Induced Topology on Subspace | https://proofwiki.org/wiki/Induced_Topology_on_Subspace_of_Subspace_Coincides_with_Induced_Topology_on_Subspace | https://proofwiki.org/wiki/Induced_Topology_on_Subspace_of_Subspace_Coincides_with_Induced_Topology_on_Subspace | [
"Topological Subspaces"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Subset",
"Definition:Topological Subspace",
"Definition:Topological Subspace",
"Definition:Topological Subspace",
"Definition:Topological Subspace",
"Definition:Set",
"Definition:Subset",
"Definition:Topological Subspace"
] | [
"Definition:Topological Subspace",
"Definition:Topological Subspace",
"Definition:Set",
"Intersection with Subset is Subset",
"Intersection is Associative"
] |
proofwiki-17952 | Real-Valued Function on Finite Set is Bounded | Let $S$ be a finite set.
Let $f: S \to \R$ be a real-valued function on $S$.
Then $f$ is bounded. | Let $K$ be defined as:
:$K = \ds \max_{x \mathop \in S} \size {\map f x}$
where $\size {\map f x}$ denotes the absolute value of $\map f x$.
Then trivially:
:$\exists K \in \R_{\ge 0}: \forall x \in S: \size {\map f x} \le K$
This is the definition of a bounded real-valued function.
{{qed}} | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $f: S \to \R$ be a [[Definition:Real-Valued Function|real-valued function]] on $S$.
Then $f$ is [[Definition:Bounded Real-Valued Function|bounded]]. | Let $K$ be defined as:
:$K = \ds \max_{x \mathop \in S} \size {\map f x}$
where $\size {\map f x}$ denotes the [[Definition:Absolute Value|absolute value]] of $\map f x$.
Then trivially:
:$\exists K \in \R_{\ge 0}: \forall x \in S: \size {\map f x} \le K$
This is the definition of a [[Definition:Bounded Real-Valued F... | Real-Valued Function on Finite Set is Bounded | https://proofwiki.org/wiki/Real-Valued_Function_on_Finite_Set_is_Bounded | https://proofwiki.org/wiki/Real-Valued_Function_on_Finite_Set_is_Bounded | [
"Bounded Real-Valued Functions",
"Real-Valued Functions",
"Finite Sets"
] | [
"Definition:Finite Set",
"Definition:Real-Valued Function",
"Definition:Bounded Mapping/Real-Valued"
] | [
"Definition:Absolute Value",
"Definition:Bounded Mapping/Real-Valued"
] |
proofwiki-17953 | Reciprocal Function is Continuous on Real Numbers without Zero | Let $\R_{\ne 0}$ denote the real numbers excluding $0$:
:$\R_{\ne 0} := \R \setminus \set 0$.
Let $f: \R_{\ne 0} \to \R$ denote the reciprocal function:
:$\forall x \in \R_{\ne 0}: \map f x = \dfrac 1 x$
Then $f$ is continuous on all real intervals which do not include $0$. | From Identity Mapping is Continuous, the real function $g$ defined as:
:$\forall x \in \R: \map g x = x$
is continuous on $\R$.
From Constant Mapping is Continuous, the real function $h$ defined as:
:$\forall x \in \R: \map x h = 1$
We note that $\map g 0 = 0$.
The result then follows from Quotient Rule for Continuous ... | Let $\R_{\ne 0}$ denote the [[Definition:Real Number|real numbers]] excluding $0$:
:$\R_{\ne 0} := \R \setminus \set 0$.
Let $f: \R_{\ne 0} \to \R$ denote the [[Definition:Reciprocal|reciprocal function]]:
:$\forall x \in \R_{\ne 0}: \map f x = \dfrac 1 x$
Then $f$ is [[Definition:Continuous Real Function|continuous]... | From [[Identity Mapping is Continuous]], the [[Definition:Real Function|real function]] $g$ defined as:
:$\forall x \in \R: \map g x = x$
is [[Definition:Continuous Real Function|continuous]] on $\R$.
From [[Constant Mapping is Continuous]], the [[Definition:Real Function|real function]] $h$ defined as:
:$\forall x \i... | Reciprocal Function is Continuous on Real Numbers without Zero | https://proofwiki.org/wiki/Reciprocal_Function_is_Continuous_on_Real_Numbers_without_Zero | https://proofwiki.org/wiki/Reciprocal_Function_is_Continuous_on_Real_Numbers_without_Zero | [
"Reciprocals",
"Examples of Continuous Real Functions"
] | [
"Definition:Real Number",
"Definition:Reciprocal",
"Definition:Continuous Real Function",
"Definition:Real Interval"
] | [
"Identity Mapping is Continuous",
"Definition:Real Function",
"Definition:Continuous Real Function",
"Constant Mapping is Continuous",
"Definition:Real Function",
"Combination Theorem for Continuous Functions/Real/Quotient Rule",
"Definition:Continuous Real Function",
"Category:Reciprocals",
"Catego... |
proofwiki-17954 | Reciprocal Function is Unbounded on Open Unit Interval | Let $A = \openint 0 1$ denote the open unit interval.
Let $f: A \to \R$ be the reciprocal function:
:$\forall x \in A: \map f x := \dfrac 1 x$
Then $f$ is unbounded. | Let $K \in \R_{>0}$.
Then:
:$\exists x \in \R: 0 < x < \dfrac 1 K$ such that $x < 1$.
Then we have:
:$\map f x = \dfrac 1 x > K$
So whatever $K$ may be, it can never be large enough to be a bound of $f$ on $\openint 0 1$.
{{qed}} | Let $A = \openint 0 1$ denote the [[Definition:Open Unit Interval|open unit interval]].
Let $f: A \to \R$ be the [[Definition:Reciprocal|reciprocal function]]:
:$\forall x \in A: \map f x := \dfrac 1 x$
Then $f$ is [[Definition:Unbounded Real-Valued Function|unbounded]]. | Let $K \in \R_{>0}$.
Then:
:$\exists x \in \R: 0 < x < \dfrac 1 K$ such that $x < 1$.
Then we have:
:$\map f x = \dfrac 1 x > K$
So whatever $K$ may be, it can never be large enough to be a [[Definition:Bound of Real-Valued Function|bound]] of $f$ on $\openint 0 1$.
{{qed}} | Reciprocal Function is Unbounded on Open Unit Interval | https://proofwiki.org/wiki/Reciprocal_Function_is_Unbounded_on_Open_Unit_Interval | https://proofwiki.org/wiki/Reciprocal_Function_is_Unbounded_on_Open_Unit_Interval | [
"Reciprocals",
"Examples of Unbounded Real-Valued Functions"
] | [
"Definition:Real Interval/Unit Interval/Open",
"Definition:Reciprocal",
"Definition:Bounded Mapping/Real-Valued/Unbounded"
] | [
"Definition:Bound of Real-Valued Function"
] |
proofwiki-17955 | Continuous Real-Valued Function is not necessarily Bounded | Let $S$ be a set.
Let $f: S \to \R$ be a continuous real-valued function.
Then $f$ is not necessarily bounded. | Let $S$ denote the (open) real interval $\openint 0 1$.
Let $f: S \to \R$ denote the reciprocal function:
:$\forall x \in S: \map f x = \dfrac 1 x$
From Reciprocal Function is Continuous on Real Numbers without Zero, $f$ is continuous on $S$.
From Reciprocal Function is Unbounded on Open Unit Interval, $f$ is unbounded... | Let $S$ be a [[Definition:Set|set]].
Let $f: S \to \R$ be a [[Definition:Continuous Real-Valued Function|continuous real-valued function]].
Then $f$ is not necessarily [[Definition:Bounded Real-Valued Function|bounded]]. | Let $S$ denote the [[Definition:Open Real Interval|(open) real interval]] $\openint 0 1$.
Let $f: S \to \R$ denote the [[Definition:Reciprocal|reciprocal function]]:
:$\forall x \in S: \map f x = \dfrac 1 x$
From [[Reciprocal Function is Continuous on Real Numbers without Zero]], $f$ is [[Definition:Continuous Real-V... | Continuous Real-Valued Function is not necessarily Bounded | https://proofwiki.org/wiki/Continuous_Real-Valued_Function_is_not_necessarily_Bounded | https://proofwiki.org/wiki/Continuous_Real-Valued_Function_is_not_necessarily_Bounded | [
"Continuous Real-Valued Functions",
"Bounded Real-Valued Functions"
] | [
"Definition:Set",
"Definition:Continuous Real-Valued Vector Function",
"Definition:Bounded Mapping/Real-Valued"
] | [
"Definition:Real Interval/Open",
"Definition:Reciprocal",
"Reciprocal Function is Continuous on Real Numbers without Zero",
"Definition:Continuous Real-Valued Vector Function",
"Reciprocal Function is Unbounded on Open Unit Interval",
"Definition:Bounded Mapping/Real-Valued/Unbounded",
"Proof by Counter... |
proofwiki-17956 | Set of Inverse Positive Integers with Zero is Compact | Let $K$ be the set of inverse positive integers with zero:
:$\ds K := \set {1, \frac 1 2, \frac 1 3, \dots} \cup \set 0$
Let $\struct {\R, \size {\, \cdot \,}}$ be the normed vector space of real numbers.
Then $K$ is compact in real numbers. | We have that $K \subset \closedint 0 1$.
Hence, $K$ is bounded.
Furthermore:
:$\ds \R \setminus K = \openint {-\infty} 0 \cup \paren {\bigcup_{n \mathop = 1}^\infty \openint {\frac 1 {n + 1}} {\frac 1 n}} \cup \openint 1 \infty$
By Union of Open Sets of Normed Vector Space is Open, $\R \setminus K$ is open.
By definiti... | Let $K$ be the [[Definition:Set|set]] of inverse [[Definition:Positive Integer|positive integers]] with [[Definition:Zero (Number)|zero]]:
:$\ds K := \set {1, \frac 1 2, \frac 1 3, \dots} \cup \set 0$
Let $\struct {\R, \size {\, \cdot \,}}$ be the [[Real Numbers with Absolute Value form Normed Vector Space|normed vec... | We have that $K \subset \closedint 0 1$.
Hence, $K$ is [[Definition:Bounded Normed Vector Space|bounded]].
Furthermore:
:$\ds \R \setminus K = \openint {-\infty} 0 \cup \paren {\bigcup_{n \mathop = 1}^\infty \openint {\frac 1 {n + 1}} {\frac 1 n}} \cup \openint 1 \infty$
By [[Union of Open Sets of Normed Vector Spa... | Set of Inverse Positive Integers with Zero is Compact | https://proofwiki.org/wiki/Set_of_Inverse_Positive_Integers_with_Zero_is_Compact | https://proofwiki.org/wiki/Set_of_Inverse_Positive_Integers_with_Zero_is_Compact | [
"Integers",
"Compact Vector Space Spaces"
] | [
"Definition:Set",
"Definition:Positive/Integer",
"Definition:Zero (Number)",
"Real Numbers with Absolute Value form Normed Vector Space",
"Definition:Compact Space/Normed Vector Space",
"Definition:Real Number"
] | [
"Definition:Bounded Subset of Normed Vector Space",
"Union of Open Sets of Normed Vector Space is Open",
"Definition:Open Set/Normed Vector Space",
"Definition:Closed Set/Normed Vector Space/Definition 1",
"Heine-Borel Theorem",
"Definition:Compact Space/Normed Vector Space"
] |
proofwiki-17957 | Space of Continuous on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space | Let $I := \closedint a b$ be a closed real interval.
Let $\map C I$ be the space of real-valued functions continuous on $I$.
Let $\struct {\R, +_\R, \times_\R}$ be the field of real numbers.
Let $\paren +$ be the pointwise addition of real-valued functions.
Let $\paren {\, \cdot \,}$ be the pointwise scalar multiplicat... | Let $f, g, h \in \map C I$ such that:
:$f, g, h : I \to \R$
Let $\lambda, \mu \in \R$.
Let $\map 0 x$ be a real-valued function such that:
:$\map 0 x : I \to 0$.
Let us use real number addition and multiplication.
$\forall x \in I$ define pointwise addition as:
:$\map {\paren {f + g}} x := \map f x +_\R \map g x$.
Defi... | Let $I := \closedint a b$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $\map C I$ be the [[Definition:Space of Real-Valued Functions Continuous on Closed Interval|space of real-valued functions continuous on $I$]].
Let $\struct {\R, +_\R, \times_\R}$ be the [[Definition:Field of Real Numbers|fie... | Let $f, g, h \in \map C I$ such that:
:$f, g, h : I \to \R$
Let $\lambda, \mu \in \R$.
Let $\map 0 x$ be a [[Definition:Real-Valued Function|real-valued function]] such that:
:$\map 0 x : I \to 0$.
Let us use [[Definition:Real Number|real number]] [[Definition:Real Addition|addition]] and [[Definition:Real Multipl... | Space of Continuous on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space | https://proofwiki.org/wiki/Space_of_Continuous_on_Closed_Interval_Real-Valued_Functions_with_Pointwise_Addition_and_Pointwise_Scalar_Multiplication_forms_Vector_Space | https://proofwiki.org/wiki/Space_of_Continuous_on_Closed_Interval_Real-Valued_Functions_with_Pointwise_Addition_and_Pointwise_Scalar_Multiplication_forms_Vector_Space | [
"Examples of Vector Spaces",
"Functional Analysis"
] | [
"Definition:Real Interval/Closed",
"Definition:Space of Real-Valued Functions Continuous on Closed Interval",
"Definition:Field of Real Numbers",
"Definition:Pointwise Addition of Real-Valued Functions",
"Definition:Pointwise Scalar Multiplication of Mappings/Real-Valued Functions",
"Definition:Vector Spa... | [
"Definition:Real-Valued Function",
"Definition:Real Number",
"Definition:Addition/Real Numbers",
"Definition:Multiplication/Real Numbers",
"Definition:Pointwise Addition of Real-Valued Functions",
"Definition:Pointwise Scalar Multiplication of Mappings/Real-Valued Functions"
] |
proofwiki-17958 | Space of Continuously Differentiable on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space | Let $I := \closedint a b$ be a closed real interval.
Let $\map C I$ be a space of real-valued functions continuous on $I$.
Let $\map {C^1} I$ be a space of continuously differentiable functions on $I$.
Let $\struct {\R, +_\R, \times_\R}$ be the field of real numbers.
Let $\paren +$ be the pointwise addition of real-val... | From Space of Continuous on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space:
:$\struct {\map C I, +, \, \cdot \,}_\R$ is a vector space.
By Differentiable Function is Continuous:
:$\map {C^1} I \subset \map C I$
Let $f, g \in \map {C^1} I$.
Let $\alph... | Let $I := \closedint a b$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $\map C I$ be a [[Definition:Space of Real-Valued Functions Continuous on Closed Interval|space of real-valued functions continuous on $I$]].
Let $\map {C^1} I$ be a [[Definition:Space of Continuous Functions of Differentiabi... | From [[Space of Continuous on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space]]:
:$\struct {\map C I, +, \, \cdot \,}_\R$ is a [[Definition:Vector Space|vector space]].
By [[Differentiable Function is Continuous]]:
:$\map {C^1} I \subset \map C I$
L... | Space of Continuously Differentiable on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space | https://proofwiki.org/wiki/Space_of_Continuously_Differentiable_on_Closed_Interval_Real-Valued_Functions_with_Pointwise_Addition_and_Pointwise_Scalar_Multiplication_forms_Vector_Space | https://proofwiki.org/wiki/Space_of_Continuously_Differentiable_on_Closed_Interval_Real-Valued_Functions_with_Pointwise_Addition_and_Pointwise_Scalar_Multiplication_forms_Vector_Space | [
"Examples of Vector Spaces",
"Functional Analysis"
] | [
"Definition:Real Interval/Closed",
"Definition:Space of Real-Valued Functions Continuous on Closed Interval",
"Definition:Space of Continuous Functions of Differentiability Class k",
"Definition:Field of Real Numbers",
"Definition:Pointwise Addition of Real-Valued Functions",
"Definition:Pointwise Scalar ... | [
"Space of Continuous on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space",
"Definition:Vector Space",
"Differentiable Function is Continuous",
"Definition:Real-Valued Function",
"Definition:Restriction/Operation",
"Definition:Restriction/... |
proofwiki-17959 | P-Sequence Space with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms Vector Space | Let $\ell^p$ be the p-sequence space.
Let $\struct {\R, +_\R, \times_\R}$ be the field of real numbers.
Let $\paren +$ be the pointwise addition on the ring of sequences.
Let $\paren {\, \cdot \,}$ be the pointwise multiplication on the ring of sequences.
Then $\struct {\ell^p, +, \, \cdot \,}_\R$ is a vector space. | Let $\sequence {a_n}_{n \mathop \in \N}, \sequence {b_n}_{n \mathop \in \N}, \sequence {c_n}_{n \mathop \in \N} \in \ell^p$.
Let $\lambda, \mu \in \R$.
Let $\sequence 0 := \tuple {0, 0, 0, \dots}$ be a real-valued function.
Let us use real number addition and multiplication.
Define pointwise addition as:
:$\sequence {a... | Let $\ell^p$ be the [[Definition:P-Sequence Space|p-sequence space]].
Let $\struct {\R, +_\R, \times_\R}$ be the [[Definition:Field of Real Numbers|field of real numbers]].
Let $\paren +$ be the [[Definition:Pointwise Addition on Ring of Sequences|pointwise addition on the ring of sequences]].
Let $\paren {\, \cdot ... | Let $\sequence {a_n}_{n \mathop \in \N}, \sequence {b_n}_{n \mathop \in \N}, \sequence {c_n}_{n \mathop \in \N} \in \ell^p$.
Let $\lambda, \mu \in \R$.
Let $\sequence 0 := \tuple {0, 0, 0, \dots}$ be a [[Definition:Real-Valued Function|real-valued function]].
Let us use [[Definition:Real Number|real number]] [[Defin... | P-Sequence Space with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms Vector Space | https://proofwiki.org/wiki/P-Sequence_Space_with_Pointwise_Addition_and_Pointwise_Scalar_Multiplication_on_Ring_of_Sequences_forms_Vector_Space | https://proofwiki.org/wiki/P-Sequence_Space_with_Pointwise_Addition_and_Pointwise_Scalar_Multiplication_on_Ring_of_Sequences_forms_Vector_Space | [
"Examples of Vector Spaces",
"Functional Analysis"
] | [
"Definition:P-Sequence Space",
"Definition:Field of Real Numbers",
"Definition:Ring of Sequences/Pointwise Addition",
"Definition:Ring of Sequences/Pointwise Multiplication",
"Definition:Vector Space"
] | [
"Definition:Real-Valued Function",
"Definition:Real Number",
"Definition:Addition/Real Numbers",
"Definition:Multiplication/Real Numbers",
"Definition:Ring of Sequences/Pointwise Addition",
"Definition:Pointwise Scalar Multiplication on Ring of Sequences",
"Definition:Ring of Sequences/Additive Inverse"... |
proofwiki-17960 | Vector Space of Continuous on Closed Interval Real Functions is not Finite Dimensional | Let $I := \closedint 0 1$ be a closed real interval.
Let $\struct {\map C I, +, \, \cdot \,}_\R$ be the continuous on closed interval real function vector space.
Then $\struct {\map C I, +, \, \cdot \,}_\R$ is not finite dimensional. | === Monomials are linearly independent ===
Let $d \in \N_{>0}$.
Consider the set of real monomials of the following form:
:$\map {x_n} t = t^n$
where $n \in \N_{>0}$ and $n \le d$.
{{AimForCont}} the set of $x_n$ is not linearly independent.
Then:
:$\forall n \in \N_{>0}: n \le d: \exists \alpha_n \in \R: \neg \forall ... | Let $I := \closedint 0 1$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $\struct {\map C I, +, \, \cdot \,}_\R$ be the [[Space of Continuous on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space|continuous on closed interval real fu... | === [[Definition:Monomial|Monomials]] are [[Definition:Linearly Independent|linearly independent]] ===
Let $d \in \N_{>0}$.
Consider the [[Definition:Set|set]] of [[Definition:Real Function|real]] [[Definition:Monomial|monomials]] of the following form:
:$\map {x_n} t = t^n$
where $n \in \N_{>0}$ and $n \le d$.
{{... | Vector Space of Continuous on Closed Interval Real Functions is not Finite Dimensional | https://proofwiki.org/wiki/Vector_Space_of_Continuous_on_Closed_Interval_Real_Functions_is_not_Finite_Dimensional | https://proofwiki.org/wiki/Vector_Space_of_Continuous_on_Closed_Interval_Real_Functions_is_not_Finite_Dimensional | [
"Vector Spaces",
"Continuous Real Functions"
] | [
"Definition:Real Interval/Closed",
"Space of Continuous on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space",
"Definition:Dimension of Vector Space/Finite"
] | [
"Definition:Monomial",
"Definition:Linearly Independent",
"Definition:Set",
"Definition:Real Function",
"Definition:Monomial",
"Definition:Set",
"Definition:Linearly Independent",
"Definition:Term of Sequence/Index",
"Definition:Limit of Real Function",
"Definition:Contradiction",
"Definition:Se... |
proofwiki-17961 | Min Operation Equals an Operand | Let $x_1, x_2, \dotsc, x_n \in S$ for some $n \in \N_{>0}$.
Then:
:$\exists i \in \closedint 1 n : x_i = \min \set {x_1, x_2, \dotsc, x_n}$ | We will prove the result by induction on the number of operands $n$.
For all $n \in \N_{>0}$, let $\map P n$ be the proposition:
:$\exists i \in \closedint 1 n : x_i = \min \set {x_1, x_2, \dotsc, x_n}$ | Let $x_1, x_2, \dotsc, x_n \in S$ for some $n \in \N_{>0}$.
Then:
:$\exists i \in \closedint 1 n : x_i = \min \set {x_1, x_2, \dotsc, x_n}$ | We will prove the result by [[Principle of Mathematical Induction|induction]] on the [[Definition:Cardinality|number]] of [[Definition:Operand|operands]] $n$.
For all $n \in \N_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\exists i \in \closedint 1 n : x_i = \min \set {x_1, x_2, \dotsc, x_n}$ | Min Operation Equals an Operand | https://proofwiki.org/wiki/Min_Operation_Equals_an_Operand | https://proofwiki.org/wiki/Min_Operation_Equals_an_Operand | [
"Min Operation"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Cardinality",
"Definition:Operation/Operand",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-17962 | Min Operation Yields Infimum of Parameters/General Case | Let $x_1, x_2, \dots ,x_n \in S$ for some $n \in \N_{>0}$.
Then:
:$\min \set {x_1, x_2, \dotsc, x_n} = \inf \set {x_1, x_2, \dotsc, x_n}$ | We will prove the result by induction on the number of operands $n$.
For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:
:$\min \set {x_1, x_2, \dotsc, x_n} = \inf \set {x_1, x_2, \dotsc, x_n}$ | Let $x_1, x_2, \dots ,x_n \in S$ for some $n \in \N_{>0}$.
Then:
:$\min \set {x_1, x_2, \dotsc, x_n} = \inf \set {x_1, x_2, \dotsc, x_n}$ | We will prove the result by [[Principle of Mathematical Induction|induction]] on the [[Definition:Cardinality|number]] of [[Definition:Operand|operands]] $n$.
For all $n \in \Z_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\min \set {x_1, x_2, \dotsc, x_n} = \inf \set {x_1, x_2, \dotsc, x_n}$ | Min Operation Yields Infimum of Parameters/General Case | https://proofwiki.org/wiki/Min_Operation_Yields_Infimum_of_Parameters/General_Case | https://proofwiki.org/wiki/Min_Operation_Yields_Infimum_of_Parameters/General_Case | [
"Min Operation"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Cardinality",
"Definition:Operation/Operand",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-17963 | Primitive of Reciprocal of Root of a x squared plus b x plus c/a greater than 0 | Let $a \in \R_{> 0}$.
Then for $x \in \R$ such that $a x^2 + b x + c > 0$:
:<nowiki>$\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \begin {cases}
\dfrac 1 {\sqrt a} \ln \size {2 \sqrt a \sqrt {a x^2 + b x + c} + 2 a x + b} + C & : b^2 - 4 a c > 0 \\ \\
\dfrac 1 {\sqrt a} \map \arsinh {\dfrac {2 a x + b} {\sqrt {4 ... | === Completing the Square ===
{{:Primitive of Reciprocal of Root of a x squared plus b x plus c/a greater than 0/Completing the Square}} | Let $a \in \R_{> 0}$.
Then for $x \in \R$ such that $a x^2 + b x + c > 0$:
:<nowiki>$\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \begin {cases}
\dfrac 1 {\sqrt a} \ln \size {2 \sqrt a \sqrt {a x^2 + b x + c} + 2 a x + b} + C & : b^2 - 4 a c > 0 \\ \\
\dfrac 1 {\sqrt a} \map \arsinh {\dfrac {2 a x + b} {\sqrt {... | === [[Primitive of Reciprocal of Root of a x squared plus b x plus c/a greater than 0/Completing the Square|Completing the Square]] ===
{{:Primitive of Reciprocal of Root of a x squared plus b x plus c/a greater than 0/Completing the Square}} | Primitive of Reciprocal of Root of a x squared plus b x plus c/a greater than 0 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_squared_plus_b_x_plus_c/a_greater_than_0 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_squared_plus_b_x_plus_c/a_greater_than_0 | [
"Primitive of Reciprocal of Root of a x squared plus b x plus c"
] | [
"Definition:Inverse Hyperbolic Sine/Real/Definition 2"
] | [
"Primitive of Reciprocal of Root of a x squared plus b x plus c/a greater than 0/Completing the Square"
] |
proofwiki-17964 | Primitive of Reciprocal of Root of a x squared plus b x plus c/a less than 0 | Let $a \in \R_{< 0}$.
Then for $x \in \R$ such that $a x^2 + b x + c > 0$:
:$\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \dfrac {-1} {\sqrt {-a} } \map \arcsin {\dfrac {2 a x + b} {\sqrt {\size {b^2 - 4 a c} } } } + C$
given that $b^2 \ne 4 a c$. | === Completing the Square ===
{{:Primitive of Reciprocal of Root of a x squared plus b x plus c/a less than 0/Completing the Square}} | Let $a \in \R_{< 0}$.
Then for $x \in \R$ such that $a x^2 + b x + c > 0$:
:$\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \dfrac {-1} {\sqrt {-a} } \map \arcsin {\dfrac {2 a x + b} {\sqrt {\size {b^2 - 4 a c} } } } + C$
given that $b^2 \ne 4 a c$. | === [[Primitive of Reciprocal of Root of a x squared plus b x plus c/a less than 0/Completing the Square|Completing the Square]] ===
{{:Primitive of Reciprocal of Root of a x squared plus b x plus c/a less than 0/Completing the Square}} | Primitive of Reciprocal of Root of a x squared plus b x plus c/a less than 0 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_squared_plus_b_x_plus_c/a_less_than_0 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_squared_plus_b_x_plus_c/a_less_than_0 | [
"Primitive of Reciprocal of Root of a x squared plus b x plus c"
] | [] | [
"Primitive of Reciprocal of Root of a x squared plus b x plus c/a less than 0/Completing the Square"
] |
proofwiki-17965 | Primitive of Reciprocal of Root of a x squared plus b x plus c/a greater than 0/Negative Discriminant | Let $a \in \R_{>0}$.
Let $b^2 - 4 a c < 0$.
Then for $x \in \R$ such that $a x^2 + b x + c > 0$:
:$\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \frac 1 {\sqrt a} \map \arsinh {\dfrac {2 a x + b} {\sqrt {4 a c - b^2} } } + C$
where $\arsinh$ denotes the area hyperbolic sine function. | Let $b^2 - 4 a c < 0$.
Then:
{{begin-eqn}}
{{eqn | l = - D
| o = >
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = -D
| r = q^2
| c = for some $q \in \R$
}}
{{eqn | ll= \leadsto
| l = q
| r = \sqrt {4 a c - b^2}
| c = by definition of $D$
}}
{{end-eqn}}
Thus:
{{begin-e... | Let $a \in \R_{>0}$.
Let $b^2 - 4 a c < 0$.
Then for $x \in \R$ such that $a x^2 + b x + c > 0$:
:$\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \frac 1 {\sqrt a} \map \arsinh {\dfrac {2 a x + b} {\sqrt {4 a c - b^2} } } + C$
where $\arsinh$ denotes the [[Definition:Real Area Hyperbolic Sine|area hyperbolic si... | Let $b^2 - 4 a c < 0$.
Then:
{{begin-eqn}}
{{eqn | l = - D
| o = >
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = -D
| r = q^2
| c = for some $q \in \R$
}}
{{eqn | ll= \leadsto
| l = q
| r = \sqrt {4 a c - b^2}
| c = by definition of $D$
}}
{{end-eqn}}
Thus:
{{be... | Primitive of Reciprocal of Root of a x squared plus b x plus c/a greater than 0/Negative Discriminant | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_squared_plus_b_x_plus_c/a_greater_than_0/Negative_Discriminant | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_squared_plus_b_x_plus_c/a_greater_than_0/Negative_Discriminant | [
"Primitive of Reciprocal of Root of a x squared plus b x plus c"
] | [
"Definition:Inverse Hyperbolic Sine/Real/Definition 2"
] | [
"Integration by Substitution",
"Primitive of Reciprocal of Root of x squared plus a squared/Inverse Hyperbolic Sine Form"
] |
proofwiki-17966 | Primitive of Reciprocal of Root of a x squared plus b x plus c/a greater than 0/Positive Discriminant | Let $a, b, c \in \R$ such that $a > 0$.
Let $b^2 - 4 a c > 0$.
Then for $x \in \R$ such that $a x^2 + b x + c > 0$:
:$\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \dfrac 1 {\sqrt a} \ln \size {2 \sqrt a \sqrt {a x^2 + b x + c} + 2 a x + b} + C$ | Let $b^2 - 4 a c > 0$.
Then:
{{begin-eqn}}
{{eqn | l = D
| o = >
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = D
| r = q^2
| c = for some $q \in \R$
}}
{{eqn | ll= \leadsto
| l = q
| r = \sqrt {b^2 - 4 a c}
| c = by definition of $D$
}}
{{end-eqn}}
Thus:
{{begin-eqn}... | Let $a, b, c \in \R$ such that $a > 0$.
Let $b^2 - 4 a c > 0$.
Then for $x \in \R$ such that $a x^2 + b x + c > 0$:
:$\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \dfrac 1 {\sqrt a} \ln \size {2 \sqrt a \sqrt {a x^2 + b x + c} + 2 a x + b} + C$ | Let $b^2 - 4 a c > 0$.
Then:
{{begin-eqn}}
{{eqn | l = D
| o = >
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = D
| r = q^2
| c = for some $q \in \R$
}}
{{eqn | ll= \leadsto
| l = q
| r = \sqrt {b^2 - 4 a c}
| c = by definition of $D$
}}
{{end-eqn}}
Thus:
{{begin... | Primitive of Reciprocal of Root of a x squared plus b x plus c/a greater than 0/Positive Discriminant | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_squared_plus_b_x_plus_c/a_greater_than_0/Positive_Discriminant | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_squared_plus_b_x_plus_c/a_greater_than_0/Positive_Discriminant | [
"Primitive of Reciprocal of Root of a x squared plus b x plus c"
] | [] | [
"Integration by Substitution",
"Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form",
"Completing the Square"
] |
proofwiki-17967 | Primitive of Reciprocal of Root of a x squared plus b x plus c/a less than 0/Positive Discriminant | Let $a \in \R_{<0}$.
Let $b^2 - 4 a c > 0$.
Then for $x \in \R$ such that $a x^2 + b x + c > 0$ and $\size {2 a x + b} < \sqrt {b^2 - 4 a c}$:
:$\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \frac {-1} {\sqrt {-a} } \map \arcsin {\frac {2 a x + b} {\sqrt {b^2 - 4 a c} } } + C$ | Let $b^2 - 4 a c > 0$.
Then:
{{begin-eqn}}
{{eqn | l = D
| o = >
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = D
| r = q^2
| c = for some $q \in \R$
}}
{{eqn | ll= \leadsto
| l = q
| r = \sqrt {b^2 - 4 a c}
| c = by definition of $D$
}}
{{end-eqn}}
Thus:
{{begin-eqn}... | Let $a \in \R_{<0}$.
Let $b^2 - 4 a c > 0$.
Then for $x \in \R$ such that $a x^2 + b x + c > 0$ and $\size {2 a x + b} < \sqrt {b^2 - 4 a c}$:
:$\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \frac {-1} {\sqrt {-a} } \map \arcsin {\frac {2 a x + b} {\sqrt {b^2 - 4 a c} } } + C$ | Let $b^2 - 4 a c > 0$.
Then:
{{begin-eqn}}
{{eqn | l = D
| o = >
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = D
| r = q^2
| c = for some $q \in \R$
}}
{{eqn | ll= \leadsto
| l = q
| r = \sqrt {b^2 - 4 a c}
| c = by definition of $D$
}}
{{end-eqn}}
Thus:
{{begin... | Primitive of Reciprocal of Root of a x squared plus b x plus c/a less than 0/Positive Discriminant | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_squared_plus_b_x_plus_c/a_less_than_0/Positive_Discriminant | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_squared_plus_b_x_plus_c/a_less_than_0/Positive_Discriminant | [
"Primitive of Reciprocal of Root of a x squared plus b x plus c"
] | [] | [
"Integration by Substitution",
"Primitive of Reciprocal of Root of a squared minus x squared/Arcsine Form"
] |
proofwiki-17968 | Primitive of Reciprocal of Root of a x squared plus b x plus c/a less than 0/Negative Discriminant | Let $a \in \R_{> 0}$.
Let $b^2 - 4 a c > 0$.
Then for $x \in \R$ such that $a x^2 + b x + c > 0$:
:$\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \frac {-1} {\sqrt {-a} } \map \arcsin {\frac {2 a x + b} {\sqrt {4 a c - b^2} } } + C$ | Let $b^2 - 4 a c < 0$.
Let $D' = -D = 4 a c - b^2$.
Then:
{{begin-eqn}}
{{eqn | l = D'
| o = >
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = D
| r = q^2
| c = for some $q \in \R$
}}
{{eqn | ll= \leadsto
| l = q
| r = \sqrt {4 a c - b^2}
| c = by definition of $D$
}}
... | Let $a \in \R_{> 0}$.
Let $b^2 - 4 a c > 0$.
Then for $x \in \R$ such that $a x^2 + b x + c > 0$:
:$\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \frac {-1} {\sqrt {-a} } \map \arcsin {\frac {2 a x + b} {\sqrt {4 a c - b^2} } } + C$ | Let $b^2 - 4 a c < 0$.
Let $D' = -D = 4 a c - b^2$.
Then:
{{begin-eqn}}
{{eqn | l = D'
| o = >
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = D
| r = q^2
| c = for some $q \in \R$
}}
{{eqn | ll= \leadsto
| l = q
| r = \sqrt {4 a c - b^2}
| c = by definition of $D$
... | Primitive of Reciprocal of Root of a x squared plus b x plus c/a less than 0/Negative Discriminant | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_squared_plus_b_x_plus_c/a_less_than_0/Negative_Discriminant | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_squared_plus_b_x_plus_c/a_less_than_0/Negative_Discriminant | [
"Primitive of Reciprocal of Root of a x squared plus b x plus c"
] | [] | [
"Integration by Substitution",
"Integration by Substitution",
"Integration by Substitution",
"Primitive of Reciprocal of Root of a squared minus x squared/Arcsine Form"
] |
proofwiki-17969 | Primitive of Reciprocal of Root of a x squared plus b x plus c/a greater than 0/Zero Discriminant | Let $a \in \R_{>0}$.
Let $b^2 - 4 a c = 0$.
Then for $x \in \R$ such that $a x^2 + b x + c > 0$:
:$\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \frac 1 {\sqrt a} \ln \size {2 a x + b} + C$ | Let $b^2 - 4 a c = 0$ {{hypothesis}}.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sqrt {a x^2 + b x + c} }
| r = \int \frac {2 \sqrt a \rd x} {\sqrt {\paren {2 a x + b}^2} }
| c = from $(1)$
}}
{{eqn | r = 2 \sqrt a \int \frac {\d x} {2 a x + b}
| c = Primitive of Constant Multiple of Function... | Let $a \in \R_{>0}$.
Let $b^2 - 4 a c = 0$.
Then for $x \in \R$ such that $a x^2 + b x + c > 0$:
:$\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \frac 1 {\sqrt a} \ln \size {2 a x + b} + C$ | Let $b^2 - 4 a c = 0$ {{hypothesis}}.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sqrt {a x^2 + b x + c} }
| r = \int \frac {2 \sqrt a \rd x} {\sqrt {\paren {2 a x + b}^2} }
| c = from $(1)$
}}
{{eqn | r = 2 \sqrt a \int \frac {\d x} {2 a x + b}
| c = [[Primitive of Constant Multiple of Funct... | Primitive of Reciprocal of Root of a x squared plus b x plus c/a greater than 0/Zero Discriminant | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_squared_plus_b_x_plus_c/a_greater_than_0/Zero_Discriminant | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_squared_plus_b_x_plus_c/a_greater_than_0/Zero_Discriminant | [
"Primitive of Reciprocal of Root of a x squared plus b x plus c"
] | [] | [
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal of a x + b"
] |
proofwiki-17970 | Primitive of Reciprocal of Root of a x squared plus b x plus c/a less than 0/Zero Discriminant | Let $a \in \R_{\ne 0}$.
Let $b^2 - 4 a c = 0$.
Then:
:$\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} }$
is not defined. | Suppose that $b^2 - 4 a c = 0$.
Then:
{{begin-eqn}}
{{eqn | l = a x^2 + b x + c
| r = \frac {\paren {2 a x + b}^2 - \paren {b^2 - 4 a c} } {4 a}
| c = Completing the Square
}}
{{eqn | r = \frac {\paren {2 a x + b}^2} {4 a}
| c = as $b^2 - 4 a c = 0$
}}
{{end-eqn}}
But we have that:
:$\paren {2 a x + b... | Let $a \in \R_{\ne 0}$.
Let $b^2 - 4 a c = 0$.
Then:
:$\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} }$
is not defined. | Suppose that $b^2 - 4 a c = 0$.
Then:
{{begin-eqn}}
{{eqn | l = a x^2 + b x + c
| r = \frac {\paren {2 a x + b}^2 - \paren {b^2 - 4 a c} } {4 a}
| c = [[Completing the Square]]
}}
{{eqn | r = \frac {\paren {2 a x + b}^2} {4 a}
| c = as $b^2 - 4 a c = 0$
}}
{{end-eqn}}
But we have that:
:$\paren {2 a... | Primitive of Reciprocal of Root of a x squared plus b x plus c/a less than 0/Zero Discriminant | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_squared_plus_b_x_plus_c/a_less_than_0/Zero_Discriminant | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_squared_plus_b_x_plus_c/a_less_than_0/Zero_Discriminant | [
"Primitive of Reciprocal of Root of a x squared plus b x plus c"
] | [] | [
"Completing the Square",
"Definition:Real Number"
] |
proofwiki-17971 | Space of Somewhere Differentiable Continuous Functions on Closed Interval is Meager in Space of Continuous Functions on Closed Interval | Let $I = \closedint a b$.
Let $\map \CC I$ be the set of continuous functions on $I$.
Let $\map \DD I$ be the set of continuous functions on $I$ that are differentiable at a point.
Let $d$ be the metric induced by the supremum norm.
Then $\map \DD I$ is meager in $\struct {\map \CC I, d}$. | Let:
:$\ds A_{n, m} = \set {f \in \map \CC I: \exists x \in I: \forall t \in I: 0 < \size {t - x} < \frac 1 m \implies \size {\frac {\map f t - \map f x} {t - x} } \le n}$
and:
:$\ds A = \bigcup_{\tuple {n, m} \mathop \in \N^2} A_{n, m}$ | Let $I = \closedint a b$.
Let $\map \CC I$ be the set of [[Definition:Continuous Function|continuous functions]] on $I$.
Let $\map \DD I$ be the set of [[Definition:Continuous Function|continuous functions]] on $I$ that are [[Definition:Differentiable Real Function at Point|differentiable at a point]].
Let $d$ be ... | Let:
:$\ds A_{n, m} = \set {f \in \map \CC I: \exists x \in I: \forall t \in I: 0 < \size {t - x} < \frac 1 m \implies \size {\frac {\map f t - \map f x} {t - x} } \le n}$
and:
:$\ds A = \bigcup_{\tuple {n, m} \mathop \in \N^2} A_{n, m}$ | Space of Somewhere Differentiable Continuous Functions on Closed Interval is Meager in Space of Continuous Functions on Closed Interval | https://proofwiki.org/wiki/Space_of_Somewhere_Differentiable_Continuous_Functions_on_Closed_Interval_is_Meager_in_Space_of_Continuous_Functions_on_Closed_Interval | https://proofwiki.org/wiki/Space_of_Somewhere_Differentiable_Continuous_Functions_on_Closed_Interval_is_Meager_in_Space_of_Continuous_Functions_on_Closed_Interval | [
"Functional Analysis",
"Meager Spaces",
"Space of Somewhere Differentiable Continuous Functions on Closed Interval is Meager in Space of Continuous Functions on Closed Interval"
] | [
"Definition:Continuous Function",
"Definition:Continuous Function",
"Definition:Differentiable Mapping/Real Function/Point",
"Definition:Metric Induced by Norm",
"Definition:Supremum Norm",
"Definition:Meager Space"
] | [] |
proofwiki-17972 | Space of Somewhere Differentiable Continuous Functions on Closed Interval is Meager in Space of Continuous Functions on Closed Interval/Lemma 1 | :$\map \DD I \subseteq A$ | Let $f \in \map \DD I$.
Then, $f$ is differentiable at some $x \in I$.
Let:
:$n = \floor {\size {\map {f'} x} } + 1$
where $\floor \cdot$ is the floor function.
Then:
:$\size {\map {f'} x} < n$
From the definition of the derivative, there exists $\delta > 0$ such that for all $t$ with $0 < \size {t - x} < \delta$, we... | :$\map \DD I \subseteq A$ | Let $f \in \map \DD I$.
Then, $f$ is [[Definition:Differentiable Real Function at Point|differentiable]] at some $x \in I$.
Let:
:$n = \floor {\size {\map {f'} x} } + 1$
where $\floor \cdot$ is the [[Definition:Floor Function|floor function]].
Then:
:$\size {\map {f'} x} < n$
From the definition of the [[Defin... | Space of Somewhere Differentiable Continuous Functions on Closed Interval is Meager in Space of Continuous Functions on Closed Interval/Lemma 1 | https://proofwiki.org/wiki/Space_of_Somewhere_Differentiable_Continuous_Functions_on_Closed_Interval_is_Meager_in_Space_of_Continuous_Functions_on_Closed_Interval/Lemma_1 | https://proofwiki.org/wiki/Space_of_Somewhere_Differentiable_Continuous_Functions_on_Closed_Interval_is_Meager_in_Space_of_Continuous_Functions_on_Closed_Interval/Lemma_1 | [
"Space of Somewhere Differentiable Continuous Functions on Closed Interval is Meager in Space of Continuous Functions on Closed Interval"
] | [] | [
"Definition:Differentiable Mapping/Real Function/Point",
"Definition:Floor Function",
"Definition:Derivative",
"Reverse Triangle Inequality",
"Category:Space of Somewhere Differentiable Continuous Functions on Closed Interval is Meager in Space of Continuous Functions on Closed Interval"
] |
proofwiki-17973 | Space of Somewhere Differentiable Continuous Functions on Closed Interval is Meager in Space of Continuous Functions on Closed Interval/Corollary | :there exists a function $f \in \map \CC I$ that is not differentiable anywhere. | Let $\map \DD I$ be the set of continuous functions on $I$ that are differentiable at a point.
Let $d$ be the metric induced by the supremum norm.
By Space of Continuous on Closed Interval Real-Valued Functions with Supremum Norm forms Banach Space:
:$\struct {\map \CC I, d}$ is a complete metric space.
By Baire Space ... | :there exists a [[Definition:Real Function|function]] $f \in \map \CC I$ that is not [[Definition:Differentiable Real Function at Point|differentiable]] anywhere. | Let $\map \DD I$ be the set of [[Definition:Continuous Function|continuous functions]] on $I$ that are [[Definition:Differentiable Real Function at Point|differentiable at a point]].
Let $d$ be the [[Definition:Metric Induced by Norm|metric induced]] by the [[Definition:Supremum Norm|supremum norm]].
By [[Space of Co... | Space of Somewhere Differentiable Continuous Functions on Closed Interval is Meager in Space of Continuous Functions on Closed Interval/Corollary | https://proofwiki.org/wiki/Space_of_Somewhere_Differentiable_Continuous_Functions_on_Closed_Interval_is_Meager_in_Space_of_Continuous_Functions_on_Closed_Interval/Corollary | https://proofwiki.org/wiki/Space_of_Somewhere_Differentiable_Continuous_Functions_on_Closed_Interval_is_Meager_in_Space_of_Continuous_Functions_on_Closed_Interval/Corollary | [
"Space of Somewhere Differentiable Continuous Functions on Closed Interval is Meager in Space of Continuous Functions on Closed Interval",
"Functional Analysis"
] | [
"Definition:Real Function",
"Definition:Differentiable Mapping/Real Function/Point"
] | [
"Definition:Continuous Function",
"Definition:Differentiable Mapping/Real Function/Point",
"Definition:Metric Induced by Norm",
"Definition:Supremum Norm",
"Space of Continuous on Closed Interval Real-Valued Functions with Supremum Norm forms Banach Space",
"Definition:Complete Metric Space",
"Baire Spa... |
proofwiki-17974 | Max Operation is Commutative | The Max operation is commutative:
:$\map \max {x, y} = \map \max {y, x}$ | To simplify our notation:
:Let $\map \max {x, y}$ be (temporarily) denoted $x \overline \wedge y$
There are three cases to consider:
:$(1): \quad x \le y$
:$(2): \quad y \le x$
:$(3): \quad x = y$
$(1): \quad$ Let $x \le y$.
Then:
{{begin-eqn}}
{{eqn | l = x \overline \wedge y
| r = y
| rr= = y \overline \w... | The [[Definition:Max Operation|Max operation]] is [[Definition:Commutative Operation|commutative]]:
:$\map \max {x, y} = \map \max {y, x}$ | To simplify our notation:
:Let $\map \max {x, y}$ be (temporarily) denoted $x \overline \wedge y$
There are three cases to consider:
:$(1): \quad x \le y$
:$(2): \quad y \le x$
:$(3): \quad x = y$
$(1): \quad$ Let $x \le y$.
Then:
{{begin-eqn}}
{{eqn | l = x \overline \wedge y
| r = y
| rr= = y \over... | Max Operation is Commutative | https://proofwiki.org/wiki/Max_Operation_is_Commutative | https://proofwiki.org/wiki/Max_Operation_is_Commutative | [
"Max Operation",
"Examples of Commutative Operations"
] | [
"Definition:Max Operation",
"Definition:Commutative/Operation"
] | [
"Definition:Commutative/Operation"
] |
proofwiki-17975 | Min Operation is Commutative | The Min operation is commutative:
:$\map \min {x, y} = \map \min {y, x}$ | To simplify our notation:
:Let $\map \min {x, y}$ be (temporarily) denoted $x \underline \vee y$.
There are three cases to consider:
:$(1): \quad x \le y$
:$(2): \quad y \le x$
:$(3): \quad x = y$
$(1): \quad$ Let $x \le y$.
Then:
{{begin-eqn}}
{{eqn | l = x \underline \vee y
| r = x
| rr= =y \underline \ve... | The [[Definition:Min Operation|Min operation]] is [[Definition:Commutative Operation|commutative]]:
:$\map \min {x, y} = \map \min {y, x}$ | To simplify our notation:
:Let $\map \min {x, y}$ be (temporarily) denoted $x \underline \vee y$.
There are three cases to consider:
:$(1): \quad x \le y$
:$(2): \quad y \le x$
:$(3): \quad x = y$
$(1): \quad$ Let $x \le y$.
Then:
{{begin-eqn}}
{{eqn | l = x \underline \vee y
| r = x
| rr= =y \underl... | Min Operation is Commutative | https://proofwiki.org/wiki/Min_Operation_is_Commutative | https://proofwiki.org/wiki/Min_Operation_is_Commutative | [
"Min Operation",
"Examples of Commutative Operations"
] | [
"Definition:Min Operation",
"Definition:Commutative/Operation"
] | [
"Definition:Commutative/Operation"
] |
proofwiki-17976 | Space of Somewhere Differentiable Continuous Functions on Closed Interval is Meager in Space of Continuous Functions on Closed Interval/Lemma 2 | :for each $\tuple {n, m} \in \N^2$, $A_{n, m}$ is nowhere dense in $\struct {\map \CC I, d}$. | === Lemma 2.1 ===
{{:Space of Somewhere Differentiable Continuous Functions on Closed Interval is Meager in Space of Continuous Functions on Closed Interval/Lemma 2/Lemma 2.1}}{{qed|lemma}}
Fix $\tuple {n, m} \in \N^2$.
By Lemma 2.1, $A_{n, m}$ is closed in $\struct {\map \CC I, d}$.
Therefore by the second definition... | :for each $\tuple {n, m} \in \N^2$, $A_{n, m}$ is [[Definition:Nowhere Dense|nowhere dense]] in $\struct {\map \CC I, d}$. | === [[Space of Somewhere Differentiable Continuous Functions on Closed Interval is Meager in Space of Continuous Functions on Closed Interval/Lemma 2/Lemma 2.1|Lemma 2.1]] ===
{{:Space of Somewhere Differentiable Continuous Functions on Closed Interval is Meager in Space of Continuous Functions on Closed Interval/Lemma... | Space of Somewhere Differentiable Continuous Functions on Closed Interval is Meager in Space of Continuous Functions on Closed Interval/Lemma 2 | https://proofwiki.org/wiki/Space_of_Somewhere_Differentiable_Continuous_Functions_on_Closed_Interval_is_Meager_in_Space_of_Continuous_Functions_on_Closed_Interval/Lemma_2 | https://proofwiki.org/wiki/Space_of_Somewhere_Differentiable_Continuous_Functions_on_Closed_Interval_is_Meager_in_Space_of_Continuous_Functions_on_Closed_Interval/Lemma_2 | [
"Space of Somewhere Differentiable Continuous Functions on Closed Interval is Meager in Space of Continuous Functions on Closed Interval"
] | [
"Definition:Nowhere Dense"
] | [
"Space of Somewhere Differentiable Continuous Functions on Closed Interval is Meager in Space of Continuous Functions on Closed Interval/Lemma 2/Lemma 2.1",
"Space of Somewhere Differentiable Continuous Functions on Closed Interval is Meager in Space of Continuous Functions on Closed Interval/Lemma 2/Lemma 2.1",
... |
proofwiki-17977 | Space of Somewhere Differentiable Continuous Functions on Closed Interval is Meager in Space of Continuous Functions on Closed Interval/Lemma 2/Lemma 2.1 | :for each $\tuple {n, m} \in \N^2$, $A_{n, m}$ is closed in $\tuple {\map \CC I, d}$. | Fix $\tuple {n, m} \in \N^2$.
From Space of Continuous on Closed Interval Real-Valued Functions with Supremum Norm forms Banach Space:
:$\tuple {\map \CC I, d}$ is complete.
Hence, from Subspace of Complete Metric Space is Closed iff Complete:
:$A_{n, m}$ is closed {{iff}} $\tuple {A_{n, m}, d}$ is complete.
Let $\seq... | :for each $\tuple {n, m} \in \N^2$, $A_{n, m}$ is [[Definition:Closed Set (Metric Space)|closed]] in $\tuple {\map \CC I, d}$. | Fix $\tuple {n, m} \in \N^2$.
From [[Space of Continuous on Closed Interval Real-Valued Functions with Supremum Norm forms Banach Space]]:
:$\tuple {\map \CC I, d}$ is [[Definition:Complete Metric Space|complete]].
Hence, from [[Subspace of Complete Metric Space is Closed iff Complete]]:
:$A_{n, m}$ is [[Definitio... | Space of Somewhere Differentiable Continuous Functions on Closed Interval is Meager in Space of Continuous Functions on Closed Interval/Lemma 2/Lemma 2.1 | https://proofwiki.org/wiki/Space_of_Somewhere_Differentiable_Continuous_Functions_on_Closed_Interval_is_Meager_in_Space_of_Continuous_Functions_on_Closed_Interval/Lemma_2/Lemma_2.1 | https://proofwiki.org/wiki/Space_of_Somewhere_Differentiable_Continuous_Functions_on_Closed_Interval_is_Meager_in_Space_of_Continuous_Functions_on_Closed_Interval/Lemma_2/Lemma_2.1 | [
"Space of Somewhere Differentiable Continuous Functions on Closed Interval is Meager in Space of Continuous Functions on Closed Interval"
] | [
"Definition:Closed Set/Metric Space"
] | [
"Space of Continuous on Closed Interval Real-Valued Functions with Supremum Norm forms Banach Space",
"Definition:Complete Metric Space",
"Subspace of Complete Metric Space is Closed iff Complete",
"Definition:Closed Set/Metric Space",
"Definition:Complete Metric Space",
"Definition:Cauchy Sequence",
"D... |
proofwiki-17978 | Decomposition of Mean Squared Error | <onlyinclude>
Let $\theta$ be a population parameter of some stochastic model.
Let $\hat \theta$ be an estimator of $\theta$.
We then have:
:$\map {\operatorname {MSE} } {\hat \theta} = \var {\hat \theta} + \paren {\map {\operatorname {bias}} {\hat \theta} }^2 $
where:
:$\map {\operatorname {MSE} } {\hat \theta}$ denot... | Let $\delta = \hat \theta - \theta$.
By {{Defof|Mean Squared Error of Estimator}}:
:$\expect {\delta ^2} = \map {\operatorname {MSE} } {\hat \theta}$
and:
{{begin-eqn}}
{{eqn | l = \expect \delta
| r = \expect {\paren {\hat \theta} - \theta}
}}
{{eqn | r = \expect {\hat \theta} - \theta
| c = Expectation of... | <onlyinclude>
Let $\theta$ be a [[Definition:Parameter of Stochastic Model|population parameter]] of some [[Definition:Stochastic Model|stochastic model]].
Let $\hat \theta$ be an [[Definition:Estimator|estimator]] of $\theta$.
We then have:
:$\map {\operatorname {MSE} } {\hat \theta} = \var {\hat \theta} + \paren {... | Let $\delta = \hat \theta - \theta$.
By {{Defof|Mean Squared Error of Estimator}}:
:$\expect {\delta ^2} = \map {\operatorname {MSE} } {\hat \theta}$
and:
{{begin-eqn}}
{{eqn | l = \expect \delta
| r = \expect {\paren {\hat \theta} - \theta}
}}
{{eqn | r = \expect {\hat \theta} - \theta
| c = [[Expectat... | Decomposition of Mean Squared Error | https://proofwiki.org/wiki/Decomposition_of_Mean_Squared_Error | https://proofwiki.org/wiki/Decomposition_of_Mean_Squared_Error | [
"Inductive Statistics",
"Variance"
] | [
"Definition:Parameter of Stochastic Model",
"Definition:Stochastic Model",
"Definition:Estimator",
"Definition:Mean Squared Error of Estimator",
"Definition:Variance",
"Definition:Bias/Estimator"
] | [
"Expectation of Linear Transformation of Random Variable",
"Variance of Linear Transformation of Random Variable",
"Variance as Expectation of Square minus Square of Expectation",
"Category:Inductive Statistics",
"Category:Variance"
] |
proofwiki-17979 | Subset of Nowhere Dense Subset is Nowhere Dense | Let $T = \struct {S, \tau}$ be a topological space.
Let $A \subseteq S$ be nowhere dense in $T$.
Let $B \subseteq A$.
Then $B$ is nowhere dense in $T$. | {{AimForCont}} it is not the case that $B$ is nowhere dense in $T$.
Then by definition of nowhere dense:
:$B^-$ contains some open set of $T$ which is non-empty.
From Set Closure Preserves Set Inclusion, we have:
:$B^- \subseteq A^-$
So:
:$A^-$ contains some open set of $T$ which is non-empty.
So $A$ is not nowhere den... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A \subseteq S$ be [[Definition:Nowhere Dense|nowhere dense]] in $T$.
Let $B \subseteq A$.
Then $B$ is [[Definition:Nowhere Dense|nowhere dense]] in $T$. | {{AimForCont}} it is not the case that $B$ is [[Definition:Nowhere Dense|nowhere dense]] in $T$.
Then by definition of [[Definition:Nowhere Dense|nowhere dense]]:
:$B^-$ contains some [[Definition:Open Set (Topology)|open set]] of $T$ which is [[Definition:Non-Empty Set|non-empty]].
From [[Set Closure Preserves Set ... | Subset of Nowhere Dense Subset is Nowhere Dense | https://proofwiki.org/wiki/Subset_of_Nowhere_Dense_Subset_is_Nowhere_Dense | https://proofwiki.org/wiki/Subset_of_Nowhere_Dense_Subset_is_Nowhere_Dense | [
"Nowhere Dense"
] | [
"Definition:Topological Space",
"Definition:Nowhere Dense",
"Definition:Nowhere Dense"
] | [
"Definition:Nowhere Dense",
"Definition:Nowhere Dense",
"Definition:Open Set/Topology",
"Definition:Non-Empty Set",
"Set Closure Preserves Set Inclusion",
"Definition:Open Set/Topology",
"Definition:Non-Empty Set",
"Definition:Nowhere Dense",
"Definition:Contradiction",
"Definition:Nowhere Dense",... |
proofwiki-17980 | Derived Set Preserves Set Inclusion | Let $T = \struct {S, \tau}$ be a topological space.
Let $B \subseteq A \subseteq S$.
Then:
:$B' \subseteq A'$
where $A'$ and $B'$ are the derived sets in $T$ of $A$ and $B$ respectively. | Let:
:$x \in B'$
By the definition of derived set:
:$x$ is a limit point of $B$.
Note that $B \subseteq A$.
From Limit Point of Subset is Limit Point of Set:
:$x$ is a limit point of $A$.
So, $x \in A'$.
That is:
:$B' \subseteq A'$
{{qed}}
Category:Limit Points
bsp2iqal0g2yrhqr0u89brrzwg4vfjq | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $B \subseteq A \subseteq S$.
Then:
:$B' \subseteq A'$
where $A'$ and $B'$ are the [[Definition:Derived Set|derived sets]] in $T$ of $A$ and $B$ respectively. | Let:
:$x \in B'$
By the definition of [[Definition:Derived Set|derived set]]:
:$x$ is a [[Definition:Limit Point (Topology)|limit point]] of $B$.
Note that $B \subseteq A$.
From [[Limit Point of Subset is Limit Point of Set]]:
:$x$ is a [[Definition:Limit Point (Topology)|limit point]] of $A$.
So, $x \in A'$.
... | Derived Set Preserves Set Inclusion | https://proofwiki.org/wiki/Derived_Set_Preserves_Set_Inclusion | https://proofwiki.org/wiki/Derived_Set_Preserves_Set_Inclusion | [
"Limit Points"
] | [
"Definition:Topological Space",
"Definition:Derived Set"
] | [
"Definition:Derived Set",
"Definition:Limit Point/Topology",
"Limit Point of Subset is Limit Point of Set",
"Definition:Limit Point/Topology",
"Category:Limit Points"
] |
proofwiki-17981 | Set Closure Preserves Set Inclusion | Let $T = \struct {S, \tau}$ be a topological space.
Let $B \subseteq A \subseteq S$.
Then:
:$B^- \subseteq A^-$
where $A^-$ and $B^-$ are the set closures in $T$ of $A$ and $B$ respectively. | By definition 1 of set closure:
:$B^- = B \cup B'$
where $B'$ is the derived set of $B$ in $T$.
Similarly:
:$A^- = A \cup A'$
where $A'$ is the derived set of $A$ in $T$.
From Derived Set Preserves Set Inclusion:
:$B' \subseteq A'$
So, by Set Union Preserves Subsets:
:$B \cup B' \subseteq A \cup A'$
That is:
:$B^- \su... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $B \subseteq A \subseteq S$.
Then:
:$B^- \subseteq A^-$
where $A^-$ and $B^-$ are the [[Definition:Closure (Topology)|set closures]] in $T$ of $A$ and $B$ respectively. | By [[Definition:Closure (Topology)/Definition 1|definition 1 of set closure]]:
:$B^- = B \cup B'$
where $B'$ is the [[Definition:Derived Set|derived set]] of $B$ in $T$.
Similarly:
:$A^- = A \cup A'$
where $A'$ is the [[Definition:Derived Set|derived set]] of $A$ in $T$.
From [[Derived Set Preserves Set Inclusio... | Set Closure Preserves Set Inclusion | https://proofwiki.org/wiki/Set_Closure_Preserves_Set_Inclusion | https://proofwiki.org/wiki/Set_Closure_Preserves_Set_Inclusion | [
"Set Closures"
] | [
"Definition:Topological Space",
"Definition:Closure (Topology)"
] | [
"Definition:Closure (Topology)/Definition 1",
"Definition:Derived Set",
"Definition:Derived Set",
"Derived Set Preserves Set Inclusion",
"Set Union Preserves Subsets",
"Category:Set Closures"
] |
proofwiki-17982 | Subset of Meager Set is Meager Set | Let $T = \struct {S, \tau}$ be a topological space.
Let $A$ be meager in $T$.
Let $B \subseteq A$.
Then $B$ is meager in $T$. | Since $A$ is meager in $T$:
:there exists a countable collection of sets $\set {U_n: n \in \N}$ nowhere dense in $T$ such that $\ds A = \bigcup_{n \mathop \in \N} U_n$.
Then, we have:
{{begin-eqn}}
{{eqn | l = B
| r = A \cap B
| c = Intersection with Subset is Subset
}}
{{eqn | r = \paren {\bigcup_{n \math... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A$ be [[Definition:Meager Space|meager]] in $T$.
Let $B \subseteq A$.
Then $B$ is [[Definition:Meager Space|meager]] in $T$. | Since $A$ is [[Definition:Meager Space|meager]] in $T$:
:there exists a countable collection of sets $\set {U_n: n \in \N}$ [[Definition:Nowhere Dense|nowhere dense]] in $T$ such that $\ds A = \bigcup_{n \mathop \in \N} U_n$.
Then, we have:
{{begin-eqn}}
{{eqn | l = B
| r = A \cap B
| c = [[Intersection... | Subset of Meager Set is Meager Set | https://proofwiki.org/wiki/Subset_of_Meager_Set_is_Meager_Set | https://proofwiki.org/wiki/Subset_of_Meager_Set_is_Meager_Set | [
"Meager Spaces"
] | [
"Definition:Topological Space",
"Definition:Meager Space",
"Definition:Meager Space"
] | [
"Definition:Meager Space",
"Definition:Nowhere Dense",
"Intersection with Subset is Subset",
"Union Distributes over Intersection",
"Intersection is Subset",
"Subset of Nowhere Dense Subset is Nowhere Dense",
"Definition:Nowhere Dense",
"Definition:Set Union",
"Definition:Nowhere Dense",
"Definiti... |
proofwiki-17983 | Functions in Vector Space of Real-Valued Functions Continuously Differentiable on Closed Interval vanish at Endpoints | Let $I := \closedint a b$ be a closed real interval.
Let $\struct {\map {C^1} I, +, \, \cdot \,}_\R$ be the continuously differentiable on closed interval real function vector space.
Let $S := \set {x \in \map {C^1} I : \map x a = y_a, \map x b = y_b}$.
Then $S$ is a vector subspace of $\struct {\map {C^1} I, +, \, \cd... | === Necessary Condition ===
Suppose $y_a = y_b = 0$. | Let $I := \closedint a b$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $\struct {\map {C^1} I, +, \, \cdot \,}_\R$ be the [[Space of Continuously Differentiable on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space|continuously dif... | === Necessary Condition ===
Suppose $y_a = y_b = 0$. | Functions in Vector Space of Real-Valued Functions Continuously Differentiable on Closed Interval vanish at Endpoints | https://proofwiki.org/wiki/Functions_in_Vector_Space_of_Real-Valued_Functions_Continuously_Differentiable_on_Closed_Interval_vanish_at_Endpoints | https://proofwiki.org/wiki/Functions_in_Vector_Space_of_Real-Valued_Functions_Continuously_Differentiable_on_Closed_Interval_vanish_at_Endpoints | [
"Vector Spaces",
"Differentiable Real-Valued Functions"
] | [
"Definition:Real Interval/Closed",
"Space of Continuously Differentiable on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space",
"Definition:Vector Subspace"
] | [] |
proofwiki-17984 | Union of Meager Sets is Meager Set | Let $T = \struct {S, \tau}$ be a topological space.
Let $A$ and $B$ be meager in $T$.
Then $A \cup B$ is meager in $T$. | Since $A$ is meager in $T$:
:there exists a countable collection of sets $\set {U_n: n \in \N}$ nowhere dense in $T$ such that $\ds A = \bigcup_{n \mathop \in \N} U_n$.
Since $B$ is meager in $T$:
:there exists a countable collection of sets $\set {V_m: m \in \N}$ nowhere dense in $T$ such that $\ds B = \bigcup_{m \mat... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A$ and $B$ be [[Definition:Meager Space|meager]] in $T$.
Then $A \cup B$ is [[Definition:Meager Space|meager]] in $T$. | Since $A$ is [[Definition:Meager Space|meager]] in $T$:
:there exists a [[Definition:Countable Set|countable]] collection of sets $\set {U_n: n \in \N}$ [[Definition:Nowhere Dense|nowhere dense]] in $T$ such that $\ds A = \bigcup_{n \mathop \in \N} U_n$.
Since $B$ is [[Definition:Meager Space|meager]] in $T$:
:there... | Union of Meager Sets is Meager Set | https://proofwiki.org/wiki/Union_of_Meager_Sets_is_Meager_Set | https://proofwiki.org/wiki/Union_of_Meager_Sets_is_Meager_Set | [
"Meager Spaces",
"Set Union"
] | [
"Definition:Topological Space",
"Definition:Meager Space",
"Definition:Meager Space"
] | [
"Definition:Meager Space",
"Definition:Countable Set",
"Definition:Nowhere Dense",
"Definition:Meager Space",
"Definition:Countable Set",
"Definition:Nowhere Dense",
"Definition:Set Union/Countable Union",
"Definition:Nowhere Dense",
"Definition:Meager Space",
"Category:Meager Spaces",
"Category... |
proofwiki-17985 | Primitive of Reciprocal of p squared plus Square of q by Hyperbolic Cosine of a x/Inverse Hyperbolic Tangent Form | :$\ds \int \frac {\d x} {p^2 + q^2 \cosh^2 a x} = \dfrac 1 {a p \sqrt {p^2 + q^2} } \tanh^{-1} \dfrac {p \tanh a x} {\sqrt {p^2 + q^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p^2 + q^2 \cosh^2 a x}
| r = \frac 1 {2 a p \sqrt {p^2 + q^2} } \ln \size {\frac {p \tanh a x + \sqrt {p^2 + q^2} } {p \tanh a x - \sqrt {p^2 + q^2} } } + C
| c = Primitive of $\dfrac {\d x} {p^2 + q^2 \cosh^2 a x}$: Logarithm Form
}}
{{eqn | r = \frac 1 {a p \sq... | :$\ds \int \frac {\d x} {p^2 + q^2 \cosh^2 a x} = \dfrac 1 {a p \sqrt {p^2 + q^2} } \tanh^{-1} \dfrac {p \tanh a x} {\sqrt {p^2 + q^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p^2 + q^2 \cosh^2 a x}
| r = \frac 1 {2 a p \sqrt {p^2 + q^2} } \ln \size {\frac {p \tanh a x + \sqrt {p^2 + q^2} } {p \tanh a x - \sqrt {p^2 + q^2} } } + C
| c = [[Primitive of Reciprocal of p squared plus Square of q by Hyperbolic Cosine of a x/Logarithm Form|P... | Primitive of Reciprocal of p squared plus Square of q by Hyperbolic Cosine of a x/Inverse Hyperbolic Tangent Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_squared_plus_Square_of_q_by_Hyperbolic_Cosine_of_a_x/Inverse_Hyperbolic_Tangent_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_squared_plus_Square_of_q_by_Hyperbolic_Cosine_of_a_x/Inverse_Hyperbolic_Tangent_Form | [
"Primitive of Reciprocal of p squared plus Square of q by Hyperbolic Cosine of a x"
] | [] | [
"Primitive of Reciprocal of p squared plus Square of q by Hyperbolic Cosine of a x/Logarithm Form"
] |
proofwiki-17986 | Weierstrass Function is Continuous | Let $a \in \openint 0 1$.
Let $b$ be a strictly positive odd integer such that:
:$\ds a b > 1 + \frac 3 2 \pi$
Let $f: \R \to \R$ be a real function defined by:
:$\ds \map f x = \sum_{n \mathop = 0}^\infty a^n \map \cos {b^n \pi x}$
for each $x \in \R$.
Then $f$ is well-defined and continuous. | Note that:
:$\ds \sup_{x \mathop \in \R} \size {a^n \map \cos {b^n \pi x} } = a^n$
Since $a \in \openint 0 1$:
:$\ds \sum_{n \mathop = 0}^\infty a^n$ converges.
So, by the Weierstrass M-Test:
:$\ds \sum_{n \mathop = 0}^\infty a^n \map \cos {b^n \pi x}$ converges uniformly on $\R$.
That is, $f$ is well-defined.
Furthe... | Let $a \in \openint 0 1$.
Let $b$ be a [[Definition:Strictly Positive Integer|strictly positive]] [[Definition:Odd Integer|odd integer]] such that:
:$\ds a b > 1 + \frac 3 2 \pi$
Let $f: \R \to \R$ be a [[Definition:Real Function|real function]] defined by:
:$\ds \map f x = \sum_{n \mathop = 0}^\infty a^n \map \... | Note that:
:$\ds \sup_{x \mathop \in \R} \size {a^n \map \cos {b^n \pi x} } = a^n$
Since $a \in \openint 0 1$:
:$\ds \sum_{n \mathop = 0}^\infty a^n$ [[Definition:Convergent Series|converges]].
So, by the [[Weierstrass M-Test]]:
:$\ds \sum_{n \mathop = 0}^\infty a^n \map \cos {b^n \pi x}$ [[Definition:Uniformly C... | Weierstrass Function is Continuous | https://proofwiki.org/wiki/Weierstrass_Function_is_Continuous | https://proofwiki.org/wiki/Weierstrass_Function_is_Continuous | [
"Weierstrass Functions"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Odd Integer",
"Definition:Real Function",
"Definition:Continuous Real Function"
] | [
"Definition:Convergent Series",
"Weierstrass M-Test",
"Definition:Uniform Convergence/Infinite Series",
"Uniformly Convergent Series of Continuous Functions Converges to Continuous Function/Corollary",
"Definition:Continuous Real Function",
"Category:Weierstrass Functions"
] |
proofwiki-17987 | Real Inverse Hyperbolic Sine Function is Bijection | The real inverse hyperbolic sine is a bijection. | From Hyperbolic Sine is Bijection over Reals and by definition of bijection, we have that $\sinh$ admits an inverse function over $\R$.
From:
:Domain of Bijection is Codomain of Inverse
:Codomain of Bijection is Domain of Inverse
the domain and image of hyperbolic sine over $\R$, is $\R$.
{{qed}}
Category:Inverse Hyper... | The [[Definition:Real Inverse Hyperbolic Sine|real inverse hyperbolic sine]] is a [[Definition:Bijection|bijection]]. | From [[Hyperbolic Sine is Bijection over Reals]] and by definition of [[Definition:Bijection/Definition 3|bijection]], we have that $\sinh$ admits an [[Definition:Inverse Mapping|inverse function]] over $\R$.
From:
:[[Domain of Bijection is Codomain of Inverse]]
:[[Codomain of Bijection is Domain of Inverse]]
the [[D... | Real Inverse Hyperbolic Sine Function is Bijection | https://proofwiki.org/wiki/Real_Inverse_Hyperbolic_Sine_Function_is_Bijection | https://proofwiki.org/wiki/Real_Inverse_Hyperbolic_Sine_Function_is_Bijection | [
"Inverse Hyperbolic Sine"
] | [
"Definition:Inverse Hyperbolic Sine/Real",
"Definition:Bijection"
] | [
"Hyperbolic Sine is Bijection over Reals",
"Definition:Bijection/Definition 3",
"Definition:Inverse Mapping",
"Domain of Bijection is Codomain of Inverse",
"Codomain of Bijection is Domain of Inverse",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definiti... |
proofwiki-17988 | Power Series Expansion for Real Area Hyperbolic Sine | The (real) area hyperbolic sine function has a Taylor series expansion:
{{begin-eqn}}
{{eqn | l = \arsinh x
| r = <nowiki>\begin {cases}
\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1} & : \size x < 1 \\ \\
\ds \ln 2 x + \paren {\sum_{n \m... | === Lemma 1 ===
{{:Power Series Expansion for Real Area Hyperbolic Sine/Lemma 1}}{{qed|lemma}}
From Power Series is Termwise Integrable within Radius of Convergence, $(1)$ can be integrated term by term:
{{begin-eqn}}
{{eqn | l = \int_0^x \frac 1 {\sqrt {t^2 + 1} } \rd t
| r = \sum_{n \mathop = 0}^\infty \int_0^x... | The [[Definition:Real Area Hyperbolic Sine|(real) area hyperbolic sine]] function has a [[Definition:Taylor Series|Taylor series expansion]]:
{{begin-eqn}}
{{eqn | l = \arsinh x
| r = <nowiki>\begin {cases}
\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n ... | === [[Power Series Expansion for Real Area Hyperbolic Sine/Lemma 1|Lemma 1]] ===
{{:Power Series Expansion for Real Area Hyperbolic Sine/Lemma 1}}{{qed|lemma}}
From [[Power Series is Termwise Integrable within Radius of Convergence]], $(1)$ can be [[Definition:Integration|integrated]] term by term:
{{begin-eqn}}
{{e... | Power Series Expansion for Real Area Hyperbolic Sine | https://proofwiki.org/wiki/Power_Series_Expansion_for_Real_Area_Hyperbolic_Sine | https://proofwiki.org/wiki/Power_Series_Expansion_for_Real_Area_Hyperbolic_Sine | [
"Power Series Expansion for Real Area Hyperbolic Sine",
"Examples of Power Series",
"Inverse Hyperbolic Sine"
] | [
"Definition:Inverse Hyperbolic Sine/Real/Definition 2",
"Definition:Taylor Series"
] | [
"Power Series Expansion for Real Area Hyperbolic Sine/Lemma 1",
"Power Series is Termwise Integrable within Radius of Convergence",
"Definition:Primitive (Calculus)/Integration",
"Derivative of Inverse Hyperbolic Sine",
"Definition:Convergent Series",
"Stirling's Formula",
"Convergence of P-Series",
"... |
proofwiki-17989 | Power Series Expansion for Real Area Hyperbolic Cosecant | The (real) area hyperbolic cosecant function has a Taylor series expansion:
{{begin-eqn}}
{{eqn | l = \arcsch x
| r = <nowiki>\begin {cases}
\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac 1 {\paren {2 n + 1} x^{2 n + 1} } & : \size x \ge 1 \\
\ds \ln \dfrac 2 x ... | From Power Series Expansion for Real Area Hyperbolic Sine:
{{begin-eqn}}
{{eqn | l = \arsinh x
| r = <nowiki>\begin {cases}
\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1} & : \size x \le 1 \\
\ds \ln 2 x - \paren {\sum_{n \mathop = 1}^\i... | The [[Definition:Real Area Hyperbolic Cosecant|(real) area hyperbolic cosecant]] function has a [[Definition:Taylor Series|Taylor series expansion]]:
{{begin-eqn}}
{{eqn | l = \arcsch x
| r = <nowiki>\begin {cases}
\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac ... | From [[Power Series Expansion for Real Area Hyperbolic Sine]]:
{{begin-eqn}}
{{eqn | l = \arsinh x
| r = <nowiki>\begin {cases}
\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1} & : \size x \le 1 \\
\ds \ln 2 x - \paren {\sum_{n \mathop = 1... | Power Series Expansion for Real Area Hyperbolic Cosecant | https://proofwiki.org/wiki/Power_Series_Expansion_for_Real_Area_Hyperbolic_Cosecant | https://proofwiki.org/wiki/Power_Series_Expansion_for_Real_Area_Hyperbolic_Cosecant | [
"Examples of Power Series",
"Inverse Hyperbolic Cosecant"
] | [
"Definition:Inverse Hyperbolic Cosecant/Real/Definition 2",
"Definition:Taylor Series"
] | [
"Power Series Expansion for Real Area Hyperbolic Sine",
"Real Area Hyperbolic Sine of Reciprocal equals Real Area Hyperbolic Cosecant"
] |
proofwiki-17990 | Power Series Expansion for Real Area Hyperbolic Secant | The (real) area hyperbolic secant function has a Taylor series expansion:
{{begin-eqn}}
{{eqn | l = \arsech x
| r = \ln \frac 2 x - \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {2 n}! x^{2 n} } {2^{2 n} \paren {n!}^2 \paren {2 n} } }
| c =
}}
{{eqn | r = \ln \frac 2 x - \paren {\dfrac 1 2 \dfrac {x^2}... | From Power Series Expansion for Real Area Hyperbolic Cosine:
{{begin-eqn}}
{{eqn | l = \arcosh x
| r = \ln 2 x - \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} x^{2 n} } }
| c =
}}
{{eqn | r = \ln 2 x - \paren {\dfrac 1 2 \dfrac 1 {2 x^2} + \dfrac {1 \times 3}... | The [[Definition:Real Area Hyperbolic Secant|(real) area hyperbolic secant]] function has a [[Definition:Taylor Series|Taylor series expansion]]:
{{begin-eqn}}
{{eqn | l = \arsech x
| r = \ln \frac 2 x - \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {2 n}! x^{2 n} } {2^{2 n} \paren {n!}^2 \paren {2 n} } }
... | From [[Power Series Expansion for Real Area Hyperbolic Cosine]]:
{{begin-eqn}}
{{eqn | l = \arcosh x
| r = \ln 2 x - \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} x^{2 n} } }
| c =
}}
{{eqn | r = \ln 2 x - \paren {\dfrac 1 2 \dfrac 1 {2 x^2} + \dfrac {1 \time... | Power Series Expansion for Real Area Hyperbolic Secant | https://proofwiki.org/wiki/Power_Series_Expansion_for_Real_Area_Hyperbolic_Secant | https://proofwiki.org/wiki/Power_Series_Expansion_for_Real_Area_Hyperbolic_Secant | [
"Examples of Power Series",
"Inverse Hyperbolic Secant"
] | [
"Definition:Inverse Hyperbolic Secant/Real/Principal Branch",
"Definition:Taylor Series"
] | [
"Power Series Expansion for Real Area Hyperbolic Cosine",
"Real Area Hyperbolic Cosine of Reciprocal equals Real Area Hyperbolic Secant"
] |
proofwiki-17991 | Primitive of Inverse Hyperbolic Cosine of x over a/Corollary | :$\ds \int -\cosh^{-1} \frac x a \rd x = x \paren {-\cosh^{-1} \dfrac x a} + \sqrt {x^2 - a^2} + C$
where $-\cosh^{-1}$ denotes the negative branch of the real inverse hyperbolic cosine multifunction. | {{begin-eqn}}
{{eqn | l = -\cosh^{-1} \frac x a
| r = -\arcosh \frac x a
| c = {{Defof|Real Inverse Hyperbolic Cosine}}
}}
{{eqn | ll= \leadsto
| l = \int -\cosh^{-1} \frac x a \rd x
| r = -\int \arcosh \frac x a \rd x
| c =
}}
{{eqn | r = -\paren {x \arcosh \dfrac x a - \sqrt {x^2 - a^2}... | :$\ds \int -\cosh^{-1} \frac x a \rd x = x \paren {-\cosh^{-1} \dfrac x a} + \sqrt {x^2 - a^2} + C$
where $-\cosh^{-1}$ denotes the [[Definition:Negative Real Number|negative]] [[Definition:Branch of Multifunction|branch]] of the [[Definition:Real Inverse Hyperbolic Cosine|real inverse hyperbolic cosine]] [[Definition... | {{begin-eqn}}
{{eqn | l = -\cosh^{-1} \frac x a
| r = -\arcosh \frac x a
| c = {{Defof|Real Inverse Hyperbolic Cosine}}
}}
{{eqn | ll= \leadsto
| l = \int -\cosh^{-1} \frac x a \rd x
| r = -\int \arcosh \frac x a \rd x
| c =
}}
{{eqn | r = -\paren {x \arcosh \dfrac x a - \sqrt {x^2 - a^2}... | Primitive of Inverse Hyperbolic Cosine of x over a/Corollary | https://proofwiki.org/wiki/Primitive_of_Inverse_Hyperbolic_Cosine_of_x_over_a/Corollary | https://proofwiki.org/wiki/Primitive_of_Inverse_Hyperbolic_Cosine_of_x_over_a/Corollary | [
"Primitive of Inverse Hyperbolic Cosine of x over a"
] | [
"Definition:Negative/Real Number",
"Definition:Multifunction/Branch",
"Definition:Inverse Hyperbolic Cosine/Real",
"Definition:Left-Total Relation/Multifunction"
] | [
"Primitive of Inverse Hyperbolic Cosine of x over a"
] |
proofwiki-17992 | Primitive of x by Inverse Hyperbolic Cosine of x over a/Corollary | :$\ds \int x \paren {-\cosh^{-1} \frac x a} \rd x = \paren {\dfrac {x^2} 2 - \dfrac {a^2} 4} \paren {-\cosh^{-1} \frac x a} + \dfrac {x \sqrt {x^2 - a^2} } 4 + C$
where $-\cosh^{-1}$ denotes the negative branch of the real inverse hyperbolic cosine multifunction. | {{begin-eqn}}
{{eqn | l = -\cosh^{-1} \frac x a
| r = -\arcosh \frac x a
| c = {{Defof|Real Inverse Hyperbolic Cosine}}
}}
{{eqn | ll= \leadsto
| l = \int x \paren {-\cosh^{-1} \frac x a} \rd x
| r = -\int x \arcosh \frac x a \rd x
| c =
}}
{{eqn | r = -\paren {\dfrac {x^2} 2 - \dfrac {a^... | :$\ds \int x \paren {-\cosh^{-1} \frac x a} \rd x = \paren {\dfrac {x^2} 2 - \dfrac {a^2} 4} \paren {-\cosh^{-1} \frac x a} + \dfrac {x \sqrt {x^2 - a^2} } 4 + C$
where $-\cosh^{-1}$ denotes the [[Definition:Negative Real Number|negative]] [[Definition:Branch of Multifunction|branch]] of the [[Definition:Real Inverse ... | {{begin-eqn}}
{{eqn | l = -\cosh^{-1} \frac x a
| r = -\arcosh \frac x a
| c = {{Defof|Real Inverse Hyperbolic Cosine}}
}}
{{eqn | ll= \leadsto
| l = \int x \paren {-\cosh^{-1} \frac x a} \rd x
| r = -\int x \arcosh \frac x a \rd x
| c =
}}
{{eqn | r = -\paren {\dfrac {x^2} 2 - \dfrac {a^... | Primitive of x by Inverse Hyperbolic Cosine of x over a/Corollary | https://proofwiki.org/wiki/Primitive_of_x_by_Inverse_Hyperbolic_Cosine_of_x_over_a/Corollary | https://proofwiki.org/wiki/Primitive_of_x_by_Inverse_Hyperbolic_Cosine_of_x_over_a/Corollary | [
"Primitive of x by Inverse Hyperbolic Cosine of x over a"
] | [
"Definition:Negative/Real Number",
"Definition:Multifunction/Branch",
"Definition:Inverse Hyperbolic Cosine/Real",
"Definition:Left-Total Relation/Multifunction"
] | [
"Primitive of x by Inverse Hyperbolic Cosine of x over a"
] |
proofwiki-17993 | Primitive of Inverse Hyperbolic Cosine of x over a over x/Corollary | {{begin-eqn}}
{{eqn | l = \int \dfrac 1 x \paren {-\cosh^{-1} \dfrac x a} \rd x
| r = -\dfrac 1 2 \ln^2 \paren {\dfrac {2 x} a} - \sum_{n \mathop \ge 1} \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n}^2} \paren {\frac a x}^{2 n} + C
| c =
}}
{{eqn | r = -\dfrac 1 2 \ln^2 \paren {\dfrac {2 x} a} -... | {{begin-eqn}}
{{eqn | l = -\cosh^{-1} \frac x a
| r = -\arcosh \frac x a
| c = {{Defof|Real Inverse Hyperbolic Cosine}}
}}
{{eqn | ll= \leadsto
| l = \int \dfrac 1 x \paren {-\cosh^{-1} \dfrac x a} \rd x
| r = -\int \dfrac 1 x \arcosh \dfrac x a \rd x
| c =
}}
{{eqn | r = -\paren {\dfrac ... | {{begin-eqn}}
{{eqn | l = \int \dfrac 1 x \paren {-\cosh^{-1} \dfrac x a} \rd x
| r = -\dfrac 1 2 \ln^2 \paren {\dfrac {2 x} a} - \sum_{n \mathop \ge 1} \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n}^2} \paren {\frac a x}^{2 n} + C
| c =
}}
{{eqn | r = -\dfrac 1 2 \ln^2 \paren {\dfrac {2 x} a} -... | {{begin-eqn}}
{{eqn | l = -\cosh^{-1} \frac x a
| r = -\arcosh \frac x a
| c = {{Defof|Real Inverse Hyperbolic Cosine}}
}}
{{eqn | ll= \leadsto
| l = \int \dfrac 1 x \paren {-\cosh^{-1} \dfrac x a} \rd x
| r = -\int \dfrac 1 x \arcosh \dfrac x a \rd x
| c =
}}
{{eqn | r = -\paren {\dfrac ... | Primitive of Inverse Hyperbolic Cosine of x over a over x/Corollary | https://proofwiki.org/wiki/Primitive_of_Inverse_Hyperbolic_Cosine_of_x_over_a_over_x/Corollary | https://proofwiki.org/wiki/Primitive_of_Inverse_Hyperbolic_Cosine_of_x_over_a_over_x/Corollary | [
"Primitive of Inverse Hyperbolic Cosine of x over a over x"
] | [
"Definition:Negative/Real Number",
"Definition:Multifunction/Branch",
"Definition:Inverse Hyperbolic Cosine/Real",
"Definition:Left-Total Relation/Multifunction"
] | [
"Primitive of Inverse Hyperbolic Cosine of x over a"
] |
proofwiki-17994 | Primitive of Inverse Hyperbolic Secant of x over a/Corollary | :$\ds \int \sech^{-1} \frac x a \rd x = -x \paren {-\sech^{-1} \dfrac x a} - a \arcsin \dfrac x a + C$
where $-\sech^{-1}$ denotes the negative branch of the real inverse hyperbolic secant multifunction. | {{begin-eqn}}
{{eqn | l = -\sech^{-1} \frac x a
| r = -\arsech \frac x a
| c = {{Defof|Real Inverse Hyperbolic Secant}}
}}
{{eqn | ll= \leadsto
| l = \int -\sech^{-1} \frac x a \rd x
| r = -\int \arsech \frac x a \rd x
| c =
}}
{{eqn | r = -\paren {x \arsech \dfrac x a + a \arcsin \dfrac ... | :$\ds \int \sech^{-1} \frac x a \rd x = -x \paren {-\sech^{-1} \dfrac x a} - a \arcsin \dfrac x a + C$
where $-\sech^{-1}$ denotes the [[Definition:Negative Real Number|negative]] [[Definition:Branch of Multifunction|branch]] of the [[Definition:Real Inverse Hyperbolic Secant|real inverse hyperbolic secant]] [[Definit... | {{begin-eqn}}
{{eqn | l = -\sech^{-1} \frac x a
| r = -\arsech \frac x a
| c = {{Defof|Real Inverse Hyperbolic Secant}}
}}
{{eqn | ll= \leadsto
| l = \int -\sech^{-1} \frac x a \rd x
| r = -\int \arsech \frac x a \rd x
| c =
}}
{{eqn | r = -\paren {x \arsech \dfrac x a + a \arcsin \dfrac ... | Primitive of Inverse Hyperbolic Secant of x over a/Corollary | https://proofwiki.org/wiki/Primitive_of_Inverse_Hyperbolic_Secant_of_x_over_a/Corollary | https://proofwiki.org/wiki/Primitive_of_Inverse_Hyperbolic_Secant_of_x_over_a/Corollary | [
"Primitive of Inverse Hyperbolic Secant of x over a"
] | [
"Definition:Negative/Real Number",
"Definition:Multifunction/Branch",
"Definition:Inverse Hyperbolic Secant/Real",
"Definition:Left-Total Relation/Multifunction"
] | [
"Primitive of Inverse Hyperbolic Secant of x over a"
] |
proofwiki-17995 | Primitive of Power of x by Inverse Hyperbolic Cosine of x over a/Corollary | :$\ds \int x^m \paren {-\cosh^{-1} \frac x a} \rd x = \dfrac {x^{m + 1} } {m + 1} \paren {-\cosh^{-1} \frac x a} + \dfrac 1 {m + 1} \int \dfrac {x^{m + 1} } {\sqrt {x^2 - a^2} } \rd x + C$
where $-\cosh^{-1}$ denotes the negative branch of the real inverse hyperbolic cosine multifunction. | {{begin-eqn}}
{{eqn | l = -\cosh^{-1} \frac x a
| r = -\arcosh \frac x a
| c = {{Defof|Real Inverse Hyperbolic Cosine}}
}}
{{eqn | ll= \leadsto
| l = \int x^m \paren {-\cosh^{-1} \frac x a} \rd x
| r = -\int x^m \arcosh \frac x a \rd x
| c =
}}
{{eqn | r = -\paren {\dfrac {x^{m + 1} } {m ... | :$\ds \int x^m \paren {-\cosh^{-1} \frac x a} \rd x = \dfrac {x^{m + 1} } {m + 1} \paren {-\cosh^{-1} \frac x a} + \dfrac 1 {m + 1} \int \dfrac {x^{m + 1} } {\sqrt {x^2 - a^2} } \rd x + C$
where $-\cosh^{-1}$ denotes the [[Definition:Negative Real Number|negative]] [[Definition:Branch of Multifunction|branch]] of the ... | {{begin-eqn}}
{{eqn | l = -\cosh^{-1} \frac x a
| r = -\arcosh \frac x a
| c = {{Defof|Real Inverse Hyperbolic Cosine}}
}}
{{eqn | ll= \leadsto
| l = \int x^m \paren {-\cosh^{-1} \frac x a} \rd x
| r = -\int x^m \arcosh \frac x a \rd x
| c =
}}
{{eqn | r = -\paren {\dfrac {x^{m + 1} } {m ... | Primitive of Power of x by Inverse Hyperbolic Cosine of x over a/Corollary | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Inverse_Hyperbolic_Cosine_of_x_over_a/Corollary | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Inverse_Hyperbolic_Cosine_of_x_over_a/Corollary | [
"Primitive of Power of x by Inverse Hyperbolic Cosine of x over a"
] | [
"Definition:Negative/Real Number",
"Definition:Multifunction/Branch",
"Definition:Inverse Hyperbolic Cosine/Real",
"Definition:Left-Total Relation/Multifunction"
] | [
"Primitive of Power of x by Inverse Hyperbolic Cosine of x over a"
] |
proofwiki-17996 | Primitive of x by Inverse Hyperbolic Secant of x over a/Corollary | :$\ds \int x \paren {-\sech^{-1} \frac x a} \rd x = -\dfrac {x^2} 2 \paren {-\sech^{-1} \frac x a} + \dfrac {a \sqrt {a^2 - x^2} } 2 + C$
where $-\sech^{-1}$ denotes the negative branch of the real inverse hyperbolic secant multifunction. | {{begin-eqn}}
{{eqn | l = -\sech^{-1} \frac x a
| r = -\arsech \frac x a
| c = {{Defof|Real Inverse Hyperbolic Secant}}
}}
{{eqn | ll= \leadsto
| l = \int x \paren {-\sech^{-1} \frac x a} \rd x
| r = -\int x \arsech \frac x a \rd x
| c =
}}
{{eqn | r = -\paren {\dfrac {x^2} 2 \arsech \dfr... | :$\ds \int x \paren {-\sech^{-1} \frac x a} \rd x = -\dfrac {x^2} 2 \paren {-\sech^{-1} \frac x a} + \dfrac {a \sqrt {a^2 - x^2} } 2 + C$
where $-\sech^{-1}$ denotes the [[Definition:Negative Real Number|negative]] [[Definition:Branch of Multifunction|branch]] of the [[Definition:Real Inverse Hyperbolic Secant|real in... | {{begin-eqn}}
{{eqn | l = -\sech^{-1} \frac x a
| r = -\arsech \frac x a
| c = {{Defof|Real Inverse Hyperbolic Secant}}
}}
{{eqn | ll= \leadsto
| l = \int x \paren {-\sech^{-1} \frac x a} \rd x
| r = -\int x \arsech \frac x a \rd x
| c =
}}
{{eqn | r = -\paren {\dfrac {x^2} 2 \arsech \dfr... | Primitive of x by Inverse Hyperbolic Secant of x over a/Corollary | https://proofwiki.org/wiki/Primitive_of_x_by_Inverse_Hyperbolic_Secant_of_x_over_a/Corollary | https://proofwiki.org/wiki/Primitive_of_x_by_Inverse_Hyperbolic_Secant_of_x_over_a/Corollary | [
"Primitive of x by Inverse Hyperbolic Secant of x over a"
] | [
"Definition:Negative/Real Number",
"Definition:Multifunction/Branch",
"Definition:Inverse Hyperbolic Secant/Real",
"Definition:Left-Total Relation/Multifunction"
] | [
"Primitive of x by Inverse Hyperbolic Secant of x over a"
] |
proofwiki-17997 | Primitive of Inverse Hyperbolic Secant of x over a over x/Corollary | {{begin-eqn}}
{{eqn | l = \int \dfrac 1 x \paren {-\sech^{-1} \dfrac x a} \rd x
| r = \frac 1 2 \map \ln {\dfrac a x} \map \ln {\dfrac {4 a} x} + \sum_{n \mathop \ge 1} \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n}^2} \paren {\frac x a}^{2 n} + C
| c =
}}
{{eqn | r = \dfrac 1 2 \map \ln {\dfrac... | {{begin-eqn}}
{{eqn | l = -\sech^{-1} \frac x a
| r = -\arsech \frac x a
| c = {{Defof|Real Inverse Hyperbolic Secant}}
}}
{{eqn | ll= \leadsto
| l = \int \dfrac 1 x \paren {-\sech^{-1} \dfrac x a} \rd x
| r = -\int \dfrac 1 x \arsech \dfrac x a \rd x
| c =
}}
{{eqn | r = -\paren {-\frac ... | {{begin-eqn}}
{{eqn | l = \int \dfrac 1 x \paren {-\sech^{-1} \dfrac x a} \rd x
| r = \frac 1 2 \map \ln {\dfrac a x} \map \ln {\dfrac {4 a} x} + \sum_{n \mathop \ge 1} \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n}^2} \paren {\frac x a}^{2 n} + C
| c =
}}
{{eqn | r = \dfrac 1 2 \map \ln {\dfrac... | {{begin-eqn}}
{{eqn | l = -\sech^{-1} \frac x a
| r = -\arsech \frac x a
| c = {{Defof|Real Inverse Hyperbolic Secant}}
}}
{{eqn | ll= \leadsto
| l = \int \dfrac 1 x \paren {-\sech^{-1} \dfrac x a} \rd x
| r = -\int \dfrac 1 x \arsech \dfrac x a \rd x
| c =
}}
{{eqn | r = -\paren {-\frac ... | Primitive of Inverse Hyperbolic Secant of x over a over x/Corollary | https://proofwiki.org/wiki/Primitive_of_Inverse_Hyperbolic_Secant_of_x_over_a_over_x/Corollary | https://proofwiki.org/wiki/Primitive_of_Inverse_Hyperbolic_Secant_of_x_over_a_over_x/Corollary | [
"Primitive of Inverse Hyperbolic Secant of x over a over x"
] | [
"Definition:Negative/Real Number",
"Definition:Multifunction/Branch",
"Definition:Inverse Hyperbolic Secant/Real",
"Definition:Left-Total Relation/Multifunction"
] | [
"Primitive of Inverse Hyperbolic Secant of x over a over x"
] |
proofwiki-17998 | Primitive of Power of x by Inverse Hyperbolic Secant of x over a/Corollary | :$\ds \int x^m \paren {-\sech^{-1} \frac x a} \rd x = \dfrac {x^{m + 1} } {m + 1} \paren {-\sech^{-1} \frac x a} - \dfrac a {m + 1} \int \dfrac {x^m} {\sqrt {a^2 - x^2} } \rd x + C$
where $-\sech^{-1}$ denotes the negative branch of the real inverse hyperbolic secant multifunction. | {{begin-eqn}}
{{eqn | l = -\sech^{-1} \frac x a
| r = -\arsech \frac x a
| c = {{Defof|Real Inverse Hyperbolic Secant}}
}}
{{eqn | ll= \leadsto
| l = \int x^m \paren {-\sech^{-1} \frac x a} \rd x
| r = -\int x^m \arsech \frac x a \rd x
| c =
}}
{{eqn | r = -\paren {\dfrac {x^{m + 1} } {m ... | :$\ds \int x^m \paren {-\sech^{-1} \frac x a} \rd x = \dfrac {x^{m + 1} } {m + 1} \paren {-\sech^{-1} \frac x a} - \dfrac a {m + 1} \int \dfrac {x^m} {\sqrt {a^2 - x^2} } \rd x + C$
where $-\sech^{-1}$ denotes the [[Definition:Negative Real Number|negative]] [[Definition:Branch of Multifunction|branch]] of the [[Defin... | {{begin-eqn}}
{{eqn | l = -\sech^{-1} \frac x a
| r = -\arsech \frac x a
| c = {{Defof|Real Inverse Hyperbolic Secant}}
}}
{{eqn | ll= \leadsto
| l = \int x^m \paren {-\sech^{-1} \frac x a} \rd x
| r = -\int x^m \arsech \frac x a \rd x
| c =
}}
{{eqn | r = -\paren {\dfrac {x^{m + 1} } {m ... | Primitive of Power of x by Inverse Hyperbolic Secant of x over a/Corollary | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Inverse_Hyperbolic_Secant_of_x_over_a/Corollary | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Inverse_Hyperbolic_Secant_of_x_over_a/Corollary | [
"Primitive of Power of x by Inverse Hyperbolic Secant of x over a"
] | [
"Definition:Negative/Real Number",
"Definition:Multifunction/Branch",
"Definition:Inverse Hyperbolic Secant/Real",
"Definition:Left-Total Relation/Multifunction"
] | [
"Primitive of Power of x by Inverse Hyperbolic Secant of x over a"
] |
proofwiki-17999 | Normed Vector Space of Bounded Sequences is not Separable | Let $\struct {\ell^\infty, \norm {\, \cdot \,}_\infty}$ be the normed vector space of bounded sequences.
$\struct {\ell^\infty, \norm {\, \cdot \,}_\infty}$ is not separable. | {{AimForCont}} $\struct {\ell^\infty, \norm {\, \cdot \,}_\infty}$ is separable.
Let $\mathbf x := \sequence {x_i}_{i \mathop \in \N}$, $\mathbf a := \sequence {a_i}_{i \mathop \in \N}$, $\mathbf b := \sequence {b_i}_{i \mathop \in \N}$ be sequences.
Let $D := \set {\mathbf x_i \in \R^\N : i \in \N}$ be a dense countab... | Let $\struct {\ell^\infty, \norm {\, \cdot \,}_\infty}$ be the [[Space of Bounded Sequences with Supremum Norm forms Normed Vector Space|normed vector space of bounded sequences]].
$\struct {\ell^\infty, \norm {\, \cdot \,}_\infty}$ is not [[Definition:Separable Normed Vector Space|separable]]. | {{AimForCont}} $\struct {\ell^\infty, \norm {\, \cdot \,}_\infty}$ is [[Definition:Separable Normed Vector Space|separable]].
Let $\mathbf x := \sequence {x_i}_{i \mathop \in \N}$, $\mathbf a := \sequence {a_i}_{i \mathop \in \N}$, $\mathbf b := \sequence {b_i}_{i \mathop \in \N}$ be [[Definition:Sequence|sequences]].... | Normed Vector Space of Bounded Sequences is not Separable | https://proofwiki.org/wiki/Normed_Vector_Space_of_Bounded_Sequences_is_not_Separable | https://proofwiki.org/wiki/Normed_Vector_Space_of_Bounded_Sequences_is_not_Separable | [
"Separable Spaces"
] | [
"Space of Bounded Sequences with Supremum Norm forms Normed Vector Space",
"Definition:Separable Space/Normed Vector Space"
] | [
"Definition:Separable Space/Normed Vector Space",
"Definition:Sequence",
"Definition:Everywhere Dense/Normed Vector Space",
"Definition:Countable Set",
"Definition:Subset",
"Definition:Set",
"Definition:Sequence",
"Definition:Term of Sequence",
"Definition:Distinct/Plural",
"Definition:Element",
... |
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