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-20800 | Fréchet Filter is Filter | Let $S$ be an infinite set.
Let $\FF$ be the Fréchet filter on $S$.
Then $\FF$ is a filter on $S$. | By definition, $\FF$ must satisfy the filter on set axioms:
{{:Axiom:Filter on Set Axioms/Axioms 1}} | Let $S$ be an [[Definition:Infinite Set|infinite set]].
Let $\FF$ be the [[Definition:Fréchet Filter|Fréchet filter]] on $S$.
Then $\FF$ is a [[Definition:Filter on Set|filter]] on $S$. | By definition, $\FF$ must satisfy the [[Axiom:Filter on Set Axioms/Axioms 1|filter on set axioms]]:
{{:Axiom:Filter on Set Axioms/Axioms 1}} | Fréchet Filter is Filter | https://proofwiki.org/wiki/Fréchet_Filter_is_Filter | https://proofwiki.org/wiki/Fréchet_Filter_is_Filter | [
"Fréchet Filters"
] | [
"Definition:Infinite Set",
"Definition:Fréchet Filter",
"Definition:Filter on Set"
] | [
"Axiom:Filter on Set Axioms/Axioms 1",
"Axiom:Filter on Set Axioms/Axioms 1"
] |
proofwiki-20801 | Existence of Nonprincipal Ultrafilter | Let $S$ be an infinite set.
Then there exists an nonprincipal ultrafilter $U$ on $S$. | Let $\FF$ be the Fréchet filter on $S$.
By Fréchet Filter is Filter, $\FF$ is a filter on $S$.
Then, by the Ultrafilter Lemma, there exists some ultrafilter $\UU \supseteq \FF$ on $S$.
The result follows by Ultrafilter is Nonprincipal iff Contains Fréchet Filter.
{{qed}}
{{BPI|Ultrafilter Lemma}}
Category:Nonprincipal ... | Let $S$ be an [[Definition:Infinite Set|infinite set]].
Then there exists an [[Definition:Nonprincipal Ultrafilter|nonprincipal ultrafilter]] $U$ on $S$. | Let $\FF$ be the [[Definition:Fréchet Filter|Fréchet filter]] on $S$.
By [[Fréchet Filter is Filter]], $\FF$ is a [[Definition:Filter on Set|filter]] on $S$.
Then, by the [[Ultrafilter Lemma]], there exists some [[Definition:Ultrafilter on Set|ultrafilter]] $\UU \supseteq \FF$ on $S$.
The result follows by [[Ultrafi... | Existence of Nonprincipal Ultrafilter | https://proofwiki.org/wiki/Existence_of_Nonprincipal_Ultrafilter | https://proofwiki.org/wiki/Existence_of_Nonprincipal_Ultrafilter | [
"Nonprincipal Ultrafilters"
] | [
"Definition:Infinite Set",
"Definition:Principal Ultrafilter/Nonprincipal"
] | [
"Definition:Fréchet Filter",
"Fréchet Filter is Filter",
"Definition:Filter on Set",
"Ultrafilter Lemma",
"Definition:Ultrafilter on Set",
"Ultrafilter is Nonprincipal iff Contains Fréchet Filter",
"Category:Nonprincipal Ultrafilters"
] |
proofwiki-20802 | Initial Topology Generated by Countable Family of Functions Separating Points is Metrizable | Let $X$ be a set.
For each $n \in \N$, let $\struct {Y_n, d_n}$ be a metric space.
Let $\family {f_n}_{n \in \N}$ be a indexed family of functions $X \to Y_n$ that separates points.
Let $\tau$ be the initial topology on $X$ generated by $\family {f_n}_{n \in \N}$.
Then $\tau$ is metrizable. | For each $x, y \in X$, define:
:$\map {\tilde {d_n} } {x, y} = \min \set {\map {d_n} {\map {f_n} x, \map {f_n} y}, 1}$
From Pointwise Minimum of Metric and Positive Real Number is Topologically Equivalent Metric, $\tilde {d_n}$ is a metric that is topologically equivalent to $d_n$.
Now define, for each $x, y \in X$:
... | Let $X$ be a [[Definition:Set|set]].
For each $n \in \N$, let $\struct {Y_n, d_n}$ be a [[Definition:Metric Space|metric space]].
Let $\family {f_n}_{n \in \N}$ be a [[Definition:Indexed Family|indexed family]] of [[Definition:Function|functions]] $X \to Y_n$ that [[Definition:Mappings Separating Points|separates p... | For each $x, y \in X$, define:
:$\map {\tilde {d_n} } {x, y} = \min \set {\map {d_n} {\map {f_n} x, \map {f_n} y}, 1}$
From [[Pointwise Minimum of Metric and Positive Real Number is Topologically Equivalent Metric]], $\tilde {d_n}$ is a [[Definition:Metric|metric]] that is [[Definition:Topologically Equivalent Metri... | Initial Topology Generated by Countable Family of Functions Separating Points is Metrizable | https://proofwiki.org/wiki/Initial_Topology_Generated_by_Countable_Family_of_Functions_Separating_Points_is_Metrizable | https://proofwiki.org/wiki/Initial_Topology_Generated_by_Countable_Family_of_Functions_Separating_Points_is_Metrizable | [
"Initial Topology",
"Metrizable Spaces"
] | [
"Definition:Set",
"Definition:Metric Space",
"Definition:Indexing Set/Family",
"Definition:Function",
"Definition:Mappings Separating Points",
"Definition:Initial Topology",
"Definition:Metrizable Space"
] | [
"Pointwise Minimum of Metric and Positive Real Number is Topologically Equivalent Metric",
"Definition:Metric Space/Metric",
"Definition:Topologically Equivalent Metrics",
"Sum of Infinite Geometric Sequence",
"Definition:Metric Space/Metric",
"Definition:Metric Space/Metric",
"Sum of Infinite Geometric... |
proofwiki-20803 | Nondeterministic Turing Machine is Equivalent to Deterministic Turing Machine | A nondeterministic turing machines and a deterministic turing machines are equivalent.
{{explain|what is meant by "equivalent"}} | === Every Deterministic Turing Machine can be run by a Nondeterministic Turing Machine ===
The machine definitions only differ in whether $\tuple {q', Y', d } \in \map \delta {q, Y}$ or $\tuple {q', Y', d} = \map \delta {q, Y}$.
Define singletons $\map \delta {q, Y} = \set {\tuple {q', Y', d} }$ from the deterministic ... | A [[Definition:Nondeterministic Turing Machine|nondeterministic turing machines]] and a [[Definition:Turing Machine|deterministic turing machines]] are equivalent.
{{explain|what is meant by "equivalent"}} | === Every Deterministic Turing Machine can be run by a Nondeterministic Turing Machine ===
The machine definitions only differ in whether $\tuple {q', Y', d } \in \map \delta {q, Y}$ or $\tuple {q', Y', d} = \map \delta {q, Y}$.
Define singletons $\map \delta {q, Y} = \set {\tuple {q', Y', d} }$ from the deterministi... | Nondeterministic Turing Machine is Equivalent to Deterministic Turing Machine | https://proofwiki.org/wiki/Nondeterministic_Turing_Machine_is_Equivalent_to_Deterministic_Turing_Machine | https://proofwiki.org/wiki/Nondeterministic_Turing_Machine_is_Equivalent_to_Deterministic_Turing_Machine | [
"Turing Machines"
] | [
"Definition:Nondeterministic Turing Machine",
"Definition:Turing Machine"
] | [] |
proofwiki-20804 | Characterization of Paracompactness in T3 Space/Lemma 12 | :$\forall A \in \AA : \set{U^* \in \UU^* : U^* \cap A \ne \O}$ is finite | ===== Lemma 4 =====
{{:Characterization of Paracompactness in T3 Space/Lemma 4}}{{qed|lemma}} | :$\forall A \in \AA : \set{U^* \in \UU^* : U^* \cap A \ne \O}$ is [[Definition:Finite Set|finite]] | ===== [[Characterization of Paracompactness in T3 Space/Lemma 4|Lemma 4]] =====
{{:Characterization of Paracompactness in T3 Space/Lemma 4}}{{qed|lemma}} | Characterization of Paracompactness in T3 Space/Lemma 12 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_12 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_12 | [
"Characterization of Paracompactness in T3 Space"
] | [
"Definition:Finite Set"
] | [
"Characterization of Paracompactness in T3 Space/Lemma 4"
] |
proofwiki-20805 | Sequence of Smallest 3 Consecutive Triangular Numbers which are Sphenic | The smallest $3$ consecutive triangular numbers which are sphenic is:
:$406$, $435$, $465$ | Let $T_n$ denote the $n$th triangular number.
The smallest sphenic number is $30$.
Hence we need investigate triangular number from where $T_n \ge 30$.
Thus:
{{begin-eqn}}
{{eqn | l = T_8
| r = 36
| rr= = 2^2 \times 3^2
| c = which is not sphenic
}}
{{eqn | l = T_9
| r = 45
| rr= = 3^2 \ti... | The [[Definition:Smallest|smallest]] $3$ consecutive [[Definition:Triangular Number|triangular numbers]] which are [[Definition:Sphenic Number|sphenic]] is:
:$406$, $435$, $465$ | Let $T_n$ denote the $n$th [[Definition:Triangular Number|triangular number]].
The smallest [[Definition:Sphenic Number|sphenic number]] is $30$.
Hence we need investigate [[Definition:Triangular Number|triangular number]] from where $T_n \ge 30$.
Thus:
{{begin-eqn}}
{{eqn | l = T_8
| r = 36
| rr= = 2^2... | Sequence of Smallest 3 Consecutive Triangular Numbers which are Sphenic | https://proofwiki.org/wiki/Sequence_of_Smallest_3_Consecutive_Triangular_Numbers_which_are_Sphenic | https://proofwiki.org/wiki/Sequence_of_Smallest_3_Consecutive_Triangular_Numbers_which_are_Sphenic | [
"Triangular Numbers",
"Sphenic Numbers",
"406"
] | [
"Definition:Smallest",
"Definition:Triangular Number",
"Definition:Sphenic Number"
] | [
"Definition:Triangular Number",
"Definition:Sphenic Number",
"Definition:Triangular Number",
"Definition:Sphenic Number",
"Definition:Sphenic Number",
"Definition:Sphenic Number",
"Definition:Sphenic Number",
"Definition:Sphenic Number",
"Definition:Sphenic Number",
"Definition:Sphenic Number",
... |
proofwiki-20806 | Characterization of Paracompactness in T3 Space/Lemma 8 | :there exists a $\sigma$-discrete refinement $\AA$ of $\UU$ | Let $\tau_{X \times X}$ denote the product topology on the cartesian product $X \times X$.
By definition of even cover there exists a neighborhood $V$ of the diagonal $\Delta_X$ of $X \times X$ in the product space $\struct {X \times X, \tau_{X \times X} }$:
:$\set{\map V x : x \in X}$ is a refinement of $\UU$
where:
:... | :there exists a [[Definition:Sigma-Discrete Set of Subsets|$\sigma$-discrete]] [[Definition:Refinement of Cover|refinement]] $\AA$ of $\UU$ | Let $\tau_{X \times X}$ denote the [[Definition:Product Topology|product topology]] on the [[Definition:Cartesian Product|cartesian product]] $X \times X$.
By definition of [[Definition:Even Cover|even cover]] there exists a [[Definition:Neighborhood of Set|neighborhood]] $V$ of the [[Definition:Diagonal Relation|dia... | Characterization of Paracompactness in T3 Space/Lemma 8 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_8 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_8 | [
"Characterization of Paracompactness in T3 Space"
] | [
"Definition:Sigma-Discrete Set of Subsets",
"Definition:Refinement of Cover"
] | [
"Definition:Product Topology",
"Definition:Cartesian Product",
"Definition:Even Cover",
"Definition:Neighborhood (Topology)/Set",
"Definition:Diagonal Relation",
"Definition:Product Space (Topology)/Two Factor Spaces",
"Definition:Refinement of Cover",
"Definition:Relation/Relation as Subset of Cartes... |
proofwiki-20807 | Union is Empty iff Sets are Empty/Set of Sets | Let $\SS$ be a set of sets.
Then:
:$\ds \bigcup \SS = \O \iff \forall S \in \SS: S = \O$ | {{begin-eqn}}
{{eqn | r = \bigcup \SS = \O
| o =
}}
{{eqn | ll= \leadstoandfrom
| o =
| q = \neg \exists x
| r = x \in \paren {\bigcup \SS}
| c = {{Defof|Empty Set}}
}}
{{eqn | ll= \leadstoandfrom
| o =
| q = \forall x
| r = \neg \paren {x \in \bigcup \SS }
| c =... | Let $\SS$ be a [[Definition:Set of Sets|set of sets]].
Then:
:$\ds \bigcup \SS = \O \iff \forall S \in \SS: S = \O$ | {{begin-eqn}}
{{eqn | r = \bigcup \SS = \O
| o =
}}
{{eqn | ll= \leadstoandfrom
| o =
| q = \neg \exists x
| r = x \in \paren {\bigcup \SS}
| c = {{Defof|Empty Set}}
}}
{{eqn | ll= \leadstoandfrom
| o =
| q = \forall x
| r = \neg \paren {x \in \bigcup \SS }
| c =... | Union is Empty iff Sets are Empty/Set of Sets | https://proofwiki.org/wiki/Union_is_Empty_iff_Sets_are_Empty/Set_of_Sets | https://proofwiki.org/wiki/Union_is_Empty_iff_Sets_are_Empty/Set_of_Sets | [
"Union is Empty iff Sets are Empty"
] | [
"Definition:Set of Sets"
] | [
"De Morgan's Laws (Predicate Logic)",
"De Morgan's Laws (Predicate Logic)/Assertion of Universality",
"Category:Union is Empty iff Sets are Empty"
] |
proofwiki-20808 | Union of Set of Sets is Non-empty iff some Set is Non-empty | Let $\SS$ be a set of sets.
Then:
:$\ds \bigcup \SS \ne \O \iff \exists S \in \SS: S \ne \O$ | Follows immediately from Union of Set of Sets is Empty iff Sets are Empty.
{{qed}}
Category:Set Union
Category:Empty Set
d7pgvuvtxh20cuxq475qz4ipp5fv37w | Let $\SS$ be a [[Definition:Set of Sets|set of sets]].
Then:
:$\ds \bigcup \SS \ne \O \iff \exists S \in \SS: S \ne \O$ | Follows immediately from [[Union of Set of Sets is Empty iff Sets are Empty]].
{{qed}}
[[Category:Set Union]]
[[Category:Empty Set]]
d7pgvuvtxh20cuxq475qz4ipp5fv37w | Union of Set of Sets is Non-empty iff some Set is Non-empty | https://proofwiki.org/wiki/Union_of_Set_of_Sets_is_Non-empty_iff_some_Set_is_Non-empty | https://proofwiki.org/wiki/Union_of_Set_of_Sets_is_Non-empty_iff_some_Set_is_Non-empty | [
"Set Union",
"Empty Set"
] | [
"Definition:Set of Sets"
] | [
"Union is Empty iff Sets are Empty/Set of Sets",
"Category:Set Union",
"Category:Empty Set"
] |
proofwiki-20809 | Limit of Hölder Mean as Exponent tends to Zero is Geometric Mean | Let $x_1, x_2, \ldots, x_n \in \R_{\ge 0}$ be real numbers.
For $p \in \R_{\ne 0}$, let $\map {M_p} {x_1, x_2, \ldots, x_n}$ denote the '''Hölder mean with exponent $p$ of $x_1, x_2, \ldots, x_n$'''.
Then:
:$\ds \lim_{p \mathop \to 0} \map {M_p} {x_1, x_2, \ldots, x_n} = \paren {x_1 x_2 \cdots x_n}^{1 / n}$
which is th... | Let $p \in \R$ such that $p \ne 0$.
{{begin-eqn}}
{{eqn | l = \map {M_p} {x_1, x_2, \ldots, x_n}
| r = \paren {\frac 1 n \sum_{k \mathop = 1}^n {x_k}^p}^{1 / p}
| c = {{Defof|Hölder Mean}}
}}
{{eqn | ll= \leadsto
| l = \map \ln {\map {M_p} {x_1, x_2, \ldots, x_n} }
| r = \map \ln {\frac 1 n \sum... | Let $x_1, x_2, \ldots, x_n \in \R_{\ge 0}$ be [[Definition:Real Number|real numbers]].
For $p \in \R_{\ne 0}$, let $\map {M_p} {x_1, x_2, \ldots, x_n}$ denote the '''[[Definition:Hölder Mean|Hölder mean with exponent $p$ of $x_1, x_2, \ldots, x_n$]]'''.
Then:
:$\ds \lim_{p \mathop \to 0} \map {M_p} {x_1, x_2, \ldots... | Let $p \in \R$ such that $p \ne 0$.
{{begin-eqn}}
{{eqn | l = \map {M_p} {x_1, x_2, \ldots, x_n}
| r = \paren {\frac 1 n \sum_{k \mathop = 1}^n {x_k}^p}^{1 / p}
| c = {{Defof|Hölder Mean}}
}}
{{eqn | ll= \leadsto
| l = \map \ln {\map {M_p} {x_1, x_2, \ldots, x_n} }
| r = \map \ln {\frac 1 n \su... | Limit of Hölder Mean as Exponent tends to Zero is Geometric Mean | https://proofwiki.org/wiki/Limit_of_Hölder_Mean_as_Exponent_tends_to_Zero_is_Geometric_Mean | https://proofwiki.org/wiki/Limit_of_Hölder_Mean_as_Exponent_tends_to_Zero_is_Geometric_Mean | [
"Hölder Mean",
"Geometric Mean"
] | [
"Definition:Real Number",
"Definition:Hölder Mean",
"Definition:Geometric Mean"
] | [
"Definition:Natural Logarithm/Positive Real",
"Logarithm of Power",
"Definition:Exponential Function/Real",
"L'Hôpital's Rule",
"Definition:Exponential Function/Real",
"Derivative of Natural Logarithm Function",
"Derivative of General Exponential Function",
"Derivative of Composite Function",
"Deriv... |
proofwiki-20810 | Hölder Mean for Exponent 1 is Arithmetic Mean | Let $x_1, x_2, \ldots, x_n \in \R_{\ge 0}$ be real numbers.
For $p \in \R_{\ne 0}$, let $\map {M_p} {x_1, x_2, \ldots, x_n}$ denote the '''Hölder mean with exponent $p$ of $x_1, x_2, \ldots, x_n$'''.
Then:
:$\map {M_1} {x_1, x_2, \ldots, x_n} = \dfrac {x_1 + x_2 + \cdots + x_n} n$
which is the arithmetic mean of $x_1, ... | Recall the definition of the '''Hölder mean with exponent $p$''':
:$\ds \map {M_p} {x_1, x_2, \ldots, x_n} = \paren {\frac 1 n \sum_{k \mathop = 1}^n {x_k}^p}^{1 / p}$
Then:
{{begin-eqn}}
{{eqn | l = \map {M_1} {x_1, x_2, \ldots, x_n}
| r = \paren {\frac 1 n \sum_{k \mathop = 1}^n {x_k}^1}^{1 / 1}
| c =
}}... | Let $x_1, x_2, \ldots, x_n \in \R_{\ge 0}$ be [[Definition:Real Number|real numbers]].
For $p \in \R_{\ne 0}$, let $\map {M_p} {x_1, x_2, \ldots, x_n}$ denote the '''[[Definition:Hölder Mean|Hölder mean with exponent $p$ of $x_1, x_2, \ldots, x_n$]]'''.
Then:
:$\map {M_1} {x_1, x_2, \ldots, x_n} = \dfrac {x_1 + x_2 ... | Recall the definition of the '''[[Definition:Hölder Mean|Hölder mean with exponent $p$]]''':
:$\ds \map {M_p} {x_1, x_2, \ldots, x_n} = \paren {\frac 1 n \sum_{k \mathop = 1}^n {x_k}^p}^{1 / p}$
Then:
{{begin-eqn}}
{{eqn | l = \map {M_1} {x_1, x_2, \ldots, x_n}
| r = \paren {\frac 1 n \sum_{k \mathop = 1}^n {x_... | Hölder Mean for Exponent 1 is Arithmetic Mean | https://proofwiki.org/wiki/Hölder_Mean_for_Exponent_1_is_Arithmetic_Mean | https://proofwiki.org/wiki/Hölder_Mean_for_Exponent_1_is_Arithmetic_Mean | [
"Hölder Mean",
"Arithmetic Mean"
] | [
"Definition:Real Number",
"Definition:Hölder Mean",
"Definition:Arithmetic Mean"
] | [
"Definition:Hölder Mean",
"Definition:Arithmetic Mean"
] |
proofwiki-20811 | Hölder Mean for Exponent -1 is Harmonic Mean | Let $x_1, x_2, \ldots, x_n \in \R_{\ge 0}$ be real numbers.
For $p \in \R_{\ne 0}$, let $\map {M_p} {x_1, x_2, \ldots, x_n}$ denote the '''Hölder mean with exponent $p$ of $x_1, x_2, \ldots, x_n$'''.
Then:
:$\map {M_{-1} } {x_1, x_2, \ldots, x_n} = \dfrac 1 {\dfrac 1 n \paren {\dfrac 1 {x_1} + \dfrac 1 {x_2} + \cdots +... | Recall the definition of the '''Hölder mean with exponent $p$''':
:$\ds \map {M_p} {x_1, x_2, \ldots, x_n} = \paren {\frac 1 n \sum_{k \mathop = 1}^n {x_k}^p}^{1 / p}$
Then:
{{begin-eqn}}
{{eqn | l = \map {M_{-1} } {x_1, x_2, \ldots, x_n}
| r = \paren {\frac 1 n \sum_{k \mathop = 1}^n {x_k}^{-1} }^{1 / -1}
... | Let $x_1, x_2, \ldots, x_n \in \R_{\ge 0}$ be [[Definition:Real Number|real numbers]].
For $p \in \R_{\ne 0}$, let $\map {M_p} {x_1, x_2, \ldots, x_n}$ denote the '''[[Definition:Hölder Mean|Hölder mean with exponent $p$ of $x_1, x_2, \ldots, x_n$]]'''.
Then:
:$\map {M_{-1} } {x_1, x_2, \ldots, x_n} = \dfrac 1 {\dfr... | Recall the definition of the '''[[Definition:Hölder Mean|Hölder mean with exponent $p$]]''':
:$\ds \map {M_p} {x_1, x_2, \ldots, x_n} = \paren {\frac 1 n \sum_{k \mathop = 1}^n {x_k}^p}^{1 / p}$
Then:
{{begin-eqn}}
{{eqn | l = \map {M_{-1} } {x_1, x_2, \ldots, x_n}
| r = \paren {\frac 1 n \sum_{k \mathop = 1}^n... | Hölder Mean for Exponent -1 is Harmonic Mean | https://proofwiki.org/wiki/Hölder_Mean_for_Exponent_-1_is_Harmonic_Mean | https://proofwiki.org/wiki/Hölder_Mean_for_Exponent_-1_is_Harmonic_Mean | [
"Hölder Mean",
"Harmonic Mean"
] | [
"Definition:Real Number",
"Definition:Hölder Mean",
"Definition:Harmonic Mean"
] | [
"Definition:Hölder Mean",
"Definition:Harmonic Mean"
] |
proofwiki-20812 | Limit of Hölder Mean as Exponent tends to Infinity | Let $x_1, x_2, \ldots, x_n \in \R_{\ge 0}$ be real numbers.
For $p \in \R_{\ne 0}$, let $\map {M_p} {x_1, x_2, \ldots, x_n}$ denote the '''Hölder mean with exponent $p$ of $x_1, x_2, \ldots, x_n$'''.
Then:
:$\ds \lim_{p \mathop \to +\infty} \map {M_p} {x_1, x_2, \ldots, x_n} = \max \set {x_1, x_2, \ldots, x_n}$ | Let $p \in \R$ such that $p \ne 0$.
Let it be assumed (or arranged) that:
:$x_1 \ge x_2 \ge \cdots \ge x_n$
Then:
{{begin-eqn}}
{{eqn | l = \lim_{p \mathop \to +\infty} \map {M_p} {x_1, x_2, \ldots, x_n}
| r = \lim_{p \mathop \to +\infty} \paren {\frac 1 n \sum_{k \mathop = 1}^n {x_k}^p}^{1 / p}
| c =
}}
{... | Let $x_1, x_2, \ldots, x_n \in \R_{\ge 0}$ be [[Definition:Real Number|real numbers]].
For $p \in \R_{\ne 0}$, let $\map {M_p} {x_1, x_2, \ldots, x_n}$ denote the '''[[Definition:Hölder Mean|Hölder mean with exponent $p$ of $x_1, x_2, \ldots, x_n$]]'''.
Then:
:$\ds \lim_{p \mathop \to +\infty} \map {M_p} {x_1, x_2, ... | Let $p \in \R$ such that $p \ne 0$.
Let it be assumed (or arranged) that:
:$x_1 \ge x_2 \ge \cdots \ge x_n$
Then:
{{begin-eqn}}
{{eqn | l = \lim_{p \mathop \to +\infty} \map {M_p} {x_1, x_2, \ldots, x_n}
| r = \lim_{p \mathop \to +\infty} \paren {\frac 1 n \sum_{k \mathop = 1}^n {x_k}^p}^{1 / p}
| c =
}}... | Limit of Hölder Mean as Exponent tends to Infinity | https://proofwiki.org/wiki/Limit_of_Hölder_Mean_as_Exponent_tends_to_Infinity | https://proofwiki.org/wiki/Limit_of_Hölder_Mean_as_Exponent_tends_to_Infinity | [
"Hölder Mean"
] | [
"Definition:Real Number",
"Definition:Hölder Mean"
] | [
"Definition:Parenthesis",
"Combination Theorem for Limits of Functions/Real/Multiple Rule"
] |
proofwiki-20813 | Limit of Hölder Mean as Exponent tends to Negative Infinity | Let $x_1, x_2, \ldots, x_n \in \R_{\ge 0}$ be real numbers.
For $p \in \R_{\ne 0}$, let $\map {M_p} {x_1, x_2, \ldots, x_n}$ denote the '''Hölder mean with exponent $p$ of $x_1, x_2, \ldots, x_n$'''.
Then:
:$\ds \lim_{p \mathop \to -\infty} \map {M_p} {x_1, x_2, \ldots, x_n} = \min \set {x_1, x_2, \ldots, x_n}$ | Let $p \in \R$ such that $p \ne 0$.
Let it be assumed (or arranged) that:
:$x_1 \ge x_2 \ge \cdots \ge x_n$
Then:
{{begin-eqn}}
{{eqn | l = \lim_{p \mathop \to -\infty} \map {M_p} {x_1, x_2, \ldots, x_n}
| r = \dfrac 1 {\ds \lim_{p \mathop \to +\infty} \map {M_p} {\dfrac 1 {x_1}, \dfrac 1 {x_2}, \ldots, \dfrac 1 ... | Let $x_1, x_2, \ldots, x_n \in \R_{\ge 0}$ be [[Definition:Real Number|real numbers]].
For $p \in \R_{\ne 0}$, let $\map {M_p} {x_1, x_2, \ldots, x_n}$ denote the '''[[Definition:Hölder Mean|Hölder mean with exponent $p$ of $x_1, x_2, \ldots, x_n$]]'''.
Then:
:$\ds \lim_{p \mathop \to -\infty} \map {M_p} {x_1, x_2, ... | Let $p \in \R$ such that $p \ne 0$.
Let it be assumed (or arranged) that:
:$x_1 \ge x_2 \ge \cdots \ge x_n$
Then:
{{begin-eqn}}
{{eqn | l = \lim_{p \mathop \to -\infty} \map {M_p} {x_1, x_2, \ldots, x_n}
| r = \dfrac 1 {\ds \lim_{p \mathop \to +\infty} \map {M_p} {\dfrac 1 {x_1}, \dfrac 1 {x_2}, \ldots, \dfrac ... | Limit of Hölder Mean as Exponent tends to Negative Infinity | https://proofwiki.org/wiki/Limit_of_Hölder_Mean_as_Exponent_tends_to_Negative_Infinity | https://proofwiki.org/wiki/Limit_of_Hölder_Mean_as_Exponent_tends_to_Negative_Infinity | [
"Hölder Mean"
] | [
"Definition:Real Number",
"Definition:Hölder Mean"
] | [
"Limit of Hölder Mean as Exponent tends to Infinity"
] |
proofwiki-20814 | Direction Angle of 2D Vector in Terms of Arctangent | Let $\mathbf a$ be a vector quantity embedded in a Cartesian plane $P$ expressed in component form as:
:$\mathbf a = x \mathbf i + y \mathbf j$
Let $\theta$ denote the direction of $\mathbf a$.
Then:
:<nowiki>$\theta = \begin{cases}
\map \arctan {\dfrac y x} & : x > 0 \\
\map \arctan {\dfrac y x} + \pi & : x < 0 \text... | Let $\mathbf a$ be such that one of the following holds:
:$\mathbf a$ is in Quadrant $\text{I}$ or Quadrant $\text{IV}$
:$\mathbf a$ is on the positive direction of the $x$-axis.
Then:
:$x > 0$
and:
:$-\dfrac \pi 2 < \theta < \dfrac \pi 2$
The components of $\mathbf a$ form the legs of a right triangle where:
{{begin-e... | Let $\mathbf a$ be a [[Definition:Vector Quantity|vector quantity]] embedded in a [[Definition:Cartesian Plane|Cartesian plane]] $P$ expressed in [[Definition:Component of Vector in Plane|component form]] as:
:$\mathbf a = x \mathbf i + y \mathbf j$
Let $\theta$ denote the [[Definition:Direction|direction]] of $\math... | Let $\mathbf a$ be such that one of the following holds:
:$\mathbf a$ is in [[Definition:First Quadrant|Quadrant $\text{I}$]] or [[Definition:Fourth Quadrant|Quadrant $\text{IV}$]]
:$\mathbf a$ is on the [[Definition:Positive Direction|positive direction]] of the [[Definition:X-Axis|$x$-axis]].
Then:
:$x > 0$
and:
:$... | Direction Angle of 2D Vector in Terms of Arctangent | https://proofwiki.org/wiki/Direction_Angle_of_2D_Vector_in_Terms_of_Arctangent | https://proofwiki.org/wiki/Direction_Angle_of_2D_Vector_in_Terms_of_Arctangent | [
"Vectors",
"Arctangent Function"
] | [
"Definition:Vector Quantity",
"Definition:Cartesian Plane",
"Definition:Vector Quantity/Component/Cartesian Plane",
"Definition:Direction",
"Definition:Inverse Tangent/Real/Arctangent",
"Definition:Real Interval/Open",
"Definition:Axis/Positive Direction",
"Definition:Axis/X-Axis",
"Definition:Real ... | [
"Definition:Cartesian Plane/Quadrants/First",
"Definition:Cartesian Plane/Quadrants/Fourth",
"Definition:Axis/Positive Direction",
"Definition:Axis/X-Axis",
"Definition:Vector Quantity/Component/Cartesian Plane",
"Definition:Triangle (Geometry)/Right-Angled/Legs",
"Definition:Triangle (Geometry)/Right-A... |
proofwiki-20815 | Minimum is Less than or Equal to Geometric Mean | :$\min \set {x_1, x_2, \ldots, x_n} \le \map G {x_1, x_2, \ldots, x_n}$
Equality holds {{iff}}:
:$x_1 = x_2 = \cdots x_n$
or $x_k = 0$ for some $k \in \set {1, 2, \ldots, n}$. | From Limit of Hölder Mean as Exponent tends to Zero is Geometric Mean:
:$\map G {x_1, x_2, \ldots, x_n} = \ds \lim_{p \mathop \to 0} \map {M_p} {x_1, x_2, \ldots, x_n}$
where $\map {M_p} {x_1, x_2, \ldots, x_n}$ denotes the Hölder mean with exponent $p$ of $x_1, x_2, \ldots, x_n$.
Then from Minimum is Less than or Equa... | :$\min \set {x_1, x_2, \ldots, x_n} \le \map G {x_1, x_2, \ldots, x_n}$
Equality holds {{iff}}:
:$x_1 = x_2 = \cdots x_n$
or $x_k = 0$ for some $k \in \set {1, 2, \ldots, n}$. | From [[Limit of Hölder Mean as Exponent tends to Zero is Geometric Mean]]:
:$\map G {x_1, x_2, \ldots, x_n} = \ds \lim_{p \mathop \to 0} \map {M_p} {x_1, x_2, \ldots, x_n}$
where $\map {M_p} {x_1, x_2, \ldots, x_n}$ denotes the [[Definition:Hölder Mean|Hölder mean with exponent $p$]] of $x_1, x_2, \ldots, x_n$.
Then... | Minimum is Less than or Equal to Geometric Mean | https://proofwiki.org/wiki/Minimum_is_Less_than_or_Equal_to_Geometric_Mean | https://proofwiki.org/wiki/Minimum_is_Less_than_or_Equal_to_Geometric_Mean | [
"Geometric Mean"
] | [] | [
"Limit of Hölder Mean as Exponent tends to Zero is Geometric Mean",
"Definition:Hölder Mean",
"Minimum is Less than or Equal to Hölder Mean"
] |
proofwiki-20816 | Maximum is Greater than or Equal to Geometric Mean | :$\max \set {x_1, x_2, \ldots, x_n} \ge \map G {x_1, x_2, \ldots, x_n}$
Equality holds {{iff}}:
:$x_1 = x_2 = \cdots x_n$
or $x_k = 0$ for some $k \in \set {1, 2, \ldots, n}$. | From Limit of Hölder Mean as Exponent tends to Zero is Geometric Mean:
:$\map G {x_1, x_2, \ldots, x_n} = \ds \lim_{p \mathop \to 0} \map {M_p} {x_1, x_2, \ldots, x_n}$
where $\map {M_p} {x_1, x_2, \ldots, x_n}$ denotes the Hölder mean with exponent $p$ of $x_1, x_2, \ldots, x_n$.
Then from Maximum is Greater than or E... | :$\max \set {x_1, x_2, \ldots, x_n} \ge \map G {x_1, x_2, \ldots, x_n}$
Equality holds {{iff}}:
:$x_1 = x_2 = \cdots x_n$
or $x_k = 0$ for some $k \in \set {1, 2, \ldots, n}$. | From [[Limit of Hölder Mean as Exponent tends to Zero is Geometric Mean]]:
:$\map G {x_1, x_2, \ldots, x_n} = \ds \lim_{p \mathop \to 0} \map {M_p} {x_1, x_2, \ldots, x_n}$
where $\map {M_p} {x_1, x_2, \ldots, x_n}$ denotes the [[Definition:Hölder Mean|Hölder mean with exponent $p$]] of $x_1, x_2, \ldots, x_n$.
Then... | Maximum is Greater than or Equal to Geometric Mean | https://proofwiki.org/wiki/Maximum_is_Greater_than_or_Equal_to_Geometric_Mean | https://proofwiki.org/wiki/Maximum_is_Greater_than_or_Equal_to_Geometric_Mean | [
"Geometric Mean"
] | [] | [
"Limit of Hölder Mean as Exponent tends to Zero is Geometric Mean",
"Definition:Hölder Mean",
"Maximum is Greater than or Equal to Hölder Mean"
] |
proofwiki-20817 | Characterization of Paracompactness in T3 Space/Lemma 20 | Let $N$ be a neighborhood of the diagonal $\Delta_X$ of $X \times X$ in the product space $\struct {X \times X, \tau_{X \times X} }$.
Then there exists an open neighborhood $W$ of the diagonal $\Delta_X$ in $\struct {X \times X, \tau_{X \times X} }$:
:$W$ is symmetric as a relation on $X \times X$, that is, $W = W^{-1}... | Let:
:$\VV = \set{V \in \tau : V \times V \subseteq N}$
From Neighborhood of Diagonal induces Open Cover:
:$\VV$ is an open cover of $T$
We have {{hypothesis}}, $\VV$ is even.
From Characterization of Even Cover, there exists an open neighborhood $R$ of the diagonal $\Delta_X$ in $\struct {X \times X, \tau_{X \times X}... | Let $N$ be a [[Definition:Neighborhood of Set|neighborhood]] of the [[Definition:Diagonal Relation|diagonal $\Delta_X$]] of $X \times X$ in the [[Definition:Product Space (Topology) of Two Factor Spaces|product space]] $\struct {X \times X, \tau_{X \times X} }$.
Then there exists an [[Definition:Open Neighborhood|ope... | Let:
:$\VV = \set{V \in \tau : V \times V \subseteq N}$
From [[Neighborhood of Diagonal induces Open Cover]]:
:$\VV$ is an [[Definition:Open Cover|open cover]] of $T$
We have {{hypothesis}}, $\VV$ is [[Definition:Even Cover|even]].
From [[Characterization of Even Cover]], there exists an [[Definition:Open Neighbo... | Characterization of Paracompactness in T3 Space/Lemma 20 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_20 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_20 | [
"Characterization of Paracompactness in T3 Space"
] | [
"Definition:Neighborhood (Topology)/Set",
"Definition:Diagonal Relation",
"Definition:Product Space (Topology)/Two Factor Spaces",
"Definition:Open Neighborhood",
"Definition:Diagonal Relation",
"Definition:Symmetric Relation",
"Definition:Relation/Relation as Subset of Cartesian Product",
"Definition... | [
"Neighborhood of Diagonal induces Open Cover",
"Definition:Open Cover",
"Definition:Even Cover",
"Characterization of Even Cover",
"Definition:Open Neighborhood",
"Definition:Diagonal Relation",
"Definition:Refinement of Cover",
"Definition:Relation/Relation as Subset of Cartesian Product",
"Definit... |
proofwiki-20818 | Reverse Triangle Inequality/Real and Complex Fields/Corollary 2 | Let $x$ and $y$ be elements of either the real numbers $\R$ or the complex numbers $\C$.
Then:
:$\size {x + y} \ge \size x - \size y$
where $\size x$ denotes either the absolute value of a real number or the complex modulus of a complex number. | Let $x$ and $y$ be elements of either the real numbers $\R$ or the complex numbers $\C$.
Let $z := -y$.
Then we have:
{{begin-eqn}}
{{eqn | l = \size {x - z}
| o = \ge
| r = \size x - \size z
| c = {{Corollary|Reverse Triangle Inequality/Real and Complex Fields|1|disp = Reverse Triangle Inequality for... | Let $x$ and $y$ be [[Definition:Element|elements]] of either the [[Definition:Real Number|real numbers]] $\R$ or the [[Definition:Complex Number|complex numbers]] $\C$.
Then:
:$\size {x + y} \ge \size x - \size y$
where $\size x$ denotes either the [[Definition:Absolute Value|absolute value]] of a [[Definition:Real Nu... | Let $x$ and $y$ be [[Definition:Element|elements]] of either the [[Definition:Real Number|real numbers]] $\R$ or the [[Definition:Complex Number|complex numbers]] $\C$.
Let $z := -y$.
Then we have:
{{begin-eqn}}
{{eqn | l = \size {x - z}
| o = \ge
| r = \size x - \size z
| c = {{Corollary|Reverse ... | Reverse Triangle Inequality/Real and Complex Fields/Corollary 2 | https://proofwiki.org/wiki/Reverse_Triangle_Inequality/Real_and_Complex_Fields/Corollary_2 | https://proofwiki.org/wiki/Reverse_Triangle_Inequality/Real_and_Complex_Fields/Corollary_2 | [
"Triangle Inequality"
] | [
"Definition:Element",
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Absolute Value",
"Definition:Real Number",
"Definition:Complex Modulus",
"Definition:Complex Number"
] | [
"Definition:Element",
"Definition:Real Number",
"Definition:Complex Number"
] |
proofwiki-20819 | Triangle Inequality/Real Numbers/General Result | Let $x_1, x_2, \dotsc, x_n \in \R$ be real numbers.
Let $\size x$ denote the absolute value of $x$.
Then:
:$\ds \size {\sum_{i \mathop = 1}^n x_i} \le \sum_{i \mathop = 1}^n \size {x_i}$ | Proof by induction:
For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
:$\ds \size {\sum_{i \mathop = 1}^n x_i} \le \sum_{i \mathop = 1}^n \size {x_i}$
$\map P 1$ is true by definition of the usual ordering on real numbers:
:$\size {x_1} \le \size {x_1}$ | Let $x_1, x_2, \dotsc, x_n \in \R$ be [[Definition:Real Number|real numbers]].
Let $\size x$ denote the [[Definition:Absolute Value|absolute value]] of $x$.
Then:
:$\ds \size {\sum_{i \mathop = 1}^n x_i} \le \sum_{i \mathop = 1}^n \size {x_i}$ | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \size {\sum_{i \mathop = 1}^n x_i} \le \sum_{i \mathop = 1}^n \size {x_i}$
$\map P 1$ is true by definition of the [[Definition:Usual Ordering|usual ordering on re... | Triangle Inequality/Real Numbers/General Result | https://proofwiki.org/wiki/Triangle_Inequality/Real_Numbers/General_Result | https://proofwiki.org/wiki/Triangle_Inequality/Real_Numbers/General_Result | [
"Triangle Inequality"
] | [
"Definition:Real Number",
"Definition:Absolute Value"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Usual Ordering",
"Principle of Mathematical Induction"
] |
proofwiki-20820 | Chebyshev's Sum Inequality/Discrete | Let $a_1, a_2, \ldots, a_n$ be real numbers such that:
:$a_1 \ge a_2 \ge \cdots \ge a_n$
Let $b_1, b_2, \ldots, b_n$ be real numbers such that:
:$b_1 \ge b_2 \ge \cdots \ge b_n$
Then:
:$\ds \dfrac 1 n \sum_{k \mathop = 1}^n a_k b_k \ge \paren {\dfrac 1 n \sum_{k \mathop = 1}^n a_k} \paren {\dfrac 1 n \sum_{k \mathop = ... | We have {{hypothesis}} that the sequences $\sequence {a_k}$ and $\sequence {b_k}$ are both decreasing.
For $j, k \in \set {1, 2, \ldots, n}$, consider:
:$\paren {a_j - a_k} \paren {b_j - b_k}$
Therefore $a_j - a_k$ and $b_j - b_k$ have the same sign for all $j, k \in \set {1, 2, \ldots, n}$.
So:
:$\forall j, k \in \set... | Let $a_1, a_2, \ldots, a_n$ be [[Definition:Real Number|real numbers]] such that:
:$a_1 \ge a_2 \ge \cdots \ge a_n$
Let $b_1, b_2, \ldots, b_n$ be [[Definition:Real Number|real numbers]] such that:
:$b_1 \ge b_2 \ge \cdots \ge b_n$
Then:
:$\ds \dfrac 1 n \sum_{k \mathop = 1}^n a_k b_k \ge \paren {\dfrac 1 n \sum_{k \... | We have {{hypothesis}} that the [[Definition:Sequence|sequences]] $\sequence {a_k}$ and $\sequence {b_k}$ are both [[Definition:Decreasing Sequence|decreasing]].
For $j, k \in \set {1, 2, \ldots, n}$, consider:
:$\paren {a_j - a_k} \paren {b_j - b_k}$
Therefore $a_j - a_k$ and $b_j - b_k$ have the same [[Definition:... | Chebyshev's Sum Inequality/Discrete/Proof | https://proofwiki.org/wiki/Chebyshev's_Sum_Inequality/Discrete | https://proofwiki.org/wiki/Chebyshev's_Sum_Inequality/Discrete/Proof | [
"Chebyshev's Sum Inequality"
] | [
"Definition:Real Number",
"Definition:Real Number"
] | [
"Definition:Sequence",
"Definition:Decreasing/Sequence",
"Definition:Sign of Number",
"General Distributivity Theorem"
] |
proofwiki-20821 | Positive Real Numbers whose Reciprocals Sum to 1 | Let $p, q \in \R_{\ge 0}$ be positive real numbers such that:
:$\dfrac 1 p + \dfrac 1 q = 1$
Then $p > 1$ and $q > 1$. | From Division by Zero it is immediate that $p > 0$ and $q > 0$.
{{AimForCont}} either $0 < p \le 1$ or $0 < q \le 1$.
{{WLOG}}, suppose $0 < p \le 1$.
Note we have that:
:$\dfrac 1 q = 1 - \dfrac 1 p$
First suppose $p = 1$.
Then:
:$\dfrac 1 q = 0$
But there exists no $q \in \R$ such that $\dfrac 1 q = 0$.
Hence it cann... | Let $p, q \in \R_{\ge 0}$ be [[Definition:Positive Real Number|positive real numbers]] such that:
:$\dfrac 1 p + \dfrac 1 q = 1$
Then $p > 1$ and $q > 1$. | From [[Division by Zero]] it is immediate that $p > 0$ and $q > 0$.
{{AimForCont}} either $0 < p \le 1$ or $0 < q \le 1$.
{{WLOG}}, suppose $0 < p \le 1$.
Note we have that:
:$\dfrac 1 q = 1 - \dfrac 1 p$
First suppose $p = 1$.
Then:
:$\dfrac 1 q = 0$
But there exists no $q \in \R$ such that $\dfrac 1 q = 0$.
He... | Positive Real Numbers whose Reciprocals Sum to 1 | https://proofwiki.org/wiki/Positive_Real_Numbers_whose_Reciprocals_Sum_to_1 | https://proofwiki.org/wiki/Positive_Real_Numbers_whose_Reciprocals_Sum_to_1 | [
"Real Analysis"
] | [
"Definition:Positive/Real Number"
] | [
"Division by Zero",
"Definition:Contradiction",
"Proof by Contradiction",
"Definition:Contradiction",
"Category:Real Analysis"
] |
proofwiki-20822 | Hölder's Inequality for Sums/Formulation 1/Equality | :$\norm {\mathbf x \mathbf y}_1 = \norm {\mathbf x}_p \norm {\mathbf y}_q$
{{iff}}:
:$\forall k \in \N: \size {y_k} = c \size {x_k}^{p - 1}$
for some real constant $c$. | {{ProofWanted}}
Category:Hölder's Inequality for Sums
49yp9senu7jy68h3ybyua5hibcz8xpc | :$\norm {\mathbf x \mathbf y}_1 = \norm {\mathbf x}_p \norm {\mathbf y}_q$
{{iff}}:
:$\forall k \in \N: \size {y_k} = c \size {x_k}^{p - 1}$
for some [[Definition:Real Number|real]] [[Definition:Constant|constant]] $c$. | {{ProofWanted}}
[[Category:Hölder's Inequality for Sums]]
49yp9senu7jy68h3ybyua5hibcz8xpc | Hölder's Inequality for Sums/Formulation 1/Equality | https://proofwiki.org/wiki/Hölder's_Inequality_for_Sums/Formulation_1/Equality | https://proofwiki.org/wiki/Hölder's_Inequality_for_Sums/Formulation_1/Equality | [
"Hölder's Inequality for Sums"
] | [
"Definition:Real Number",
"Definition:Constant"
] | [
"Category:Hölder's Inequality for Sums"
] |
proofwiki-20823 | Plancherel Theorem | For all $f \in \map {L^1} {\R^n} \cap \map {L^2} {\R^n}$, we have:
:$\norm {\map \FF f}_2 = \norm {\map {\FF ^{-1} } f}_2 = \norm f_2$
where:
:$\map \FF f$ is the Fourier transform of $f$
:$\map {\FF ^{-1} } f$ is the inverse Fourier transform of $f$
:$\norm \cdot_2$ denotes the $L^2$ norm | {{ProofWanted}}
{{Namedfor|Michel Plancherel}} | For all $f \in \map {L^1} {\R^n} \cap \map {L^2} {\R^n}$, we have:
:$\norm {\map \FF f}_2 = \norm {\map {\FF ^{-1} } f}_2 = \norm f_2$
where:
:$\map \FF f$ is the [[Definition:Fourier Transform|Fourier transform]] of $f$
:$\map {\FF ^{-1} } f$ is the [[Definition:Inverse Fourier Transform|inverse Fourier transform]] of... | {{ProofWanted}}
{{Namedfor|Michel Plancherel}} | Plancherel Theorem | https://proofwiki.org/wiki/Plancherel_Theorem | https://proofwiki.org/wiki/Plancherel_Theorem | [] | [
"Definition:Fourier Transform",
"Definition:Inverse Fourier Transform",
"Definition:Lp Norm"
] | [] |
proofwiki-20824 | Reverse Triangle Inequality/Real and Complex Fields/Corollary 3 | Let $x$ and $y$ be elements of either the real numbers $\R$ or the complex numbers $\C$.
Then:
:$\size {x + y} \ge \size {\size x - \size y}$
where $\size x$ denotes either the absolute value of a real number or the complex modulus of a complex number. | Let $x$ and $y$ be elements of either the real numbers $\R$ or the complex numbers $\C$.
Let $z := -y$.
Then we have:
{{begin-eqn}}
{{eqn | l = \size {x - z}
| o = \ge
| r = \size {\size x - \size z}
| c = Reverse Triangle Inequality for Real and Complex Fields
}}
{{eqn | l = \size {x - \paren {-y} }
... | Let $x$ and $y$ be [[Definition:Element|elements]] of either the [[Definition:Real Number|real numbers]] $\R$ or the [[Definition:Complex Number|complex numbers]] $\C$.
Then:
:$\size {x + y} \ge \size {\size x - \size y}$
where $\size x$ denotes either the [[Definition:Absolute Value|absolute value]] of a [[Definition... | Let $x$ and $y$ be [[Definition:Element|elements]] of either the [[Definition:Real Number|real numbers]] $\R$ or the [[Definition:Complex Number|complex numbers]] $\C$.
Let $z := -y$.
Then we have:
{{begin-eqn}}
{{eqn | l = \size {x - z}
| o = \ge
| r = \size {\size x - \size z}
| c = [[Reverse Tr... | Reverse Triangle Inequality/Real and Complex Fields/Corollary 3 | https://proofwiki.org/wiki/Reverse_Triangle_Inequality/Real_and_Complex_Fields/Corollary_3 | https://proofwiki.org/wiki/Reverse_Triangle_Inequality/Real_and_Complex_Fields/Corollary_3 | [
"Triangle Inequality"
] | [
"Definition:Element",
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Absolute Value",
"Definition:Real Number",
"Definition:Complex Modulus",
"Definition:Complex Number"
] | [
"Definition:Element",
"Definition:Real Number",
"Definition:Complex Number",
"Reverse Triangle Inequality/Real and Complex Fields"
] |
proofwiki-20825 | T3 Space with Sigma-Locally Finite Basis is Perfectly T4 Space | Let $T = \struct {S, \tau}$ be a $T_3$ topological space.
Let $\BB$ be a $\sigma$-locally finite basis.
Then:
:$T$ is a perfectly $T_4$ space | From T3 Space with Sigma-Locally Finite Basis is T4 Space:
:$T$ is a $T_4$ space
It remains to show that every closed set in $T$ is a $G_\delta$ set
Let $\BB = \ds \bigcup_{n \mathop \in \N} \BB_n$ be a $\sigma$-locally finite basis where $\BB_n$ is a locally finite set of subsets for each $n \in \N$.
Let $F$ be closed... | Let $T = \struct {S, \tau}$ be a [[Definition:T3 Space|$T_3$ topological space]].
Let $\BB$ be a [[Definition:Sigma-Locally Finite Basis|$\sigma$-locally finite basis]].
Then:
:$T$ is a [[Definition:Perfectly T4 Space|perfectly $T_4$ space]] | From [[T3 Space with Sigma-Locally Finite Basis is T4 Space]]:
:$T$ is a [[Definition:T4 Space|$T_4$ space]]
It remains to show that every [[Definition:Closed Set (Topology)|closed set]] in $T$ is a [[Definition:G-Delta Set|$G_\delta$ set]]
Let $\BB = \ds \bigcup_{n \mathop \in \N} \BB_n$ be a [[Definition:Sigma-Loc... | T3 Space with Sigma-Locally Finite Basis is Perfectly T4 Space | https://proofwiki.org/wiki/T3_Space_with_Sigma-Locally_Finite_Basis_is_Perfectly_T4_Space | https://proofwiki.org/wiki/T3_Space_with_Sigma-Locally_Finite_Basis_is_Perfectly_T4_Space | [
"T3 Spaces",
"Perfectly T4 Spaces",
"T3 Space with Sigma-Locally Finite Basis is Perfectly T4 Space"
] | [
"Definition:T3 Space",
"Definition:Sigma-Locally Finite Basis",
"Definition:Perfectly T4 Space"
] | [
"T3 Space with Sigma-Locally Finite Basis is T4 Space",
"Definition:T4 Space",
"Definition:Closed Set/Topology",
"Definition:G-Delta Set",
"Definition:Sigma-Locally Finite Basis",
"Definition:Locally Finite Set of Subsets",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definit... |
proofwiki-20826 | Localization of Module Homomorphism is Multiplicative | Let $A$ be a commutative ring with unity.
Let $S \subseteq A$ be a multiplicatively closed subset.
Let $S^{-1}A$ be the localization of $A$ at $S$.
Let $f_1$ and $f_2$ be $A$-homomorphisms:
:$M_1 \stackrel {f_1} \longrightarrow M_2 \stackrel {f_2} \longrightarrow M_3$
For $i=1,2,3$, let $\struct { S^{-1}M_i, \iota_i}$ ... | Let $g : S^{-1}M_1 \to S^{-1}M_3$ be defined by:
:$g:= \paren {S^{-1} f_2} \circ \paren {S^{-1} f_1}$
Then:
{{begin-eqn}}
{{eqn | l = g \circ \iota_1
| r = \paren {S^{-1}f_2} \circ \paren {S^{-1}f_1 } \circ \iota_1
}}
{{eqn | r = \paren {S^{-1}f_2} \circ \iota_2 \circ f_1
| c = {{Defof|Localization of Modul... | Let $A$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $S \subseteq A$ be a [[Definition:Multiplicatively Closed Subset of Ring|multiplicatively closed subset]].
Let $S^{-1}A$ be the [[Definition:Localization of Ring|localization]] of $A$ at $S$.
Let $f_1$ and $f_2$ be $A$-[[Definit... | Let $g : S^{-1}M_1 \to S^{-1}M_3$ be defined by:
:$g:= \paren {S^{-1} f_2} \circ \paren {S^{-1} f_1}$
Then:
{{begin-eqn}}
{{eqn | l = g \circ \iota_1
| r = \paren {S^{-1}f_2} \circ \paren {S^{-1}f_1 } \circ \iota_1
}}
{{eqn | r = \paren {S^{-1}f_2} \circ \iota_2 \circ f_1
| c = {{Defof|Localization of Modu... | Localization of Module Homomorphism is Multiplicative | https://proofwiki.org/wiki/Localization_of_Module_Homomorphism_is_Multiplicative | https://proofwiki.org/wiki/Localization_of_Module_Homomorphism_is_Multiplicative | [
"Commutative Algebra"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Multiplicatively Closed Subset of Ring",
"Definition:Localization of Ring",
"Definition:Linear Transformation",
"Definition:Localization of Module",
"Definition:Linear Transformation",
"Definition:Linear Transformation"
] | [] |
proofwiki-20827 | Matrix Equation of Plane Rotation | Let $r_\alpha$ be a rotation of the plane about the origin through an angle of $\alpha$.
Let $r_\alpha$ rotate an arbitrary point in the plane $P = \tuple {x, y}$ onto $P' = \tuple {x', y'}$
Then the rotation matrix representing $r_\alpha$ can be presented as:
:$\begin {bmatrix} x' \\ y' \end {bmatrix} = \begin {bmatri... | Let the coordinates of $P'$ be encoded as the elements of a $2 \times 1$ matrix.
We have:
{{begin-eqn}}
{{eqn | l = \begin {bmatrix} x' \\ y' \end {bmatrix}
| r = P'
}}
{{eqn | r = \map {r_\alpha} P
| c = {{Defof|Plane Rotation}}
}}
{{eqn | r = \begin {bmatrix} x \cos \alpha - y \sin \alpha \\ x \sin \alpha... | Let $r_\alpha$ be a [[Definition:Plane Rotation|rotation]] of [[Definition:The Plane|the plane]] about the [[Definition:Origin|origin]] through an [[Definition:Angle|angle]] of $\alpha$.
Let $r_\alpha$ [[Definition:Plane Rotation|rotate]] an arbitrary [[Definition:Point|point]] in [[Definition:The Plane|the plane]] $P... | Let the [[Definition:Coordinate of Vector|coordinates]] of $P'$ be encoded as the [[Definition:Element of Matrix|elements]] of a $2 \times 1$ [[Definition:Matrix|matrix]].
We have:
{{begin-eqn}}
{{eqn | l = \begin {bmatrix} x' \\ y' \end {bmatrix}
| r = P'
}}
{{eqn | r = \map {r_\alpha} P
| c = {{Defof|Pla... | Matrix Equation of Plane Rotation | https://proofwiki.org/wiki/Matrix_Equation_of_Plane_Rotation | https://proofwiki.org/wiki/Matrix_Equation_of_Plane_Rotation | [
"Geometric Rotations",
"Rotation Matrices",
"Equations defining Plane Rotation"
] | [
"Definition:Rotation (Geometry)/Plane",
"Definition:Plane Surface/The Plane",
"Definition:Coordinate System/Origin",
"Definition:Angle",
"Definition:Rotation (Geometry)/Plane",
"Definition:Point",
"Definition:Plane Surface/The Plane",
"Definition:Rotation Matrix"
] | [
"Definition:Coordinate System/Coordinate",
"Definition:Matrix/Element",
"Definition:Matrix",
"Equations defining Plane Rotation/Cartesian"
] |
proofwiki-20828 | Inverse of Nonsingular 2 x 2 Real Square Matrix | Let $\mathbf A$ be an nonsingular $2 \times 2$ real square matrix defined as:
:$\mathbf A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$
Then its inverse matrix $\mathbf A^{-1}$ is:
:$\mathbf A^{-1} = \dfrac 1 {\map \det {\mathbf A} } \begin {pmatrix} d & -b \\ -c & a \end {pmatrix} = \dfrac 1 {a d - b c} \begin {pmat... | === Case: $a \ne 0$ ===
We construct $\begin {pmatrix} \mathbf A & \mathbf I \end {pmatrix}$:
:$\begin {pmatrix} \mathbf A & \mathbf I \end {pmatrix} = \paren {\begin {array} {cc|cc} a & b & 1 & 0 \\ c & d & 0 & 1 \\ \end {array} }$
In the following, $\sequence {e_n}_{n \mathop \ge 1}$ denotes the sequence of elementar... | Let $\mathbf A$ be an [[Definition:Nonsingular Matrix|nonsingular]] [[Square Matrix/Examples/Real 2 x 2|$2 \times 2$ real square matrix]] defined as:
:$\mathbf A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$
Then its [[Definition:Inverse Matrix|inverse matrix]] $\mathbf A^{-1}$ is:
:$\mathbf A^{-1} = \dfrac 1 {\m... | === Case: $a \ne 0$ ===
We construct $\begin {pmatrix} \mathbf A & \mathbf I \end {pmatrix}$:
:$\begin {pmatrix} \mathbf A & \mathbf I \end {pmatrix} = \paren {\begin {array} {cc|cc} a & b & 1 & 0 \\ c & d & 0 & 1 \\ \end {array} }$
In the following, $\sequence {e_n}_{n \mathop \ge 1}$ denotes the [[Definition:Sequ... | Inverse of Nonsingular 2 x 2 Real Square Matrix | https://proofwiki.org/wiki/Inverse_of_Nonsingular_2_x_2_Real_Square_Matrix | https://proofwiki.org/wiki/Inverse_of_Nonsingular_2_x_2_Real_Square_Matrix | [
"Matrix Inverse Algorithm",
"Inverse Matrices"
] | [
"Definition:Nonsingular Matrix",
"Square Matrix/Examples/Real 2 x 2",
"Definition:Inverse Matrix"
] | [
"Definition:Sequence",
"Definition:Elementary Operation/Row",
"Definition:Matrix",
"Definition:Echelon Matrix/Reduced Echelon Form",
"Matrix Inverse Algorithm",
"Determinant/Examples/Order 2"
] |
proofwiki-20829 | Element of Localization of Module is Represented as Quotient over S | Let $A$ be a commutative ring with unity.
Let $S \subseteq A$ be a multiplicatively closed subset.
Let $S^{-1}A$ be the localization of $A$ at $S$.
Let $M$ be a $A$-module.
Let $\struct { S^{-1}M, \iota}$ be the localization of $M$ at $S$.
Let $x \in S^{-1}M$.
Then there exist $m \in M$ and $s \in S$ such that:
:$x = s... | Recall that $S \subseteq \paren {S^{-1}A}^\times$ by {{Defof|Localization of Ring}}.
{{ProofWanted}}
Category:Commutative Algebra
bxdbbuto9481clltm3bz01s8h4li7pt | Let $A$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $S \subseteq A$ be a [[Definition:Multiplicatively Closed Subset of Ring|multiplicatively closed subset]].
Let $S^{-1}A$ be the [[Definition:Localization of Ring|localization]] of $A$ at $S$.
Let $M$ be a [[Definition:Module ove... | Recall that $S \subseteq \paren {S^{-1}A}^\times$ by {{Defof|Localization of Ring}}.
{{ProofWanted}}
[[Category:Commutative Algebra]]
bxdbbuto9481clltm3bz01s8h4li7pt | Element of Localization of Module is Represented as Quotient over S | https://proofwiki.org/wiki/Element_of_Localization_of_Module_is_Represented_as_Quotient_over_S | https://proofwiki.org/wiki/Element_of_Localization_of_Module_is_Represented_as_Quotient_over_S | [
"Commutative Algebra"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Multiplicatively Closed Subset of Ring",
"Definition:Localization of Ring",
"Definition:Module over Ring",
"Definition:Localization of Module"
] | [
"Category:Commutative Algebra"
] |
proofwiki-20830 | Evaluation Linear Transformation on Normed Vector Space is Weak to Weak-* Continuous Embedding into Second Normed Dual | Let $\Bbb F \in \set {\R, \C}$.
Let $X$ be a normed vector space over $\Bbb F$.
Let $X^\ast$ be the normed dual of $X$.
Let $X^{\ast \ast}$ be the second normed dual of $X$.
Let $w$ be the weak topology on $X$.
Let $w^\ast$ be the weak-$\ast$ topology on $X^{\ast \ast}$.
Let $\iota : X \to X^{\ast \ast}$ be the eval... | From the definition of the initial topology, $w^\ast$ is generated by the mappings $f^\wedge : X^{\ast \ast} \to \Bbb F$ defined by:
:$\map {f^{\wedge} } \Phi = \map \Phi f$
for each $f \in X^\ast$ and $\Phi \in X^{\ast \ast}$.
From Continuity in Initial Topology, it therefore suffices to verify that $f^\wedge \circ \... | Let $\Bbb F \in \set {\R, \C}$.
Let $X$ be a [[Definition:Normed Vector Space|normed vector space]] over $\Bbb F$.
Let $X^\ast$ be the [[Definition:Normed Dual Space|normed dual]] of $X$.
Let $X^{\ast \ast}$ be the [[Definition:Second Normed Dual|second normed dual]] of $X$.
Let $w$ be the [[Definition:Weak Topo... | From the definition of the [[Definition:Initial Topology|initial topology]], $w^\ast$ is [[Definition:Initial Topology|generated]] by the [[Definition:Mapping|mappings]] $f^\wedge : X^{\ast \ast} \to \Bbb F$ defined by:
:$\map {f^{\wedge} } \Phi = \map \Phi f$
for each $f \in X^\ast$ and $\Phi \in X^{\ast \ast}$.
F... | Evaluation Linear Transformation on Normed Vector Space is Weak to Weak-* Continuous Embedding into Second Normed Dual | https://proofwiki.org/wiki/Evaluation_Linear_Transformation_on_Normed_Vector_Space_is_Weak_to_Weak-*_Continuous_Embedding_into_Second_Normed_Dual | https://proofwiki.org/wiki/Evaluation_Linear_Transformation_on_Normed_Vector_Space_is_Weak_to_Weak-*_Continuous_Embedding_into_Second_Normed_Dual | [
"Weak Topologies on Topological Vector Spaces",
"Weak-* Topologies",
"Evaluation Linear Transformations (Normed Vector Spaces)"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Second Normed Dual",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Weak-* Topology",
"Definition:Evaluation Linear Transformation/Normed Vector Space",
"Definition:Continuous Mapping"
] | [
"Definition:Initial Topology",
"Definition:Initial Topology",
"Definition:Mapping",
"Continuity in Initial Topology",
"Definition:Continuous Mapping",
"Characterization of Continuity of Linear Functional in Weak Topology",
"Category:Weak Topologies on Topological Vector Spaces",
"Category:Weak-* Topol... |
proofwiki-20831 | Determinant of Plane Rotation Matrix | The matrix associated with a rotation of the plane has a determinant of $1$. | From Matrix Equation of Plane Rotation, we have:
{{begin-eqn}}
{{eqn | l = \begin {vmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end {vmatrix}
| r = \map \cos \alpha \map \cos \alpha - \paren {-\map \sin \alpha} \map \sin \alpha
| c = Determinant of Order 2
}}
{{eqn | l =
| r = \co... | The [[Definition:Matrix|matrix]] associated with a [[Definition:Plane Rotation|rotation]] of [[Definition:The Plane|the plane]] has a [[Definition:Determinant|determinant]] of $1$. | From [[Matrix Equation of Plane Rotation]], we have:
{{begin-eqn}}
{{eqn | l = \begin {vmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end {vmatrix}
| r = \map \cos \alpha \map \cos \alpha - \paren {-\map \sin \alpha} \map \sin \alpha
| c = [[Determinant of Order 2]]
}}
{{eqn | l =
... | Determinant of Plane Rotation Matrix | https://proofwiki.org/wiki/Determinant_of_Plane_Rotation_Matrix | https://proofwiki.org/wiki/Determinant_of_Plane_Rotation_Matrix | [
"Determinants"
] | [
"Definition:Matrix",
"Definition:Rotation (Geometry)/Plane",
"Definition:Plane Surface/The Plane",
"Definition:Determinant"
] | [
"Matrix Equation of Plane Rotation",
"Determinant/Examples/Order 2",
"Sum of Squares of Sine and Cosine",
"Category:Determinants"
] |
proofwiki-20832 | T3 Space with Sigma-Locally Finite Basis is Perfectly T4 Space/Lemma 2 | :$\CC$ is a is a cover of $G$ | Let $x \in G$.
From Characterization of T3 Space:
:$\exists U \in \tau : x \in U : U^- \subseteq G$
where $U^-$ denotes the closure of $U$ in $T$.
By definition of basis:
:$\exists B \in \BB : x \in B : B \subseteq U$
From Topological Closure of Subset is Subset of Topological Closure:
:$B^- \subseteq U^-$
From Subset ... | :$\CC$ is a is a [[Definition:Cover of Set|cover]] of $G$ | Let $x \in G$.
From [[Characterization of T3 Space]]:
:$\exists U \in \tau : x \in U : U^- \subseteq G$
where $U^-$ denotes the [[Definition:Closure (Topology)|closure]] of $U$ in $T$.
By definition of [[Definition:Analytic Basis|basis]]:
:$\exists B \in \BB : x \in B : B \subseteq U$
From [[Topological Closure o... | T3 Space with Sigma-Locally Finite Basis is Perfectly T4 Space/Lemma 2 | https://proofwiki.org/wiki/T3_Space_with_Sigma-Locally_Finite_Basis_is_Perfectly_T4_Space/Lemma_2 | https://proofwiki.org/wiki/T3_Space_with_Sigma-Locally_Finite_Basis_is_Perfectly_T4_Space/Lemma_2 | [
"T3 Space with Sigma-Locally Finite Basis is Perfectly T4 Space"
] | [
"Definition:Cover of Set"
] | [
"Characterization of T3 Space",
"Definition:Closure (Topology)",
"Definition:Basis (Topology)/Analytic Basis",
"Topological Closure of Subset is Subset of Topological Closure",
"Subset Relation is Transitive",
"Definition:Cover of Set",
"Category:T3 Space with Sigma-Locally Finite Basis is Perfectly T4 ... |
proofwiki-20833 | T3 Space with Sigma-Locally Finite Basis is Perfectly T4 Space/Lemma 1 | :$G$ is an $F_\sigma$ set | Let:
:$\CC = \set{B \in \BB : B^- \subseteq G}$
where $B^-$ denotes the closure of $B$ in $T$. | :$G$ is an [[Definition:F-Sigma Set|$F_\sigma$ set]] | Let:
:$\CC = \set{B \in \BB : B^- \subseteq G}$
where $B^-$ denotes the [[Definition:Closure (Topology)|closure]] of $B$ in $T$. | T3 Space with Sigma-Locally Finite Basis is Perfectly T4 Space/Lemma 1 | https://proofwiki.org/wiki/T3_Space_with_Sigma-Locally_Finite_Basis_is_Perfectly_T4_Space/Lemma_1 | https://proofwiki.org/wiki/T3_Space_with_Sigma-Locally_Finite_Basis_is_Perfectly_T4_Space/Lemma_1 | [
"T3 Space with Sigma-Locally Finite Basis is Perfectly T4 Space"
] | [
"Definition:F-Sigma Set"
] | [
"Definition:Closure (Topology)"
] |
proofwiki-20834 | Regular Space with Sigma-Locally Finite Basis is Perfectly Normal | Let $T = \struct {S, \tau}$ be a regular topological space.
Let $\BB$ be a $\sigma$-locally finite basis.
Then:
:$T$ is a perfectly normal space. | {{Recall|Normal Space|normal space|index = 1}}
{{:Definition:Normal Space/Definition 1}}
{{Recall|Regular Space|regular space|index = 1}}
{{:Definition:Regular Space/Definition 2}}
From $T_3$ Space with $\sigma$-Locally Finite Basis is Perfectly $T_4$ Space:
:$T$ is a perfectly $T_4$ space
Hence $T$ is a perfectly norm... | Let $T = \struct {S, \tau}$ be a [[Definition:Regular Space|regular topological space]].
Let $\BB$ be a [[Definition:Sigma-Locally Finite Basis|$\sigma$-locally finite basis]].
Then:
:$T$ is a [[Definition:Perfectly Normal Space|perfectly normal space]]. | {{Recall|Normal Space|normal space|index = 1}}
{{:Definition:Normal Space/Definition 1}}
{{Recall|Regular Space|regular space|index = 1}}
{{:Definition:Regular Space/Definition 2}}
From [[T3 Space with Sigma-Locally Finite Basis is Perfectly T4 Space|$T_3$ Space with $\sigma$-Locally Finite Basis is Perfectly $T_4$ S... | Regular Space with Sigma-Locally Finite Basis is Perfectly Normal | https://proofwiki.org/wiki/Regular_Space_with_Sigma-Locally_Finite_Basis_is_Perfectly_Normal | https://proofwiki.org/wiki/Regular_Space_with_Sigma-Locally_Finite_Basis_is_Perfectly_Normal | [
"Regular Spaces",
"Sigma-Locally Finite Bases",
"Perfectly Normal Spaces"
] | [
"Definition:Regular Space",
"Definition:Sigma-Locally Finite Basis",
"Definition:Perfectly Normal Space"
] | [
"T3 Space with Sigma-Locally Finite Basis is Perfectly T4 Space",
"Definition:Perfectly T4 Space",
"Definition:Perfectly Normal Space",
"Category:Regular Spaces",
"Category:Sigma-Locally Finite Bases",
"Category:Perfectly Normal Spaces"
] |
proofwiki-20835 | Neighborhood of Diagonal induces Open Cover | :$\VV$ is an open cover of $T$ | By definition of product topology:
:$\BB = \set {V_1 \times V_2: V_1, V_2 \in \tau}$ is a basis on $T \times T$
Let $x \in X$.
From Set is Neighborhood of Subset iff Neighborhood of all Points of Subset:
:$U$ is a neighborhood of the point $\tuple{x,x}$
From Characterization of Neighborhood by Basis:
:$\exists V_1, V_2... | :$\VV$ is an [[Definition:Open Cover|open cover]] of $T$ | By definition of [[Definition:Product Topology on Two Factor Spaces|product topology]]:
:$\BB = \set {V_1 \times V_2: V_1, V_2 \in \tau}$ is a [[Definition:Analytic Basis|basis]] on $T \times T$
Let $x \in X$.
From [[Set is Neighborhood of Subset iff Neighborhood of all Points of Subset]]:
:$U$ is a [[Definition:Nei... | Neighborhood of Diagonal induces Open Cover | https://proofwiki.org/wiki/Neighborhood_of_Diagonal_induces_Open_Cover | https://proofwiki.org/wiki/Neighborhood_of_Diagonal_induces_Open_Cover | [
"Product Spaces",
"Neighborhoods",
"Covers"
] | [
"Definition:Open Cover"
] | [
"Definition:Product Topology/Two Factor Spaces",
"Definition:Basis (Topology)/Analytic Basis",
"Set is Neighborhood of Subset iff Neighborhood of all Points of Subset",
"Definition:Neighborhood (Topology)/Point",
"Definition:Element",
"Characterization of Neighborhood by Basis",
"Definition:Cartesian Pr... |
proofwiki-20836 | Characterization of Neighborhood by Basis | Let $\struct {S, \tau}$ be a topological space.
Let $\BB$ be an analytic basis for $\tau$.
Let $N \subseteq S$.
Let $x \in N$.
Then $N$ is a neighborhood at $x$ {{iff}}:
:$\exists B \in \BB : x \in B : B \subseteq N$ | From Basis induces Local Basis:
:$\BB_x = \set {B \in \BB: x \in B}$ is a local basis at $x$
By definition of local basis, $N$ is a neighborhood at $x$ {{iff}}:
:$\exists B \in \BB_x : B \subseteq N$
By definition of $\BB_x$, $N$ is a neighborhood at $x$ {{iff}}:
:$\exists B \in \BB : x \in B : B \subseteq N$
{{qed}}
... | Let $\struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\BB$ be an [[Definition:Analytic Basis|analytic basis]] for $\tau$.
Let $N \subseteq S$.
Let $x \in N$.
Then $N$ is a [[Definition:Neighborhood (Topology)|neighborhood]] at $x$ {{iff}}:
:$\exists B \in \BB : x \in B : B \subseteq... | From [[Basis induces Local Basis]]:
:$\BB_x = \set {B \in \BB: x \in B}$ is a [[Definition:Local Basis|local basis]] at $x$
By definition of [[Definition:Local Basis|local basis]], $N$ is a [[Definition:Neighborhood (Topology)|neighborhood]] at $x$ {{iff}}:
:$\exists B \in \BB_x : B \subseteq N$
By definition of $\BB... | Characterization of Neighborhood by Basis | https://proofwiki.org/wiki/Characterization_of_Neighborhood_by_Basis | https://proofwiki.org/wiki/Characterization_of_Neighborhood_by_Basis | [
"Neighborhoods"
] | [
"Definition:Topological Space",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Neighborhood (Topology)"
] | [
"Basis induces Local Basis",
"Definition:Local Basis",
"Definition:Local Basis",
"Definition:Neighborhood (Topology)",
"Definition:Neighborhood (Topology)",
"Category:Neighborhoods"
] |
proofwiki-20837 | Inverse of Plane Rotation Matrix | Let $\mathbf R$ be the matrix associated with a rotation of the plane about the origin through an angle of $\alpha$:
:<nowiki>$\mathbf R = \begin{bmatrix}
\cos \alpha & -\sin \alpha \\
\sin \alpha & \cos \alpha
\end{bmatrix}$</nowiki>
Then its inverse matrix $\mathbf R^{-1}$ is:
:<nowiki>$\mathbf R^{-1} = \begin{bmatri... | Let:
:<nowiki>$\mathbf A = \begin{bmatrix}
\cos \alpha & \sin \alpha \\
-\sin \alpha & \cos \alpha
\end{bmatrix}$</nowiki>
Consider $\mathbf R \mathbf A$:
{{begin-eqn}}
{{eqn | l = \mathbf R \mathbf A
| r = <nowiki>\begin{bmatrix}
\cos \alpha & -\sin \alpha \\
\sin \alpha & \cos \alpha
\end{bmatrix} \begin{bmatri... | Let $\mathbf R$ be the [[Definition:Matrix|matrix]] associated with a [[Definition:Plane Rotation|rotation]] of [[Definition:The Plane|the plane]] about the [[Definition:Origin|origin]] through an [[Definition:Angle|angle]] of $\alpha$:
:<nowiki>$\mathbf R = \begin{bmatrix}
\cos \alpha & -\sin \alpha \\
\sin \alpha & ... | Let:
:<nowiki>$\mathbf A = \begin{bmatrix}
\cos \alpha & \sin \alpha \\
-\sin \alpha & \cos \alpha
\end{bmatrix}$</nowiki>
Consider $\mathbf R \mathbf A$:
{{begin-eqn}}
{{eqn | l = \mathbf R \mathbf A
| r = <nowiki>\begin{bmatrix}
\cos \alpha & -\sin \alpha \\
\sin \alpha & \cos \alpha
\end{bmatrix} \begin{bma... | Inverse of Plane Rotation Matrix | https://proofwiki.org/wiki/Inverse_of_Plane_Rotation_Matrix | https://proofwiki.org/wiki/Inverse_of_Plane_Rotation_Matrix | [
"Inverse Matrices",
"Geometric Rotations"
] | [
"Definition:Matrix",
"Definition:Rotation (Geometry)/Plane",
"Definition:Plane Surface/The Plane",
"Definition:Coordinate System/Origin",
"Definition:Angle",
"Definition:Inverse Matrix"
] | [
"Sum of Squares of Sine and Cosine",
"Sum of Squares of Sine and Cosine",
"Definition:Inverse Matrix",
"Definition:Inverse Matrix",
"Category:Inverse Matrices",
"Category:Geometric Rotations"
] |
proofwiki-20838 | Continued Fraction Expansion via Gauss Map | Let $T : \closedint 0 1 \to \closedint 0 1$ be the Gauss map.
Let $x \in \closedint 0 1 \setminus \Q$.
Then $x$ has the simple infinite continued fraction:
{{begin-eqn}}
{{eqn | l = x
| r = \sqbrk {0; \map {a_1} x, \map {a_2} x, \ldots}
}}
{{eqn | r = 0 + \cfrac 1 {\map {a_1} x + \cfrac 1 {\map {a_2} x + \cfrac 1... | For $x \in \closedint 0 1 \setminus \Q$, we have:
:$\map {a_1} x = \floor {\dfrac 1 x}$
and:
:$\forall n \in \N_{>0} : \map {a_n} x = \map {a_1} { T^{n-1} x }$
{{ProofWanted}}
Category:Continued Fractions
hjrdfdd1e6jlws39rj068ieocsek5ge | Let $T : \closedint 0 1 \to \closedint 0 1$ be the [[Definition:Gauss Map|Gauss map]].
Let $x \in \closedint 0 1 \setminus \Q$.
Then $x$ has the [[Definition:Simple Infinite Continued Fraction|simple infinite continued fraction]]:
{{begin-eqn}}
{{eqn | l = x
| r = \sqbrk {0; \map {a_1} x, \map {a_2} x, \ldots}... | For $x \in \closedint 0 1 \setminus \Q$, we have:
:$\map {a_1} x = \floor {\dfrac 1 x}$
and:
:$\forall n \in \N_{>0} : \map {a_n} x = \map {a_1} { T^{n-1} x }$
{{ProofWanted}}
[[Category:Continued Fractions]]
hjrdfdd1e6jlws39rj068ieocsek5ge | Continued Fraction Expansion via Gauss Map | https://proofwiki.org/wiki/Continued_Fraction_Expansion_via_Gauss_Map | https://proofwiki.org/wiki/Continued_Fraction_Expansion_via_Gauss_Map | [
"Continued Fractions"
] | [
"Definition:Gauss Map",
"Definition:Simple Continued Fraction/Infinite",
"Definition:Floor Function"
] | [
"Category:Continued Fractions"
] |
proofwiki-20839 | Composition of Symmetric Relation with Itself is Union of Products of Images | Let $\RR$ be a symmetric relation on a set $S$.
Then:
:$\RR \circ \RR = \ds \map {\bigcup_{s \mathop \in S} } {\map \RR s \times \map \RR s}$
where
:$\RR \circ \RR$ is the composition of $\RR$ with itself
:$\map \RR s$ is the image of $s$ under $\RR$
:$\map \RR s \times \map \RR s$ is the Cartesian product of $\map \RR... | We have:
{{begin-eqn}}
{{eqn | l = \tuple {x, y} \in \RR \circ \RR
| o = \leadstoandfrom
| r = \exists s \in S : \tuple {x, s}, \tuple {s, y} \in \RR
| c = {{Defof|Composite Relation}}
}}
{{eqn | o = \leadstoandfrom
| r = \exists s \in S : \tuple {s, x}, \tuple {s, y} \in \RR
| c = {{Defof... | Let $\RR$ be a [[Definition:Symmetric Relation|symmetric relation]] on a [[Definition:Set|set]] $S$.
Then:
:$\RR \circ \RR = \ds \map {\bigcup_{s \mathop \in S} } {\map \RR s \times \map \RR s}$
where
:$\RR \circ \RR$ is the [[Definition:Composite Relation|composition]] of $\RR$ with itself
:$\map \RR s$ is the [[Def... | We have:
{{begin-eqn}}
{{eqn | l = \tuple {x, y} \in \RR \circ \RR
| o = \leadstoandfrom
| r = \exists s \in S : \tuple {x, s}, \tuple {s, y} \in \RR
| c = {{Defof|Composite Relation}}
}}
{{eqn | o = \leadstoandfrom
| r = \exists s \in S : \tuple {s, x}, \tuple {s, y} \in \RR
| c = {{Defof... | Composition of Symmetric Relation with Itself is Union of Products of Images | https://proofwiki.org/wiki/Composition_of_Symmetric_Relation_with_Itself_is_Union_of_Products_of_Images | https://proofwiki.org/wiki/Composition_of_Symmetric_Relation_with_Itself_is_Union_of_Products_of_Images | [
"Set Union",
"Cartesian Product",
"Symmetric Relations"
] | [
"Definition:Symmetric Relation",
"Definition:Set",
"Definition:Composition of Relations",
"Definition:Image (Set Theory)/Relation/Element",
"Definition:Cartesian Product"
] | [
"Category:Set Union",
"Category:Cartesian Product",
"Category:Symmetric Relations"
] |
proofwiki-20840 | Intersection of Neighborhood of Diagonal with Inverse is Neighborhood | Let $T = \struct{X, \tau}$ be a topological space.
Let $\tau_{X \times X}$ denote the product topology on the cartesian product $X \times X$.
Let $R$ be a neighborhood of the diagonal $\Delta_X$ of $X \times X$ in the product space $\struct{X \times X, \tau_{X \times X}}$.
Let $R^{-1}$ denote the inverse relation of $R... | From Set is Neighborhood of Subset iff Neighborhood of all Points of Subset:
:$\forall \tuple{x, x} \in \Delta_X : R$ is a neighborhood of $\tuple{x, x}$
From Inverse of Neighborhood of Diagonal Point is Neighborhood:
:$\forall \tuple{x, x} \in \Delta_X : R^{-1}$ is a neighborhood of $\tuple{x, x}$
From Intersection of... | Let $T = \struct{X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\tau_{X \times X}$ denote the [[Definition:Product Topology|product topology]] on the [[Definition:Cartesian Product|cartesian product]] $X \times X$.
Let $R$ be a [[Definition:Neighborhood of Set|neighborhood]] of the [[Defini... | From [[Set is Neighborhood of Subset iff Neighborhood of all Points of Subset]]:
:$\forall \tuple{x, x} \in \Delta_X : R$ is a [[Definition:Neighborhood of Point|neighborhood]] of $\tuple{x, x}$
From [[Inverse of Neighborhood of Diagonal Point is Neighborhood]]:
:$\forall \tuple{x, x} \in \Delta_X : R^{-1}$ is a [[De... | Intersection of Neighborhood of Diagonal with Inverse is Neighborhood | https://proofwiki.org/wiki/Intersection_of_Neighborhood_of_Diagonal_with_Inverse_is_Neighborhood | https://proofwiki.org/wiki/Intersection_of_Neighborhood_of_Diagonal_with_Inverse_is_Neighborhood | [
"Neighborhoods",
"Diagonal Relation",
"Relations"
] | [
"Definition:Topological Space",
"Definition:Product Topology",
"Definition:Cartesian Product",
"Definition:Neighborhood (Topology)/Set",
"Definition:Diagonal Relation",
"Definition:Product Space",
"Definition:Inverse Relation",
"Definition:Relation/Relation as Subset of Cartesian Product",
"Definiti... | [
"Set is Neighborhood of Subset iff Neighborhood of all Points of Subset",
"Definition:Neighborhood (Topology)/Point",
"Inverse of Neighborhood of Diagonal Point is Neighborhood",
"Definition:Neighborhood (Topology)/Point",
"Intersection of Neighborhoods in Topological Space is Neighborhood",
"Definition:N... |
proofwiki-20841 | Inverse of Neighborhood of Diagonal Point is Neighborhood | Let $T = \struct{X, \tau}$ be a topological Space.
Let $\tau_{X \times X}$ denote the product topology on the cartesian product $X \times X$.
Let $x \in X$.
Let $R$ be a neighborhood of $\tuple{x,x}$ in the product space $\struct{X \times X, \tau_{X \times X}}$.
Let $R^{-1}$ denote the inverse relation of $R$ where $R$... | By definition of neighborhood:
:$\tuple{x, x} \in R$
By definition of product topology:
:$\BB = \set {V_1 \times V_2: V_1, V_2 \in \tau}$ is a basis for $\tau_{X \times X}$
From Characterization of Neighborhood by Basis:
:$\exists V_1, V_2 \in \tau : \tuple{x, x} \in V_1 \times V_2 : V_1 \times V_2 \subseteq R$
By defi... | Let $T = \struct{X, \tau}$ be a [[Definition:Topological Space|topological Space]].
Let $\tau_{X \times X}$ denote the [[Definition:Product Topology|product topology]] on the [[Definition:Cartesian Product|cartesian product]] $X \times X$.
Let $x \in X$.
Let $R$ be a [[Definition:Neighborhood of Set|neighborhood]]... | By definition of [[Definition:Neighborhood (Topology)|neighborhood]]:
:$\tuple{x, x} \in R$
By definition of [[Definition:Product Topology on Two Factor Spaces|product topology]]:
:$\BB = \set {V_1 \times V_2: V_1, V_2 \in \tau}$ is a [[Definition:Analytic Basis|basis]] for $\tau_{X \times X}$
From [[Characterizati... | Inverse of Neighborhood of Diagonal Point is Neighborhood | https://proofwiki.org/wiki/Inverse_of_Neighborhood_of_Diagonal_Point_is_Neighborhood | https://proofwiki.org/wiki/Inverse_of_Neighborhood_of_Diagonal_Point_is_Neighborhood | [
"Neighborhoods",
"Inverse Relations"
] | [
"Definition:Topological Space",
"Definition:Product Topology",
"Definition:Cartesian Product",
"Definition:Neighborhood (Topology)/Set",
"Definition:Product Space",
"Definition:Inverse Relation",
"Definition:Relation/Relation as Subset of Cartesian Product",
"Definition:Neighborhood (Topology)/Set"
] | [
"Definition:Neighborhood (Topology)",
"Definition:Product Topology/Two Factor Spaces",
"Definition:Basis (Topology)/Analytic Basis",
"Characterization of Neighborhood by Basis",
"Definition:Symmetric Relation",
"Definition:Product Topology/Two Factor Spaces",
"Definition:Neighborhood (Topology)",
"Cat... |
proofwiki-20842 | Cauchy's Inequality/Vector Form | Let $\mathbf a$ and $\mathbf b$ be vectors in a vector space $V$.
Then:
:$\paren {\mathbf a \cdot \mathbf b}^2 \le \paren {\mathbf a \cdot \mathbf a} \paren {\mathbf b \cdot \mathbf b}$
where $\cdot$ denotes dot product. | Let us express $\mathbf a$ and $\mathbf b$ in component form:
{{begin-eqn}}
{{eqn | l = \mathbf a
| o = :=
| r = \sum_{i \mathop = 1}^n a_i \mathbf e_i
}}
{{eqn | l = \mathbf b
| o = :=
| r = \sum_{i \mathop = 1}^n b_i \mathbf e_i
}}
{{end-eqn}}
where the ordered $n$-tuple $\tuple {\mathbf e_1, ... | Let $\mathbf a$ and $\mathbf b$ be [[Definition:Vector|vectors]] in a [[Definition:Vector Space|vector space]] $V$.
Then:
:$\paren {\mathbf a \cdot \mathbf b}^2 \le \paren {\mathbf a \cdot \mathbf a} \paren {\mathbf b \cdot \mathbf b}$
where $\cdot$ denotes [[Definition:Dot Product|dot product]]. | Let us express $\mathbf a$ and $\mathbf b$ in [[Definition:Component of Vector|component form]]:
{{begin-eqn}}
{{eqn | l = \mathbf a
| o = :=
| r = \sum_{i \mathop = 1}^n a_i \mathbf e_i
}}
{{eqn | l = \mathbf b
| o = :=
| r = \sum_{i \mathop = 1}^n b_i \mathbf e_i
}}
{{end-eqn}}
where the [[D... | Cauchy's Inequality/Vector Form | https://proofwiki.org/wiki/Cauchy's_Inequality/Vector_Form | https://proofwiki.org/wiki/Cauchy's_Inequality/Vector_Form | [
"Cauchy's Inequality",
"Vector Analysis"
] | [
"Definition:Vector",
"Definition:Vector Space",
"Definition:Dot Product"
] | [
"Definition:Vector Quantity/Component",
"Definition:Ordered Tuple",
"Definition:Standard Ordered Basis/Vector Space",
"Cauchy's Inequality"
] |
proofwiki-20843 | Hölder's Inequality for Sums/Formulation 1 | Let $\mathbf x$ and $\mathbf y$ denote the vectors consisting of the sequences:
:$\mathbf x = \sequence {x_n} \in {\ell^p}_\GF$
:$\mathbf y = \sequence {y_n} \in {\ell^q}_\GF$
where ${\ell^p}_\GF$ denotes the $p$-sequence space in $\GF$.
Let $\norm {\mathbf x}_p$ denote the $p$-norm of $\mathbf x$.
Then:
:$\mathbf x \m... | Define:
{{begin-eqn}}
{{eqn | l = \mathbf u
| r = \sequence {u_n}
| rr= = \dfrac {\mathbf x} {\norm {\mathbf x}_p}
| c =
}}
{{eqn | l = \mathbf v
| r = \sequence {v_n}
| rr= = \dfrac {\mathbf y} {\norm {\mathbf y}_q}
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | q =
| l =... | Let $\mathbf x$ and $\mathbf y$ denote the [[Definition:Vector|vectors]] consisting of the [[Definition:Sequence|sequences]]:
:$\mathbf x = \sequence {x_n} \in {\ell^p}_\GF$
:$\mathbf y = \sequence {y_n} \in {\ell^q}_\GF$
where ${\ell^p}_\GF$ denotes the [[Definition:P-Sequence Space|$p$-sequence space in $\GF$]].
Le... | Define:
{{begin-eqn}}
{{eqn | l = \mathbf u
| r = \sequence {u_n}
| rr= = \dfrac {\mathbf x} {\norm {\mathbf x}_p}
| c =
}}
{{eqn | l = \mathbf v
| r = \sequence {v_n}
| rr= = \dfrac {\mathbf y} {\norm {\mathbf y}_q}
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | q =
| l... | Hölder's Inequality for Sums/Formulation 1 | https://proofwiki.org/wiki/Hölder's_Inequality_for_Sums/Formulation_1 | https://proofwiki.org/wiki/Hölder's_Inequality_for_Sums/Formulation_1 | [
"Hölder's Inequality for Sums"
] | [
"Definition:Vector",
"Definition:Sequence",
"Definition:P-Sequence Space",
"Definition:P-Norm",
"Definition:P-Norm",
"Definition:Taxicab Norm"
] | [
"Absolute Value Function is Completely Multiplicative",
"Young's Inequality for Products",
"Definition:Strictly Positive/Real Number",
"Comparison Test"
] |
proofwiki-20844 | Hölder's Inequality for Sums/Formulation 2 | Let $\sequence {x_n}_{n \mathop \in \N}$ and $\sequence {y_n}_{n \mathop \in \N}$ be sequences in $\GF$ such that $\ds \sum_{k \mathop \in \N} \size {x_k}^p$ and $\ds \sum_{k \mathop \in \N} \size {y_k}^q$ are convergent.
Then:
:$\ds \sum_{k \mathop \in \N} \size {x_k y_k} \le \paren {\sum_{k \mathop \in \N} \size {x_k... | Let:
{{begin-eqn}}
{{eqn | l = X
| r = \paren {\sum_{k \mathop \in \N} \size {x_k}^p}^{1 / p}
}}
{{eqn | l = Y
| r = \paren {\sum_{k \mathop \in \N} \size {y_k}^q}^{1 / q}
}}
{{end-eqn}}
We note that $X, Y \in \R_{\ge 0}$.
Let:
{{begin-eqn}}
{{eqn | q = \forall k \in \N
| l = u_k
| r = \dfrac {x... | Let $\sequence {x_n}_{n \mathop \in \N}$ and $\sequence {y_n}_{n \mathop \in \N}$ be [[Definition:Sequence|sequences]] in $\GF$ such that $\ds \sum_{k \mathop \in \N} \size {x_k}^p$ and $\ds \sum_{k \mathop \in \N} \size {y_k}^q$ are [[Definition:Convergent Sequence|convergent]].
Then:
:$\ds \sum_{k \mathop \in \N} \... | Let:
{{begin-eqn}}
{{eqn | l = X
| r = \paren {\sum_{k \mathop \in \N} \size {x_k}^p}^{1 / p}
}}
{{eqn | l = Y
| r = \paren {\sum_{k \mathop \in \N} \size {y_k}^q}^{1 / q}
}}
{{end-eqn}}
We note that $X, Y \in \R_{\ge 0}$.
Let:
{{begin-eqn}}
{{eqn | q = \forall k \in \N
| l = u_k
| r = \dfr... | Hölder's Inequality for Sums/Formulation 2 | https://proofwiki.org/wiki/Hölder's_Inequality_for_Sums/Formulation_2 | https://proofwiki.org/wiki/Hölder's_Inequality_for_Sums/Formulation_2 | [
"Hölder's Inequality for Sums"
] | [
"Definition:Sequence",
"Definition:Convergent Sequence"
] | [
"Absolute Value Function is Completely Multiplicative",
"Young's Inequality for Products",
"Definition:Summation",
"Definition:Summation"
] |
proofwiki-20845 | Characterization of Paracompactness in T3 Space/Lemma 13 | :there exists a sequence $\sequence{V_n}_{n \in \N}$ of neighborhoods of the diagonal $\Delta_X$ in $\struct {X \times X, \tau_{X \times X} }$:
::$V_0 = V$
::$\forall n \in \N_{> 0} : V_n$ is symmetric as a relation on $X \times X$
::$\forall n \in \N_{> 0}$ the composite relation $V_n \circ V_n$ is a subset of $V_{n -... | ==== Lemma 20 ====
{{:Characterization of Paracompactness in T3 Space/Lemma 20}}{{qed|lemma}}
The sequence $\sequence{V_n}_{n \in \N}$ is now constructed using the Principle of Recursive Definition and Zermelo's Well-Ordering Theorem.
Let:
:$\NN = \leftset{U \subseteq X \times X : U }$ is a neighborhood of the diagonal... | :there exists a [[Definition:Sequence|sequence]] $\sequence{V_n}_{n \in \N}$ of [[Definition:Neighborhood of Set|neighborhoods]] of the [[Definition:Diagonal Relation|diagonal $\Delta_X$]] in $\struct {X \times X, \tau_{X \times X} }$:
::$V_0 = V$
::$\forall n \in \N_{> 0} : V_n$ is [[Definition:Symmetric Relation|symm... | ==== [[Characterization of Paracompactness in T3 Space/Lemma 20|Lemma 20]] ====
{{:Characterization of Paracompactness in T3 Space/Lemma 20}}{{qed|lemma}}
The [[Definition:Sequence|sequence]] $\sequence{V_n}_{n \in \N}$ is now constructed using the [[Principle of Recursive Definition]] and [[Zermelo's Well-Ordering T... | Characterization of Paracompactness in T3 Space/Lemma 13 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_13 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_13 | [
"Characterization of Paracompactness in T3 Space"
] | [
"Definition:Sequence",
"Definition:Neighborhood (Topology)/Set",
"Definition:Diagonal Relation",
"Definition:Symmetric Relation",
"Definition:Relation/Relation as Subset of Cartesian Product",
"Definition:Composition of Relations",
"Definition:Subset"
] | [
"Characterization of Paracompactness in T3 Space/Lemma 20",
"Definition:Sequence",
"Principle of Recursive Definition",
"Zermelo's Well-Ordering Theorem",
"Definition:Neighborhood (Topology)/Set",
"Definition:Diagonal Relation",
"Zermelo's Well-Ordering Theorem",
"Definition:Well-Ordering",
"Definit... |
proofwiki-20846 | Characterization of Set Equals Union of Sets | Let $A$ be a set.
Let $\BB$ be a set of sets.
Then $A = \ds \bigcup \BB$ {{iff}}:
:$\forall a \in A : \exists B \in \BB : a \in B$
:$\forall B \in \BB : B \subseteq A$ | === Necessary Condition ===
Let $A = \ds \bigcup \BB$.
By definition of set union:
:$\forall a \in A = \ds \bigcup \BB : \exists B \in \BB : a \in B$
From Set is Subset of Union:
:$\forall B \in \BB : B \subseteq \ds \bigcup \BB = A$
{{qed|lemma}} | Let $A$ be a [[Definition:Set|set]].
Let $\BB$ be a [[Definition:Set of Sets|set of sets]].
Then $A = \ds \bigcup \BB$ {{iff}}:
:$\forall a \in A : \exists B \in \BB : a \in B$
:$\forall B \in \BB : B \subseteq A$ | === Necessary Condition ===
Let $A = \ds \bigcup \BB$.
By definition of [[Definition:Set Union|set union]]:
:$\forall a \in A = \ds \bigcup \BB : \exists B \in \BB : a \in B$
From [[Set is Subset of Union]]:
:$\forall B \in \BB : B \subseteq \ds \bigcup \BB = A$
{{qed|lemma}} | Characterization of Set Equals Union of Sets | https://proofwiki.org/wiki/Characterization_of_Set_Equals_Union_of_Sets | https://proofwiki.org/wiki/Characterization_of_Set_Equals_Union_of_Sets | [
"Set Union"
] | [
"Definition:Set",
"Definition:Set of Sets"
] | [
"Definition:Set Union",
"Set is Subset of Union",
"Definition:Set Union"
] |
proofwiki-20847 | Subset of Discrete Set of Subsets is Discrete | Let $T = \struct {S, \tau}$ be a topological space.
Let $\FF \subseteq \powerset S$ be a set of subsets of $S$.
Let $\GG \subseteq \FF$.
If $\FF$ is discrete then $\GG$ is discrete. | We prove the contrapositive statement:
:If $\GG$ is not discrete then $\FF$ is not discrete.
Let $\GG$ not be discrete.
By definition of discrete:
:$\exists x \in S : \forall N \subseteq S : N$ is a neighborhood of $x : \exists X_1, X_2 \in \GG : X_1 \ne X_2: X_1 \cap N \ne \O, X_2 \cap N \ne \O$
By definition of subse... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\FF \subseteq \powerset S$ be a [[Definition:Set of Sets|set of subsets]] of $S$.
Let $\GG \subseteq \FF$.
If $\FF$ is [[Definition:Discrete Set of Subsets|discrete]] then $\GG$ is [[Definition:Discrete Set of Subsets|discret... | We prove the [[Definition:Contrapositive Statement|contrapositive statement]]:
:If $\GG$ is not [[Definition:Discrete Set of Subsets|discrete]] then $\FF$ is not [[Definition:Discrete Set of Subsets|discrete]].
Let $\GG$ not be [[Definition:Discrete Set of Subsets|discrete]].
By definition of [[Definition:Discrete S... | Subset of Discrete Set of Subsets is Discrete | https://proofwiki.org/wiki/Subset_of_Discrete_Set_of_Subsets_is_Discrete | https://proofwiki.org/wiki/Subset_of_Discrete_Set_of_Subsets_is_Discrete | [
"Discrete Sets of Subsets"
] | [
"Definition:Topological Space",
"Definition:Set of Sets",
"Definition:Discrete Set of Subsets",
"Definition:Discrete Set of Subsets"
] | [
"Definition:Contrapositive Statement",
"Definition:Discrete Set of Subsets",
"Definition:Discrete Set of Subsets",
"Definition:Discrete Set of Subsets",
"Definition:Discrete Set of Subsets",
"Definition:Neighborhood (Topology)",
"Definition:Subset",
"Definition:Neighborhood (Topology)",
"Definition:... |
proofwiki-20848 | Double Orthocomplement of Closed Linear Subspace | Let $H$ be a Hilbert space.
Let $A \subseteq H$ be a closed linear subspace of $H$.
Then:
:$\paren {A^\perp}^\perp = A$ | {{MissingLinks|identity operator}}
Let $I : H \to H$ be the identity operator (viz., $I h = h$).
Also, let $P : H \to A$ be the orthogonal projection.
Then $I - P : H \to A^\perp$ is the Orthogonal Projection onto Orthocomplement.
By Kernel of Orthogonal Projection:
:$\map \ker {I - P} = \paren {A^\perp}^\perp$
{{qed|l... | Let $H$ be a [[Definition:Hilbert Space|Hilbert space]].
Let $A \subseteq H$ be a [[Definition:Closed Linear Subspace|closed linear subspace]] of $H$.
Then:
:$\paren {A^\perp}^\perp = A$ | {{MissingLinks|identity operator}}
Let $I : H \to H$ be the identity operator (viz., $I h = h$).
Also, let $P : H \to A$ be the [[Definition:Orthogonal Projection|orthogonal projection]].
Then $I - P : H \to A^\perp$ is the [[Orthogonal Projection onto Orthocomplement]].
By [[Kernel of Orthogonal Projection]]:
:$\m... | Double Orthocomplement of Closed Linear Subspace | https://proofwiki.org/wiki/Double_Orthocomplement_of_Closed_Linear_Subspace | https://proofwiki.org/wiki/Double_Orthocomplement_of_Closed_Linear_Subspace | [
"Hilbert Spaces"
] | [
"Definition:Hilbert Space",
"Definition:Closed Linear Subspace"
] | [
"Definition:Orthogonal Projection",
"Orthogonal Projection onto Orthocomplement",
"Kernel of Orthogonal Projection",
"Orthogonal Projection is Projection",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Range of Orthogonal Projection"
] |
proofwiki-20849 | Löwenheim-Skolem Theorem | Let $\FF$ be a formula which is valid in a countably infinite domain.
Then $\FF$ is universally valid. | {{ProofWanted}}
{{Namedfor|Leopold Löwenheim|name2 = Thoralf Albert Skolem|cat = Löwenheim|cat2 = Skolem}} | Let $\FF$ be a [[Definition:Formula|formula]] which is [[Definition:Valid Formula|valid]] in a [[Definition:Countably Infinite Set|countably infinite]] [[Definition:Domain|domain]].
Then $\FF$ is [[Definition:Universally Valid Formula|universally valid]]. | {{ProofWanted}}
{{Namedfor|Leopold Löwenheim|name2 = Thoralf Albert Skolem|cat = Löwenheim|cat2 = Skolem}} | Löwenheim-Skolem Theorem | https://proofwiki.org/wiki/Löwenheim-Skolem_Theorem | https://proofwiki.org/wiki/Löwenheim-Skolem_Theorem | [
"Model Theory for Predicate Logic"
] | [
"Definition:Formula",
"Definition:Valid Formula",
"Definition:Countably Infinite/Set",
"Definition:Domain",
"Definition:Universally Valid Formula"
] | [] |
proofwiki-20850 | Second Derivative of Inverse Function | Let $f$ be a real function which is of differentiability class $2$.
Let $f$ have an inverse $f^{-1}$, likewise of differentiability class $2$.
Then:
:$\dfrac {\d^2 x} {\d y^2} = -\dfrac {\d^2 y} {\d x^2} \paren {\dfrac {\d y} {\d x} }^{-3}$ | {{begin-eqn}}
{{eqn | l = \dfrac {\d^2 x} {\d y^2}
| r = \map {\dfrac \d {\d y} } {\dfrac {\d x} {\d y} }
| c = {{Defof|Second Derivative}}
}}
{{eqn | r = \map {\dfrac \d {\d y} } {\dfrac 1 {\d y / \d x} }
| c = Derivative of Inverse Function
}}
{{eqn | r = \dfrac {-1} {\paren {\d y / \d x}^2} \map {\... | Let $f$ be a [[Definition:Real Function|real function]] which is of [[Definition:Differentiability Class|differentiability class]] $2$.
Let $f$ have an [[Definition:Inverse Mapping|inverse]] $f^{-1}$, likewise of [[Definition:Differentiability Class|differentiability class]] $2$.
Then:
:$\dfrac {\d^2 x} {\d y^2} = -\... | {{begin-eqn}}
{{eqn | l = \dfrac {\d^2 x} {\d y^2}
| r = \map {\dfrac \d {\d y} } {\dfrac {\d x} {\d y} }
| c = {{Defof|Second Derivative}}
}}
{{eqn | r = \map {\dfrac \d {\d y} } {\dfrac 1 {\d y / \d x} }
| c = [[Derivative of Inverse Function]]
}}
{{eqn | r = \dfrac {-1} {\paren {\d y / \d x}^2} \ma... | Second Derivative of Inverse Function | https://proofwiki.org/wiki/Second_Derivative_of_Inverse_Function | https://proofwiki.org/wiki/Second_Derivative_of_Inverse_Function | [
"Derivative of Inverse Function"
] | [
"Definition:Real Function",
"Definition:Differentiability Class",
"Definition:Inverse Mapping",
"Definition:Differentiability Class"
] | [
"Derivative of Inverse Function",
"Derivative of Composite Function",
"Derivative of Composite Function",
"Derivative of Inverse Function"
] |
proofwiki-20851 | Third Derivative of Inverse Function | Let $f$ be a real function which is of differentiability class $3$.
Let $f$ have an inverse $f^{-1}$, likewise of differentiability class $3$.
Then:
:$\dfrac {\d^3 x} {\d y^3} = -\paren {\dfrac {\d^3 y} {\d x^3} \dfrac {\d y} {\d x} - 3 \paren {\dfrac {\d^2 y} {\d x^2} }^2} \paren {\dfrac {\d y} {\d x} }^{-5}$ | {{begin-eqn}}
{{eqn | l = \dfrac {\d^3 x} {\d y^3}
| r = \map {\dfrac \d \d y } {\dfrac {\d^2 x} {\d y^2} }
| c = {{Defof|Third Derivative}}
}}
{{eqn | r = \map {\dfrac \d {\d y} } {-\dfrac {\d^2 y} {\d x^2} \paren {\dfrac {\d y} {\d x} }^{-3} }
| c = Derivative of Inverse Function
}}
{{eqn | r = -\pa... | Let $f$ be a [[Definition:Real Function|real function]] which is of [[Definition:Differentiability Class|differentiability class]] $3$.
Let $f$ have an [[Definition:Inverse Mapping|inverse]] $f^{-1}$, likewise of [[Definition:Differentiability Class|differentiability class]] $3$.
Then:
:$\dfrac {\d^3 x} {\d y^3} = -\... | {{begin-eqn}}
{{eqn | l = \dfrac {\d^3 x} {\d y^3}
| r = \map {\dfrac \d \d y } {\dfrac {\d^2 x} {\d y^2} }
| c = {{Defof|Third Derivative}}
}}
{{eqn | r = \map {\dfrac \d {\d y} } {-\dfrac {\d^2 y} {\d x^2} \paren {\dfrac {\d y} {\d x} }^{-3} }
| c = [[Derivative of Inverse Function]]
}}
{{eqn | r = ... | Third Derivative of Inverse Function | https://proofwiki.org/wiki/Third_Derivative_of_Inverse_Function | https://proofwiki.org/wiki/Third_Derivative_of_Inverse_Function | [
"Derivative of Inverse Function"
] | [
"Definition:Real Function",
"Definition:Differentiability Class",
"Definition:Inverse Mapping",
"Definition:Differentiability Class"
] | [
"Derivative of Inverse Function",
"Product Rule for Derivatives",
"Derivative of Composite Function",
"Derivative of Composite Function",
"Derivative of Composite Function"
] |
proofwiki-20852 | Relation is Equivalence iff Reflexive and Circular | Let $\RR \subseteq S \times S$ be a relation in $S$.
Then:
:$\RR$ is reflexive and circular
{{iff}}:
:$\RR$ is an equivalence relation. | === Sufficient Condition ===
Let $\RR$ be reflexive and circular.
Then from Reflexive Circular Relation is Equivalence:
:$\RR$ is an equivalence relation.
{{qed|lemma}} | Let $\RR \subseteq S \times S$ be a [[Definition:Relation|relation]] in $S$.
Then:
:$\RR$ is [[Definition:Reflexive Relation|reflexive]] and [[Definition:Circular Relation|circular]]
{{iff}}:
:$\RR$ is an [[Definition:Equivalence Relation|equivalence relation]]. | === Sufficient Condition ===
Let $\RR$ be [[Definition:Reflexive Relation|reflexive]] and [[Definition:Circular Relation|circular]].
Then from [[Reflexive Circular Relation is Equivalence]]:
:$\RR$ is an [[Definition:Equivalence Relation|equivalence relation]].
{{qed|lemma}} | Relation is Equivalence iff Reflexive and Circular | https://proofwiki.org/wiki/Relation_is_Equivalence_iff_Reflexive_and_Circular | https://proofwiki.org/wiki/Relation_is_Equivalence_iff_Reflexive_and_Circular | [
"Equivalence Relations",
"Circular Relations",
"Reflexive Relations"
] | [
"Definition:Relation",
"Definition:Reflexive Relation",
"Definition:Circular Relation",
"Definition:Equivalence Relation"
] | [
"Definition:Reflexive Relation",
"Definition:Circular Relation",
"Reflexive Circular Relation is Equivalence",
"Definition:Equivalence Relation",
"Definition:Equivalence Relation",
"Definition:Reflexive Relation",
"Definition:Circular Relation"
] |
proofwiki-20853 | Equivalence Relation is Circular | Let $\RR \subseteq S \times S$ be an equivalence relation.
Then $\RR$ is also a circular relation. | Let $x, y, z \in S$ be arbitrary such that:
:$\tuple {x, y} \in \RR$ and $\tuple {y, z} \in \RR$
We have {{afortiori}} that $\RR$ is transitive.
Hence:
:$\tuple {x, z} \in \RR$
We also have {{afortiori}} that $\RR$ is symmetric.
Hence:
:$\tuple {z, x} \in \RR$
We have demonstrated that:
:$\tuple {x, y} \in \RR$ and $\t... | Let $\RR \subseteq S \times S$ be an [[Definition:Equivalence Relation|equivalence relation]].
Then $\RR$ is also a [[Definition:Circular Relation|circular relation]]. | Let $x, y, z \in S$ be arbitrary such that:
:$\tuple {x, y} \in \RR$ and $\tuple {y, z} \in \RR$
We have {{afortiori}} that $\RR$ is [[Definition:Transitive Relation|transitive]].
Hence:
:$\tuple {x, z} \in \RR$
We also have {{afortiori}} that $\RR$ is [[Definition:Symmetric Relation|symmetric]].
Hence:
:$\tuple {z... | Equivalence Relation is Circular | https://proofwiki.org/wiki/Equivalence_Relation_is_Circular | https://proofwiki.org/wiki/Equivalence_Relation_is_Circular | [
"Equivalence Relations",
"Circular Relations"
] | [
"Definition:Equivalence Relation",
"Definition:Circular Relation"
] | [
"Definition:Transitive Relation",
"Definition:Symmetric Relation",
"Definition:Circular Relation"
] |
proofwiki-20854 | Primitive of x over x squared plus a squared/Corollary | :$\ds \int \frac {x \rd x} {a^2 + b^2 x^2} = \frac 1 {2 b^2} \map \ln {a^2 + b^2 x^2} + C$ | Let $z = b x$.
Then:
:$\dfrac {\d x} {\d z} = \dfrac 1 b$
Hence:
{{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {a^2 + b^2 x^2}
| r = \int \dfrac 1 b \frac {\paren {z / b} \d z} {a^2 + z^2}
| c = Integration by Substitution
}}
{{eqn | r = \dfrac 1 {b^2} \int \frac {z \d z} {a^2 + z^2}
| c = simplifyin... | :$\ds \int \frac {x \rd x} {a^2 + b^2 x^2} = \frac 1 {2 b^2} \map \ln {a^2 + b^2 x^2} + C$ | Let $z = b x$.
Then:
:$\dfrac {\d x} {\d z} = \dfrac 1 b$
Hence:
{{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {a^2 + b^2 x^2}
| r = \int \dfrac 1 b \frac {\paren {z / b} \d z} {a^2 + z^2}
| c = [[Integration by Substitution]]
}}
{{eqn | r = \dfrac 1 {b^2} \int \frac {z \d z} {a^2 + z^2}
| c = sim... | Primitive of x over x squared plus a squared/Corollary | https://proofwiki.org/wiki/Primitive_of_x_over_x_squared_plus_a_squared/Corollary | https://proofwiki.org/wiki/Primitive_of_x_over_x_squared_plus_a_squared/Corollary | [
"Primitive of x over x squared plus a squared"
] | [] | [
"Integration by Substitution",
"Linear Combination of Integrals/Indefinite",
"Primitive of x over x squared plus a squared"
] |
proofwiki-20855 | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 2/Corollary | :$\ds \int \frac {\d x} {a^2 - b^2 x^2} = \dfrac 1 {2 a b} \ln \size {\dfrac {a + b x} {a - b x} } + C$ | Let $z = b x$.
Then:
:$\dfrac {\d x} {\d z} = \dfrac 1 b$
Hence:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - b^2 x^2}
| r = \int \dfrac 1 b \frac {\d z} {a^2 - z^2}
| c = Integration by Substitution
}}
{{eqn | r = \dfrac 1 b \cdot \dfrac 1 {2 a} \ln \size {\dfrac {a + x} {a - x} } + C
| c = Pri... | :$\ds \int \frac {\d x} {a^2 - b^2 x^2} = \dfrac 1 {2 a b} \ln \size {\dfrac {a + b x} {a - b x} } + C$ | Let $z = b x$.
Then:
:$\dfrac {\d x} {\d z} = \dfrac 1 b$
Hence:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - b^2 x^2}
| r = \int \dfrac 1 b \frac {\d z} {a^2 - z^2}
| c = [[Integration by Substitution]]
}}
{{eqn | r = \dfrac 1 b \cdot \dfrac 1 {2 a} \ln \size {\dfrac {a + x} {a - x} } + C
| ... | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 2/Corollary | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_2/Corollary | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_2/Corollary | [
"Primitive of Reciprocal of a squared minus x squared"
] | [] | [
"Integration by Substitution",
"Primitive of Reciprocal of a squared minus x squared/Logarithm Form 2"
] |
proofwiki-20856 | Determinant of Plane Reflection Matrix | The matrix associated with a reflection of the plane has a determinant of $-1$. | From Matrix Equation of Plane Reflection, we have:
{{begin-eqn}}
{{eqn | l = \begin {vmatrix} \cos 2 \alpha & \sin 2 \alpha \\ \sin 2 \alpha & -\cos 2 \alpha \end {vmatrix}
| r = -\map \cos {2 \alpha} \map \cos {2 \alpha} - \map \sin {2 \alpha} \map \sin {2 \alpha}
| c = Determinant of Order 2
}}
{{eqn | r ... | The [[Definition:Matrix|matrix]] associated with a [[Definition:Plane Reflection|reflection]] of [[Definition:The Plane|the plane]] has a [[Definition:Determinant|determinant]] of $-1$. | From [[Matrix Equation of Plane Reflection]], we have:
{{begin-eqn}}
{{eqn | l = \begin {vmatrix} \cos 2 \alpha & \sin 2 \alpha \\ \sin 2 \alpha & -\cos 2 \alpha \end {vmatrix}
| r = -\map \cos {2 \alpha} \map \cos {2 \alpha} - \map \sin {2 \alpha} \map \sin {2 \alpha}
| c = [[Determinant of Order 2]]
}}
{... | Determinant of Plane Reflection Matrix | https://proofwiki.org/wiki/Determinant_of_Plane_Reflection_Matrix | https://proofwiki.org/wiki/Determinant_of_Plane_Reflection_Matrix | [
"Geometric Reflections",
"Determinants"
] | [
"Definition:Matrix",
"Definition:Reflection (Geometry)/Plane",
"Definition:Plane Surface/The Plane",
"Definition:Determinant"
] | [
"Matrix Equation of Plane Reflection",
"Determinant/Examples/Order 2",
"Sum of Squares of Sine and Cosine",
"Category:Geometric Reflections",
"Category:Determinants"
] |
proofwiki-20857 | Inverse of Plane Reflection Matrix | Let $\mathbf R$ be the matrix associated with a reflection in the plane.
:<nowiki>$\mathbf R = \begin{bmatrix}
\cos 2\alpha & \sin 2\alpha \\
\sin 2\alpha & -\cos 2\alpha
\end{bmatrix}$</nowiki>
Then its inverse matrix $\mathbf R^{-1}$ is itself. | Consider $\mathbf R \mathbf R$:
{{begin-eqn}}
{{eqn | l = \mathbf R \mathbf R
| r = \begin {bmatrix} \cos 2 \alpha & \sin 2 \alpha \\ \sin 2 \alpha & -\cos 2 \alpha \end {bmatrix} \begin {bmatrix} \cos 2 \alpha & \sin 2 \alpha \\ \sin 2 \alpha & -\cos 2 \alpha \end {bmatrix}
}}
{{eqn | r = \begin {bmatrix} \map \... | Let $\mathbf R$ be the [[Definition:Matrix|matrix]] associated with a [[Definition:Plane Reflection|reflection]] in [[Definition:The Plane|the plane]].
:<nowiki>$\mathbf R = \begin{bmatrix}
\cos 2\alpha & \sin 2\alpha \\
\sin 2\alpha & -\cos 2\alpha
\end{bmatrix}$</nowiki>
Then its [[Definition:Inverse Matrix|inver... | Consider $\mathbf R \mathbf R$:
{{begin-eqn}}
{{eqn | l = \mathbf R \mathbf R
| r = \begin {bmatrix} \cos 2 \alpha & \sin 2 \alpha \\ \sin 2 \alpha & -\cos 2 \alpha \end {bmatrix} \begin {bmatrix} \cos 2 \alpha & \sin 2 \alpha \\ \sin 2 \alpha & -\cos 2 \alpha \end {bmatrix}
}}
{{eqn | r = \begin {bmatrix} \map ... | Inverse of Plane Reflection Matrix | https://proofwiki.org/wiki/Inverse_of_Plane_Reflection_Matrix | https://proofwiki.org/wiki/Inverse_of_Plane_Reflection_Matrix | [
"Inverse Matrices"
] | [
"Definition:Matrix",
"Definition:Reflection (Geometry)/Plane",
"Definition:Plane Surface/The Plane",
"Definition:Inverse Matrix"
] | [
"Sum of Squares of Sine and Cosine",
"Definition:Inverse Matrix",
"Definition:Inverse Matrix",
"Category:Inverse Matrices"
] |
proofwiki-20858 | Primitive of Reciprocal of Root of a x + b by Root of p x + q/a p less than 0 | Let $a p < 0$.
Then:
:$\ds \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } } = \dfrac 2 {\sqrt {-a p} } \map \arctan {\sqrt {\dfrac {-p \paren {a x + b} } {a \paren {p x + q} } } } + C$
for all $x \in \R$ such that $\paren {a x + b} \paren {p x + q} > 0$. | First let us express the integrand in the following form:
{{begin-eqn}}
{{eqn | n = 1
| l = \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } }
| r = \int \frac {\d x} {\sqrt {a p \paren {x - \paren {-\frac b a} } \paren {x - \paren {-\frac q p} } } }
| c =
}}
{{end-eqn}}
Recall the defin... | Let $a p < 0$.
Then:
:$\ds \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } } = \dfrac 2 {\sqrt {-a p} } \map \arctan {\sqrt {\dfrac {-p \paren {a x + b} } {a \paren {p x + q} } } } + C$
for all $x \in \R$ such that $\paren {a x + b} \paren {p x + q} > 0$. | First let us express the [[Definition:Integrand|integrand]] in the following form:
{{begin-eqn}}
{{eqn | n = 1
| l = \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } }
| r = \int \frac {\d x} {\sqrt {a p \paren {x - \paren {-\frac b a} } \paren {x - \paren {-\frac q p} } } }
| c =
}}
{{... | Primitive of Reciprocal of Root of a x + b by Root of p x + q/a p less than 0/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_+_b_by_Root_of_p_x_+_q/a_p_less_than_0 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_+_b_by_Root_of_p_x_+_q/a_p_less_than_0/Proof_2 | [
"Primitive of Reciprocal of Root of a x + b by Root of p x + q"
] | [] | [
"Definition:Integration/Integrand",
"Definition:Euler Substitution/Third",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Integration by Substitution",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form",
"Arccotangent of Reciprocal equals Arctangent",
"Sum of A... |
proofwiki-20859 | Arcsine in terms of Twice Arctangent | :$\arcsin x = 2 \map \arctan {\dfrac x {1 + \sqrt {1 - x^2} } }$ | Let:
:$\theta = \arcsin x$
Then by the definition of arcsine:
:$x = \sin \theta$
and:
:$-\dfrac \pi 2 < \theta < \dfrac \pi 2$
Then:
{{begin-eqn}}
{{eqn | l = \tan \dfrac \theta 2
| r = \dfrac {\sin \theta} {1 + \cos \theta}
| c = {{Corollary|Half Angle Formula for Tangent|1}}
}}
{{eqn | r = \dfrac {\sin ... | :$\arcsin x = 2 \map \arctan {\dfrac x {1 + \sqrt {1 - x^2} } }$ | Let:
:$\theta = \arcsin x$
Then by the definition of [[Definition:Real Arcsine|arcsine]]:
:$x = \sin \theta$
and:
:$-\dfrac \pi 2 < \theta < \dfrac \pi 2$
Then:
{{begin-eqn}}
{{eqn | l = \tan \dfrac \theta 2
| r = \dfrac {\sin \theta} {1 + \cos \theta}
| c = {{Corollary|Half Angle Formula for Tangent... | Arcsine in terms of Twice Arctangent | https://proofwiki.org/wiki/Arcsine_in_terms_of_Twice_Arctangent | https://proofwiki.org/wiki/Arcsine_in_terms_of_Twice_Arctangent | [
"Arcsine Function",
"Arctangent Function"
] | [] | [
"Definition:Inverse Sine/Real/Arcsine",
"Sum of Squares of Sine and Cosine",
"Sum of Squares of Sine and Cosine",
"Definition:Inverse Tangent/Real/Arctangent",
"Category:Arcsine Function",
"Category:Arctangent Function"
] |
proofwiki-20860 | Fekete's Subadditive Lemma | Let $\sequence {a_n}_{n \mathop \ge 1}$ be a subadditive sequence.
Then:
:$\ds \lim_{n \mathop \to \infty} \frac {a_n} n = \inf_{n \mathop \ge 1} \frac {a_n} n$ | Let $k \ge 1$.
Let $n \ge 1$.
By Division Theorem, there exist $q \in \N$ and $r \in \set {0, 1, \ldots , k - 1}$ such that:
:$n = k q + r$
Thus:
{{begin-eqn}}
{{eqn | l = \frac {a_n} n
| r = \frac {a_{k q + r} } n
}}
{{eqn | o = \le
| r = \frac {a_{k q} + a_r} n
| c = {{Defof|Subadditive Sequence}}
}... | Let $\sequence {a_n}_{n \mathop \ge 1}$ be a [[Definition:Subadditive Sequence|subadditive sequence]].
Then:
:$\ds \lim_{n \mathop \to \infty} \frac {a_n} n = \inf_{n \mathop \ge 1} \frac {a_n} n$ | Let $k \ge 1$.
Let $n \ge 1$.
By [[Division Theorem]], there exist $q \in \N$ and $r \in \set {0, 1, \ldots , k - 1}$ such that:
:$n = k q + r$
Thus:
{{begin-eqn}}
{{eqn | l = \frac {a_n} n
| r = \frac {a_{k q + r} } n
}}
{{eqn | o = \le
| r = \frac {a_{k q} + a_r} n
| c = {{Defof|Subadditive Seque... | Fekete's Subadditive Lemma | https://proofwiki.org/wiki/Fekete's_Subadditive_Lemma | https://proofwiki.org/wiki/Fekete's_Subadditive_Lemma | [
"Fekete's Subadditive Lemma",
"Real Sequences"
] | [
"Definition:Subadditive Sequence"
] | [
"Division Theorem",
"Definition:Infimum of Set/Real Numbers",
"Convergence of Limsup and Liminf"
] |
proofwiki-20861 | Supremum Operator Norm on Continuous Linear Transformation Space is Submultiplicative | Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$, $\struct {Z, \norm {\, \cdot \,}_Z}$ be normed vector spaces.
Let $A : Y \to Z$ and $B : X \to Y$ be continuous linear transformations.
Let $\norm {\, \cdot \,}$ be the supremum operator norm.
Let $\circ$ denote the composition.
Then $\n... | {{begin-eqn}}
{{eqn | l = \norm {A \circ B}
| r = \sup_{x \mathop \in X \mathop : \norm x_X \mathop \le 1} \norm {\map {\paren {A \circ B} } x}_Z
| c = {{Defof|Supremum Operator Norm}}
}}
{{eqn | o = \le
| r = \sup_{x \mathop \in X \mathop : \norm x_X \mathop \le 1} \norm A \cdot \norm {\map B x}_Y
... | Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$, $\struct {Z, \norm {\, \cdot \,}_Z}$ be [[Definition:Normed Vector Space|normed vector spaces]].
Let $A : Y \to Z$ and $B : X \to Y$ be [[Definition:Continuous Linear Transformation|continuous linear transformations]].
Let $\norm {\, \... | {{begin-eqn}}
{{eqn | l = \norm {A \circ B}
| r = \sup_{x \mathop \in X \mathop : \norm x_X \mathop \le 1} \norm {\map {\paren {A \circ B} } x}_Z
| c = {{Defof|Supremum Operator Norm}}
}}
{{eqn | o = \le
| r = \sup_{x \mathop \in X \mathop : \norm x_X \mathop \le 1} \norm A \cdot \norm {\map B x}_Y
... | Supremum Operator Norm on Continuous Linear Transformation Space is Submultiplicative | https://proofwiki.org/wiki/Supremum_Operator_Norm_on_Continuous_Linear_Transformation_Space_is_Submultiplicative | https://proofwiki.org/wiki/Supremum_Operator_Norm_on_Continuous_Linear_Transformation_Space_is_Submultiplicative | [
"Continuous Linear Transformations",
"Supremum Operator Norm",
"Submultiplicative Functions"
] | [
"Definition:Normed Vector Space",
"Definition:Continuous Linear Transformation",
"Definition:Supremum Operator Norm",
"Definition:Composition of Mappings",
"Definition:Submultiplicative Function on Ring"
] | [
"Supremum Operator Norm as Universal Upper Bound",
"Supremum Operator Norm as Universal Upper Bound"
] |
proofwiki-20862 | Characterization of Paracompactness in T3 Space/Lemma 14 | :$\forall n \in \N_{>0}: U_n \subseteq V_0$ | === Proof: $V_n \subseteq V_{n - 1}$ ===
We have:
{{begin-eqn}}
{{eqn | q = \forall n \in \N_{> 0}
| l = \tuple{x, y} \in V_n
| o = \leadsto
| r = \tuple{x, y}, \tuple{y, y} \in V_n
| c = as $\Delta_X \subseteq V_n$
}}
{{eqn | o = \leadsto
| r = \tuple{x, y} \in V_n \circ V_n
| c = ... | :$\forall n \in \N_{>0}: U_n \subseteq V_0$ | === Proof: $V_n \subseteq V_{n - 1}$ ===
We have:
{{begin-eqn}}
{{eqn | q = \forall n \in \N_{> 0}
| l = \tuple{x, y} \in V_n
| o = \leadsto
| r = \tuple{x, y}, \tuple{y, y} \in V_n
| c = as $\Delta_X \subseteq V_n$
}}
{{eqn | o = \leadsto
| r = \tuple{x, y} \in V_n \circ V_n
| c =... | Characterization of Paracompactness in T3 Space/Lemma 14 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_14 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_14 | [
"Characterization of Paracompactness in T3 Space"
] | [] | [
"Subset Relation is Transitive",
"Subset Relation is Transitive",
"Subset Relation is Transitive"
] |
proofwiki-20863 | Existence of Chebyshev Polynomials of the Second Kind | There exists a '''Chebyshev polynomial of the second kind''' for all natural numbers $n$. | {{tidy}}
For $n = 0$:
{{begin-eqn}}
{{eqn | l = \map \sin {\paren {0 + 1} \theta}
| r = \map \sin \theta
}}
{{eqn | r = 1 \cdot \sin \theta
}}
{{end-eqn}}
:$\map {U_0} x = 1$, $U_0 \in \Bbb P$
{{MissingLinks|$\Bbb P$ as the set/space of polynomials}}
For $n = 1$:
{{begin-eqn}}
{{eqn | l = \map \sin {\paren {1 + 1... | There exists a '''[[Definition:Chebyshev Polynomial of the Second Kind|Chebyshev polynomial of the second kind]]''' for all [[Definition:Natural Numbers|natural numbers]] $n$. | {{tidy}}
For $n = 0$:
{{begin-eqn}}
{{eqn | l = \map \sin {\paren {0 + 1} \theta}
| r = \map \sin \theta
}}
{{eqn | r = 1 \cdot \sin \theta
}}
{{end-eqn}}
:$\map {U_0} x = 1$, $U_0 \in \Bbb P$
{{MissingLinks|$\Bbb P$ as the set/space of polynomials}}
For $n = 1$:
{{begin-eqn}}
{{eqn | l = \map \sin {\paren {... | Existence of Chebyshev Polynomials of the Second Kind | https://proofwiki.org/wiki/Existence_of_Chebyshev_Polynomials_of_the_Second_Kind | https://proofwiki.org/wiki/Existence_of_Chebyshev_Polynomials_of_the_Second_Kind | [
"Chebyshev Polynomials of the Second Kind"
] | [
"Definition:Chebyshev Polynomials/Second Kind",
"Definition:Natural Numbers"
] | [
"Double Angle Formulas/Sine",
"Sine of Sum",
"Sine of Difference",
"Second Principle of Mathematical Induction",
"Category:Chebyshev Polynomials of the Second Kind"
] |
proofwiki-20864 | Composition of Relations Preserves Subsets | Let $A, B, S, T$ be relations as subsets of Cartesian products.
Let $A \subseteq B$ and $S \subset T$.
Then:
:$A \circ S \subseteq B \circ T$ | We have:
{{begin-eqn}}
{{eqn | q = \forall \tuple{x, y}
| l = \tuple{x, y} \in A \circ S
| o = \leadsto
| r = \exists z : \tuple{x, z} \in S, \tuple{z, y} \in A
| c = {{Defof|Composite Relation}}
}}
{{eqn | o = \leadsto
| r = \exists z : \tuple{x, z} \in T, \tuple{z, y} \in B
| c = ... | Let $A, B, S, T$ be [[Definition:Relation as Subset of Cartesian Product|relations as subsets]] of [[Definition:Cartesian Product|Cartesian products]].
Let $A \subseteq B$ and $S \subset T$.
Then:
:$A \circ S \subseteq B \circ T$ | We have:
{{begin-eqn}}
{{eqn | q = \forall \tuple{x, y}
| l = \tuple{x, y} \in A \circ S
| o = \leadsto
| r = \exists z : \tuple{x, z} \in S, \tuple{z, y} \in A
| c = {{Defof|Composite Relation}}
}}
{{eqn | o = \leadsto
| r = \exists z : \tuple{x, z} \in T, \tuple{z, y} \in B
| c =... | Composition of Relations Preserves Subsets | https://proofwiki.org/wiki/Composition_of_Relations_Preserves_Subsets | https://proofwiki.org/wiki/Composition_of_Relations_Preserves_Subsets | [
"Relations",
"Cartesian Product",
"Composite Relations"
] | [
"Definition:Relation/Relation as Subset of Cartesian Product",
"Definition:Cartesian Product"
] | [
"Definition:Subset",
"Category:Relations",
"Category:Cartesian Product",
"Category:Composite Relations"
] |
proofwiki-20865 | Primitive of Reciprocal of Root of a x + b by Root of p x + q/a greater than 0, p less than 0 | Let $a > 0$ and $p < 0$.
Then:
:$\ds \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } } = \dfrac {-1} {\sqrt {-a p} } \map \arcsin {\dfrac {2 a p x + b p + a q} {a q - b p} } + C$
for all $x \in \R$ such that $\paren {a x + b} \paren {p x + q} > 0$. | === Completing the Square ===
{{:Primitive of Reciprocal of Root of a x + b by Root of p x + q/Completing the Square}}{{qed|lemma}}
Hence:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } }
| r = \frac 1 {\sqrt {-a p} } \int \frac {\d u} {\sqrt {\paren {b p - a q}^2 - u^2} }... | Let $a > 0$ and $p < 0$.
Then:
:$\ds \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } } = \dfrac {-1} {\sqrt {-a p} } \map \arcsin {\dfrac {2 a p x + b p + a q} {a q - b p} } + C$
for all $x \in \R$ such that $\paren {a x + b} \paren {p x + q} > 0$. | === [[Primitive of Reciprocal of Root of a x + b by Root of p x + q/Completing the Square|Completing the Square]] ===
{{:Primitive of Reciprocal of Root of a x + b by Root of p x + q/Completing the Square}}{{qed|lemma}}
Hence:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } }
... | Primitive of Reciprocal of Root of a x + b by Root of p x + q/a greater than 0, p less than 0 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_+_b_by_Root_of_p_x_+_q/a_greater_than_0,_p_less_than_0 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_+_b_by_Root_of_p_x_+_q/a_greater_than_0,_p_less_than_0 | [
"Primitive of Reciprocal of Root of a x + b by Root of p x + q"
] | [] | [
"Primitive of Reciprocal of Root of a x + b by Root of p x + q/Completing the Square",
"Primitive of Reciprocal of Root of a x + b by Root of p x + q/Completing the Square",
"Primitive of Reciprocal of Root of a squared minus x squared",
"Primitive of Reciprocal of Root of a x + b by Root of p x + q/Completin... |
proofwiki-20866 | Primitive of Reciprocal of Root of a x + b by Root of p x + q/a p greater than 0 | Let $a, b, p, q \in \R$ such that $a p \ne b q$.
Let $a p > 0$.
Then:
:$\ds \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } } = \dfrac 2 {\sqrt {a p} } \ln \size {\sqrt {p \paren {a x + b} } + \sqrt {a \paren {p x + q} } } + C$
for all $x \in \R$ such that $\paren {a x + b} \paren {p x + q} > 0$. | === Lemma $1$ ===
{{:Primitive of Reciprocal of Root of a x + b by Root of p x + q/Lemma 1}}{{qed|lemma}} | Let $a, b, p, q \in \R$ such that $a p \ne b q$.
Let $a p > 0$.
Then:
:$\ds \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } } = \dfrac 2 {\sqrt {a p} } \ln \size {\sqrt {p \paren {a x + b} } + \sqrt {a \paren {p x + q} } } + C$
for all $x \in \R$ such that $\paren {a x + b} \paren {p x + q} > 0$. | === [[Primitive of Reciprocal of Root of a x + b by Root of p x + q/Lemma 1|Lemma $1$]] ===
{{:Primitive of Reciprocal of Root of a x + b by Root of p x + q/Lemma 1}}{{qed|lemma}} | Primitive of Reciprocal of Root of a x + b by Root of p x + q/a p greater than 0 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_+_b_by_Root_of_p_x_+_q/a_p_greater_than_0 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_+_b_by_Root_of_p_x_+_q/a_p_greater_than_0 | [
"Primitive of Reciprocal of Root of a x + b by Root of p x + q"
] | [] | [
"Primitive of Reciprocal of Root of a x + b by Root of p x + q/Lemma 1",
"Primitive of Reciprocal of Root of a x + b by Root of p x + q/Lemma 1"
] |
proofwiki-20867 | Characterization of Paracompactness in T3 Space/Lemma 15 | :$\forall n \in \N_{>0}: \set{\map {U_n} x : x \in X}$ refines $\set{\map {V_0} x : x \in X}$ | From Composite of Reflexive Relations is Reflexive:
:$\forall n \in \N_{>0} : U_n$ is reflexive.
From Set of Images of Reflexive Relation is Cover of Set:
:$\set{\map {V_0} x : x \in X}$ is a cover of $X$.
and
:$\forall n \in \N_{>0} : \set{\map {U_n} x : x \in X}$ is a cover of $X$. | :$\forall n \in \N_{>0}: \set{\map {U_n} x : x \in X}$ [[Definition:Refinement of Cover|refines]] $\set{\map {V_0} x : x \in X}$ | From [[Composite of Reflexive Relations is Reflexive]]:
:$\forall n \in \N_{>0} : U_n$ is [[Definition:Reflexive Relation|reflexive]].
From [[Set of Images of Reflexive Relation is Cover of Set]]:
:$\set{\map {V_0} x : x \in X}$ is a [[Definition:Cover of Set|cover]] of $X$.
and
:$\forall n \in \N_{>0} : \set{\map {U... | Characterization of Paracompactness in T3 Space/Lemma 15 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_15 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_15 | [
"Characterization of Paracompactness in T3 Space"
] | [
"Definition:Refinement of Cover"
] | [
"Composite of Reflexive Relations is Reflexive",
"Definition:Reflexive Relation",
"Set of Images of Reflexive Relation is Cover of Set",
"Definition:Cover of Set",
"Definition:Cover of Set"
] |
proofwiki-20868 | Set of Images of Reflexive Relation is Cover of Set | Let $\RR \subseteq S \times S$ be a reflexive relation in $S$.
Let $\II = \set{\map \RR x : x \in S}$ be the set of images under $\RR$.
Then:
:$\II$ is a cover of $S$ | By definition of reflexive relation:
:$\forall x \in S : \tuple{x, x} \in \RR$
By definition of image:
:$\forall x \in S : x \in \map \RR x$
Hence, $\II$ is a cover of $S$ by definition.
{{qed}}
Category:Reflexive Relations
Category:Covers
pkki8rprjko873pvq5fy5roibdawvle | Let $\RR \subseteq S \times S$ be a [[Definition:Reflexive Relation|reflexive relation in $S$]].
Let $\II = \set{\map \RR x : x \in S}$ be the [[Definition:Set|set]] of [[Definition:Image of Element under Relation|images under $\RR$]].
Then:
:$\II$ is a [[Definition:Cover of Set|cover]] of $S$ | By definition of [[Definition:Reflexive Relation|reflexive relation]]:
:$\forall x \in S : \tuple{x, x} \in \RR$
By definition of [[Definition:Image of Element under Relation|image]]:
:$\forall x \in S : x \in \map \RR x$
Hence, $\II$ is a [[Definition:Cover of Set|cover]] of $S$ by definition.
{{qed}}
[[Category:... | Set of Images of Reflexive Relation is Cover of Set | https://proofwiki.org/wiki/Set_of_Images_of_Reflexive_Relation_is_Cover_of_Set | https://proofwiki.org/wiki/Set_of_Images_of_Reflexive_Relation_is_Cover_of_Set | [
"Reflexive Relations",
"Covers"
] | [
"Definition:Reflexive Relation",
"Definition:Set",
"Definition:Image (Set Theory)/Relation/Element",
"Definition:Cover of Set"
] | [
"Definition:Reflexive Relation",
"Definition:Image (Set Theory)/Relation/Element",
"Definition:Cover of Set",
"Category:Reflexive Relations",
"Category:Covers"
] |
proofwiki-20869 | Image of Point under Neighborhood of Diagonal is Neighborhood of Point | Let $T = \struct{X, \tau}$ be a topological space.
Let $\tau_{X \times X}$ denote the product topology on the cartesian product $X \times X$.
Let $V$ be a neighborhood of the diagonal $\Delta_X$ of $X \times X$ in the product space $\struct {X \times X, \tau_{X \times X} }$.
Then:
:$\forall x \in X : \map V x$ is a nei... | Let $x \in X$.
By definition of diagonal:
:$\tuple{x, x} \in \Delta_X$
From Set is Neighborhood of Subset iff Neighborhood of all Points of Subset:
:$V$ is a neighborhood of $\tuple{x, x}$
By definition of the product topology:
:$\BB = \set {U_1 \times U_2: U_1, U_2 \in \tau}$ is a basis for $\tau_{X \times X}$
From Ch... | Let $T = \struct{X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\tau_{X \times X}$ denote the [[Definition:Product Topology|product topology]] on the [[Definition:Cartesian Product|cartesian product]] $X \times X$.
Let $V$ be a [[Definition:Neighborhood of Set|neighborhood]] of the [[Definit... | Let $x \in X$.
By definition of [[Definition:Diagonal Relation|diagonal]]:
:$\tuple{x, x} \in \Delta_X$
From [[Set is Neighborhood of Subset iff Neighborhood of all Points of Subset]]:
:$V$ is a [[Definition:Neighborhood of Point|neighborhood]] of $\tuple{x, x}$
By definition of the [[Definition:Product Topology o... | Image of Point under Neighborhood of Diagonal is Neighborhood of Point | https://proofwiki.org/wiki/Image_of_Point_under_Neighborhood_of_Diagonal_is_Neighborhood_of_Point | https://proofwiki.org/wiki/Image_of_Point_under_Neighborhood_of_Diagonal_is_Neighborhood_of_Point | [
"Neighborhoods",
"Product Spaces",
"Relations"
] | [
"Definition:Topological Space",
"Definition:Product Topology",
"Definition:Cartesian Product",
"Definition:Neighborhood (Topology)/Set",
"Definition:Diagonal Relation",
"Definition:Product Space (Topology)/Two Factor Spaces",
"Definition:Neighborhood (Topology)/Point"
] | [
"Definition:Diagonal Relation",
"Set is Neighborhood of Subset iff Neighborhood of all Points of Subset",
"Definition:Neighborhood (Topology)/Point",
"Definition:Product Topology/Two Factor Spaces",
"Definition:Basis (Topology)/Synthetic Basis",
"Characterization of Neighborhood by Basis",
"Definition:C... |
proofwiki-20870 | Image of Element under Cartesian Product of Subsets | Let $S$ and $T$ be sets.
Let $A \subseteq S$ and $B \subseteq T$.
Let $\RR$ be the relation defined by the Cartesian product $A \times B$.
Then:
:$\forall x \in A: \map \RR x = B$ | We have:
{{begin-eqn}}
{{eqn | q = \forall x \in A
| l = \map \RR s
| r = \set {t \in T: \tuple {s, t} \in \RR}
| c = {{Defof|Image of Element under Relation}}
}}
{{eqn | r = \set {t \in T: \tuple {s, t} \in A \times B}
| c = Definition of $\RR$
}}
{{eqn | r = \set {t \in T: s \in A, t \in B}
... | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $A \subseteq S$ and $B \subseteq T$.
Let $\RR$ be the [[Definition:Relation as Subset of Cartesian Product|relation]] defined by the [[Definition:Cartesian Product|Cartesian product]] $A \times B$.
Then:
:$\forall x \in A: \map \RR x = B$ | We have:
{{begin-eqn}}
{{eqn | q = \forall x \in A
| l = \map \RR s
| r = \set {t \in T: \tuple {s, t} \in \RR}
| c = {{Defof|Image of Element under Relation}}
}}
{{eqn | r = \set {t \in T: \tuple {s, t} \in A \times B}
| c = Definition of $\RR$
}}
{{eqn | r = \set {t \in T: s \in A, t \in B}
... | Image of Element under Cartesian Product of Subsets | https://proofwiki.org/wiki/Image_of_Element_under_Cartesian_Product_of_Subsets | https://proofwiki.org/wiki/Image_of_Element_under_Cartesian_Product_of_Subsets | [
"Cartesian Product",
"Relations"
] | [
"Definition:Set",
"Definition:Relation/Relation as Subset of Cartesian Product",
"Definition:Cartesian Product"
] | [
"Category:Cartesian Product",
"Category:Relations"
] |
proofwiki-20871 | Image under Subset of Relation is Subset of Image under Relation/Corollary | Let $x \in S$.
Then:
:$\map {\RR_2} x \subseteq \map {\RR_1} x$
where $\map {\RR_1} x$ denotes the image of $x$ under $\RR_1$. | {{begin-eqn}}
{{eqn | l = \map {\RR_2} x
| r = \RR_2 \sqbrk {\set x}
| c = Image of Singleton under Relation
}}
{{eqn | o = \subseteq
| r = \RR_1 \sqbrk {\set x}
| c = Image under Subset of Relation is Subset of Image under Relation
}}
{{eqn | r = \map {\RR_1} x
| c = Image of Singleton u... | Let $x \in S$.
Then:
:$\map {\RR_2} x \subseteq \map {\RR_1} x$
where $\map {\RR_1} x$ denotes the [[Definition:Image of Element under Relation|image]] of $x$ under $\RR_1$. | {{begin-eqn}}
{{eqn | l = \map {\RR_2} x
| r = \RR_2 \sqbrk {\set x}
| c = [[Image of Singleton under Relation]]
}}
{{eqn | o = \subseteq
| r = \RR_1 \sqbrk {\set x}
| c = [[Image under Subset of Relation is Subset of Image under Relation]]
}}
{{eqn | r = \map {\RR_1} x
| c = [[Image of S... | Image under Subset of Relation is Subset of Image under Relation/Corollary | https://proofwiki.org/wiki/Image_under_Subset_of_Relation_is_Subset_of_Image_under_Relation/Corollary | https://proofwiki.org/wiki/Image_under_Subset_of_Relation_is_Subset_of_Image_under_Relation/Corollary | [
"Relation Theory"
] | [
"Definition:Image (Set Theory)/Relation/Element"
] | [
"Image of Singleton under Relation",
"Image under Subset of Relation is Subset of Image under Relation",
"Image of Singleton under Relation",
"Category:Relation Theory"
] |
proofwiki-20872 | Image of Subset under Neighborhood of Diagonal is Neighborhood of Subset | Let $T = \struct{X, \tau}$ be a topological space.
Let $\tau_{X \times X}$ denote the product topology on the cartesian product $X \times X$.
Let $V$ be a neighborhood of the diagonal $\Delta_X$ of $X \times X$ in the product space $\struct {X \times X, \tau_{X \times X} }$.
Then:
:$\forall A \subseteq X : V \sqbrk A$ ... | Let $A \subseteq X$.
From Image of Subset under Relation equals Union of Images of Elements:
:$V \sqbrk A = \ds \bigcup_{x \in A} \map V x$
From Subset of Union:
:$\forall x \in A : \map V x \subseteq V \sqbrk A$
From Image of Point under Neighborhood of Diagonal is Neighborhood of Point:
:$\forall x \in A : \map V x$ ... | Let $T = \struct{X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\tau_{X \times X}$ denote the [[Definition:Product Topology|product topology]] on the [[Definition:Cartesian Product|cartesian product]] $X \times X$.
Let $V$ be a [[Definition:Neighborhood of Set|neighborhood]] of the [[Definit... | Let $A \subseteq X$.
From [[Image of Subset under Relation equals Union of Images of Elements]]:
:$V \sqbrk A = \ds \bigcup_{x \in A} \map V x$
From [[Subset of Union]]:
:$\forall x \in A : \map V x \subseteq V \sqbrk A$
From [[Image of Point under Neighborhood of Diagonal is Neighborhood of Point]]:
:$\forall x \... | Image of Subset under Neighborhood of Diagonal is Neighborhood of Subset | https://proofwiki.org/wiki/Image_of_Subset_under_Neighborhood_of_Diagonal_is_Neighborhood_of_Subset | https://proofwiki.org/wiki/Image_of_Subset_under_Neighborhood_of_Diagonal_is_Neighborhood_of_Subset | [
"Neighborhoods",
"Product Spaces",
"Relations"
] | [
"Definition:Topological Space",
"Definition:Product Topology",
"Definition:Cartesian Product",
"Definition:Neighborhood (Topology)/Set",
"Definition:Diagonal Relation",
"Definition:Product Space (Topology)/Two Factor Spaces",
"Definition:Neighborhood (Topology)/Set"
] | [
"Image of Subset under Relation equals Union of Images of Elements",
"Set is Subset of Union",
"Image of Point under Neighborhood of Diagonal is Neighborhood of Point",
"Definition:Neighborhood (Topology)/Point",
"Superset of Neighborhood in Topological Space is Neighborhood",
"Definition:Neighborhood (To... |
proofwiki-20873 | Primitive of Reciprocal of Root of a x + b by Root of p x + q/a p less than 0/Proof 2 | Let $a, b, p, q \in \R$ such that $a p \ne b q$.
{{:Primitive of Reciprocal of Root of a x + b by Root of p x + q/a p less than 0}} | First let us express the integrand in the following form:
{{begin-eqn}}
{{eqn | n = 1
| l = \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } }
| r = \int \frac {\d x} {\sqrt {a p \paren {x - \paren {-\frac b a} } \paren {x - \paren {-\frac q p} } } }
| c =
}}
{{end-eqn}}
Recall the defin... | Let $a, b, p, q \in \R$ such that $a p \ne b q$.
{{:Primitive of Reciprocal of Root of a x + b by Root of p x + q/a p less than 0}} | First let us express the [[Definition:Integrand|integrand]] in the following form:
{{begin-eqn}}
{{eqn | n = 1
| l = \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } }
| r = \int \frac {\d x} {\sqrt {a p \paren {x - \paren {-\frac b a} } \paren {x - \paren {-\frac q p} } } }
| c =
}}
{{... | Primitive of Reciprocal of Root of a x + b by Root of p x + q/a p less than 0/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_+_b_by_Root_of_p_x_+_q/a_p_less_than_0/Proof_2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_+_b_by_Root_of_p_x_+_q/a_p_less_than_0/Proof_2 | [
"Primitive of Reciprocal of Root of a x + b by Root of p x + q"
] | [] | [
"Definition:Integration/Integrand",
"Definition:Euler Substitution/Third",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Integration by Substitution",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form",
"Arccotangent of Reciprocal equals Arctangent",
"Sum of A... |
proofwiki-20874 | Lemmata for Euler's Third Substitution/Lemma 1 | :$x = \dfrac {a \beta - \alpha t^2} {a - t^2}$ | {{begin-eqn}}
{{eqn | l = \sqrt {a x^2 + b x + c}
| r = \sqrt {a \paren {x - \alpha} \paren {x - \beta} }
| c =
}}
{{eqn | r = \paren {x - \alpha} t
| c = {{hypothesis}}
}}
{{eqn | ll= \leadsto
| l = a \paren {x - \alpha} \paren {x - \beta}
| r = \paren {x - \alpha}^2 t^2
| c = squa... | :$x = \dfrac {a \beta - \alpha t^2} {a - t^2}$ | {{begin-eqn}}
{{eqn | l = \sqrt {a x^2 + b x + c}
| r = \sqrt {a \paren {x - \alpha} \paren {x - \beta} }
| c =
}}
{{eqn | r = \paren {x - \alpha} t
| c = {{hypothesis}}
}}
{{eqn | ll= \leadsto
| l = a \paren {x - \alpha} \paren {x - \beta}
| r = \paren {x - \alpha}^2 t^2
| c = [[De... | Lemmata for Euler's Third Substitution/Lemma 1 | https://proofwiki.org/wiki/Lemmata_for_Euler's_Third_Substitution/Lemma_1 | https://proofwiki.org/wiki/Lemmata_for_Euler's_Third_Substitution/Lemma_1 | [
"Euler's Third Substitution"
] | [] | [
"Definition:Square/Function"
] |
proofwiki-20875 | Filtration's Lp Spaces are Dense in Limit Filtration's Lp Space | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $\sequence {\FF_n}_{n \mathop \in \N}$ be a filtration of $\Sigma$.
Let $\FF_\infty$ be the limit of $\sequence {\FF_n}_{n \mathop \in \N}$.
Let $p \ge 1$.
Let $\map {L^p} {\cdot}$ denote the $L^p$ spaces.
Then $\ds \bigcup_{n \mathop \ge 0} \map {L^p} {X, \FF_n, ... | First, it is a subset, since
:$\forall n \ge 0 : \map {L^p} {X, \FF_n, \mu} \subseteq \map {L^p} {X, \FF_\infty, \mu}$
in view of $\FF_n \subseteq \FF_\infty$.
In the following, we show its density.
Let:
:$\ds \AA_0 := \bigcup_{n \mathop \ge 0} \FF_n$
Then $\AA_0$ is an algebra.
By Sigma-Algebra extended from Algebra ... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $\sequence {\FF_n}_{n \mathop \in \N}$ be a [[Definition:Filtration of Sigma-Algebra/Discrete Time|filtration]] of $\Sigma$.
Let $\FF_\infty$ be the [[Definition:Limit of Filtration of Sigma-Algebra/Discrete Time|limit]] of $\sequenc... | First, it is a [[Definition:Subset|subset]], since
:$\forall n \ge 0 : \map {L^p} {X, \FF_n, \mu} \subseteq \map {L^p} {X, \FF_\infty, \mu}$
in view of $\FF_n \subseteq \FF_\infty$.
In the following, we show its [[Definition:Everywhere Dense in Normed Vector Space|density]].
Let:
:$\ds \AA_0 := \bigcup_{n \mathop \... | Filtration's Lp Spaces are Dense in Limit Filtration's Lp Space | https://proofwiki.org/wiki/Filtration's_Lp_Spaces_are_Dense_in_Limit_Filtration's_Lp_Space | https://proofwiki.org/wiki/Filtration's_Lp_Spaces_are_Dense_in_Limit_Filtration's_Lp_Space | [
"Filtrations of Sigma-Algebras",
"Lp Spaces"
] | [
"Definition:Measure Space",
"Definition:Filtration of Sigma-Algebra/Discrete Time",
"Definition:Limit of Filtration of Sigma-Algebra/Discrete Time",
"Definition:Lp Space",
"Definition:Everywhere Dense/Normed Vector Space",
"Definition:Subset"
] | [
"Definition:Subset",
"Definition:Everywhere Dense/Normed Vector Space",
"Definition:Algebra of Sets",
"Sigma-Algebra extended from Algebra by Measure",
"Definition:Sigma-Algebra",
"Space of Simple P-Integrable Functions is Everywhere Dense in Lebesgue Space",
"Category:Filtrations of Sigma-Algebras",
... |
proofwiki-20876 | Lemmata for Euler's Third Substitution/Lemma 2 | :$x - \alpha = \dfrac {a \paren {\alpha - \beta} } {t^2 - a}$ | {{begin-eqn}}
{{eqn | l = x
| r = \dfrac {a \beta - \alpha t^2} {a - t^2}
| c = {{Lemma|Lemmata for Euler's Third Substitution|1|proof = yes}}
}}
{{eqn | ll= \leadsto
| l = x - \alpha
| r = \dfrac {a \beta - \alpha t^2} {a - t^2} - \alpha
| c =
}}
{{eqn | r = \dfrac {a \beta - \alpha t^2 ... | :$x - \alpha = \dfrac {a \paren {\alpha - \beta} } {t^2 - a}$ | {{begin-eqn}}
{{eqn | l = x
| r = \dfrac {a \beta - \alpha t^2} {a - t^2}
| c = {{Lemma|Lemmata for Euler's Third Substitution|1|proof = yes}}
}}
{{eqn | ll= \leadsto
| l = x - \alpha
| r = \dfrac {a \beta - \alpha t^2} {a - t^2} - \alpha
| c =
}}
{{eqn | r = \dfrac {a \beta - \alpha t^2 ... | Lemmata for Euler's Third Substitution/Lemma 2 | https://proofwiki.org/wiki/Lemmata_for_Euler's_Third_Substitution/Lemma_2 | https://proofwiki.org/wiki/Lemmata_for_Euler's_Third_Substitution/Lemma_2 | [
"Euler's Third Substitution"
] | [] | [
"Definition:Common Denominator"
] |
proofwiki-20877 | Lemmata for Euler's Third Substitution/Lemma 4 | :$t = \pm \sqrt {\dfrac {a \paren {x - \beta} } {x - \alpha} }$ | {{begin-eqn}}
{{eqn | l = x
| r = \dfrac {a \beta - \alpha t^2} {a - t^2}
| c = {{Lemma|Lemmata for Euler's Third Substitution|1|proof = yes}}
}}
{{eqn | ll= \leadsto
| l = x \paren {a - t^2}
| r = a \beta - \alpha t^2
| c =
}}
{{eqn | ll= \leadsto
| l = t^2 \paren {x - \alpha}
... | :$t = \pm \sqrt {\dfrac {a \paren {x - \beta} } {x - \alpha} }$ | {{begin-eqn}}
{{eqn | l = x
| r = \dfrac {a \beta - \alpha t^2} {a - t^2}
| c = {{Lemma|Lemmata for Euler's Third Substitution|1|proof = yes}}
}}
{{eqn | ll= \leadsto
| l = x \paren {a - t^2}
| r = a \beta - \alpha t^2
| c =
}}
{{eqn | ll= \leadsto
| l = t^2 \paren {x - \alpha}
... | Lemmata for Euler's Third Substitution/Lemma 4 | https://proofwiki.org/wiki/Lemmata_for_Euler's_Third_Substitution/Lemma_4 | https://proofwiki.org/wiki/Lemmata_for_Euler's_Third_Substitution/Lemma_4 | [
"Euler's Third Substitution"
] | [] | [] |
proofwiki-20878 | Lemmata for Euler's Third Substitution/Lemma 3 | :$\dfrac {\d x} {\d t} = \dfrac {2 t a \paren {\beta - \alpha} } {\paren {a - t^2}^2}$ | {{begin-eqn}}
{{eqn | l = \dfrac {\d x} {\d t}
| r = \map {\dfrac \d {\d t} } {x - \alpha}
| c = Derivative of Function plus Constant
}}
{{eqn | r = \map {\dfrac \d {\d t} } {\dfrac {a \paren {\alpha - \beta} } {t^2 - a} }
| c = {{Lemma|Lemmata for Euler's Third Substitution|2|proof = yes}}
}}
{{eqn |... | :$\dfrac {\d x} {\d t} = \dfrac {2 t a \paren {\beta - \alpha} } {\paren {a - t^2}^2}$ | {{begin-eqn}}
{{eqn | l = \dfrac {\d x} {\d t}
| r = \map {\dfrac \d {\d t} } {x - \alpha}
| c = [[Derivative of Function plus Constant]]
}}
{{eqn | r = \map {\dfrac \d {\d t} } {\dfrac {a \paren {\alpha - \beta} } {t^2 - a} }
| c = {{Lemma|Lemmata for Euler's Third Substitution|2|proof = yes}}
}}
{{e... | Lemmata for Euler's Third Substitution/Lemma 3 | https://proofwiki.org/wiki/Lemmata_for_Euler's_Third_Substitution/Lemma_3 | https://proofwiki.org/wiki/Lemmata_for_Euler's_Third_Substitution/Lemma_3 | [
"Euler's Third Substitution"
] | [] | [
"Derivative of Function plus Constant",
"Power Rule for Derivatives",
"Derivative of Composite Function"
] |
proofwiki-20879 | Increasing Martingale Theorem | Let $\struct {X, \Sigma, \mu}$ be a probability space.
Let $f$ be a $\mu$-integrable function.
Given sub-$\sigma$-algebra $\CC \subseteq \Sigma$, let $\expect {f \mid \CC}$ denote the conditional expectation of $f$ on $\CC$.
Let $\sequence {\FF_n}_{n \mathop \in \N}$ be a filtration of $\Sigma$.
Let $\FF_\infty$ be th... | By Tower Property of Conditional Expectation:
:$\expect {f \mid \FF_n} = \expect {\expect {f \mid \FF_\infty} \mid \FF_n}$
Let $\tilde f := \expect {f \mid \FF_\infty}$.
Then we need to show that:
:$\ds \lim_{n \mathop \to \infty} \expect {\tilde f \mid \FF_n} = \tilde f$
holds in $L^1$ norm, and $\mu$-almost surely.
T... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Probability Space|probability space]].
Let $f$ be a $\mu$-[[Definition:Measure-Integrable Function|integrable function]].
Given [[Definition:Sub-Sigma-Algebra|sub-$\sigma$-algebra]] $\CC \subseteq \Sigma$, let $\expect {f \mid \CC}$ denote the [[Definition:Conditional ... | By [[Tower Property of Conditional Expectation]]:
:$\expect {f \mid \FF_n} = \expect {\expect {f \mid \FF_\infty} \mid \FF_n}$
Let $\tilde f := \expect {f \mid \FF_\infty}$.
Then we need to show that:
:$\ds \lim_{n \mathop \to \infty} \expect {\tilde f \mid \FF_n} = \tilde f$
holds in [[Definition:Lp Norm|$L^1$ norm]... | Increasing Martingale Theorem | https://proofwiki.org/wiki/Increasing_Martingale_Theorem | https://proofwiki.org/wiki/Increasing_Martingale_Theorem | [
"Filtrations of Sigma-Algebras",
"Martingales"
] | [
"Definition:Probability Space",
"Definition:Integrable Function/Measure Space",
"Definition:Sub-Sigma-Algebra",
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Definition:Filtration of Sigma-Algebra/Discrete Time",
"Definition:Limit of Filtration of Sigma-Algebra/Discrete Time",
"Defin... | [
"Tower Property of Conditional Expectation",
"Definition:Lp Norm",
"Definition:Almost Everywhere",
"Filtration's Lp Spaces are Dense in Limit Filtration's Lp Space",
"Markov's Inequality",
"Measure of Limit of Increasing Sequence of Measurable Sets",
"Doob's Maximal Inequality",
"Measure of Limit of I... |
proofwiki-20880 | Primitive of Reciprocal of p x + q by Root of a x + b/p (b p - a q) greater than 0 | Let $p \paren {b p - a q} > 0$.
Then:
:$\ds \int \frac {\d x} {\paren {p x + q} \sqrt {a x + b} } = \frac 1 {\sqrt {p \paren {b p - a q} } } \ln \size {\frac {\sqrt {p \paren {a x + b} } - \sqrt {b p - a q} } {\sqrt {p \paren {a x + b} } + \sqrt {b p - a q} } } + C$ | === Lemma ===
{{:Primitive of Reciprocal of p x + q by Root of a x + b/Lemma}}{{qed|lemma}}
We have {{hypothesis}} that:
:$p \paren {b p - a q} > 0$
which means:
:$\dfrac {b p - a q} p > 0$
Hence let:
:$d^2 = \dfrac {b p - a q} p$
Thus:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {p x + q} \sqrt {a x + b} }
... | Let $p \paren {b p - a q} > 0$.
Then:
:$\ds \int \frac {\d x} {\paren {p x + q} \sqrt {a x + b} } = \frac 1 {\sqrt {p \paren {b p - a q} } } \ln \size {\frac {\sqrt {p \paren {a x + b} } - \sqrt {b p - a q} } {\sqrt {p \paren {a x + b} } + \sqrt {b p - a q} } } + C$ | === [[Primitive of Reciprocal of p x + q by Root of a x + b/Lemma|Lemma]] ===
{{:Primitive of Reciprocal of p x + q by Root of a x + b/Lemma}}{{qed|lemma}}
We have {{hypothesis}} that:
:$p \paren {b p - a q} > 0$
which means:
:$\dfrac {b p - a q} p > 0$
Hence let:
:$d^2 = \dfrac {b p - a q} p$
Thus:
{{begin-eqn}}
{... | Primitive of Reciprocal of p x + q by Root of a x + b/p (b p - a q) greater than 0 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_x_+_q_by_Root_of_a_x_+_b/p_(b_p_-_a_q)_greater_than_0 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_x_+_q_by_Root_of_a_x_+_b/p_(b_p_-_a_q)_greater_than_0 | [
"Primitive of Reciprocal of p x + q by Root of a x + b"
] | [] | [
"Primitive of Reciprocal of p x + q by Root of a x + b/Lemma",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form"
] |
proofwiki-20881 | Primitive of Reciprocal of p x + q by Root of a x + b/p (b p - a q) less than 0 | Let $p \paren {b p - a q} > 0$.
Then:
:$\ds \int \frac {\d x} {\paren {p x + q} \sqrt {a x + b} } = \dfrac 2 {\sqrt {p \paren {a q - b p} } } \arctan \sqrt {\dfrac {p \paren {a x + b} } {a q - b p} } + C$ | === Lemma ===
{{:Primitive of Reciprocal of p x + q by Root of a x + b/Lemma}}{{qed|lemma}}
We have {{hypothesis}} that:
:$p \paren {b p - a q} < 0$
which means:
:$\dfrac {b p - a q} p < 0$
Hence let:
:$d^2 = -\dfrac {b p - a q} p$
Thus:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {p x + q} \sqrt {a x + b} }
... | Let $p \paren {b p - a q} > 0$.
Then:
:$\ds \int \frac {\d x} {\paren {p x + q} \sqrt {a x + b} } = \dfrac 2 {\sqrt {p \paren {a q - b p} } } \arctan \sqrt {\dfrac {p \paren {a x + b} } {a q - b p} } + C$ | === [[Primitive of Reciprocal of p x + q by Root of a x + b/Lemma|Lemma]] ===
{{:Primitive of Reciprocal of p x + q by Root of a x + b/Lemma}}{{qed|lemma}}
We have {{hypothesis}} that:
:$p \paren {b p - a q} < 0$
which means:
:$\dfrac {b p - a q} p < 0$
Hence let:
:$d^2 = -\dfrac {b p - a q} p$
Thus:
{{begin-eqn}}
... | Primitive of Reciprocal of p x + q by Root of a x + b/p (b p - a q) less than 0 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_x_+_q_by_Root_of_a_x_+_b/p_(b_p_-_a_q)_less_than_0 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_x_+_q_by_Root_of_a_x_+_b/p_(b_p_-_a_q)_less_than_0 | [
"Primitive of Reciprocal of p x + q by Root of a x + b"
] | [] | [
"Primitive of Reciprocal of p x + q by Root of a x + b/Lemma",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form",
"Inverse Tangent is Odd Function"
] |
proofwiki-20882 | Primitive of Reciprocal of x by Root of a x squared plus b x plus c/Lemma | :$\ds \int \frac {\d x} {x \sqrt {a x^2 + b x + c} } = -\int \frac {\d u} {\pm \sqrt {a + b u + c u^2} }$
where $u := \dfrac 1 x$, according to whether $u > 0$ or $u < 0$. | {{begin-eqn}}
{{eqn | l = x
| r = \frac 1 u
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d u}
| r = \frac {-1} {u^2}
| c = Primitive of Power
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {x \sqrt {a x^2 + b x + c} }
| r = \int {\frac 1 {\frac 1 u \sqrt {a \paren {\fra... | :$\ds \int \frac {\d x} {x \sqrt {a x^2 + b x + c} } = -\int \frac {\d u} {\pm \sqrt {a + b u + c u^2} }$
where $u := \dfrac 1 x$, according to whether $u > 0$ or $u < 0$. | {{begin-eqn}}
{{eqn | l = x
| r = \frac 1 u
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d u}
| r = \frac {-1} {u^2}
| c = [[Primitive of Power]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {x \sqrt {a x^2 + b x + c} }
| r = \int {\frac 1 {\frac 1 u \sqrt {a \paren {... | Primitive of Reciprocal of x by Root of a x squared plus b x plus c/Lemma | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_a_x_squared_plus_b_x_plus_c/Lemma | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_a_x_squared_plus_b_x_plus_c/Lemma | [
"Primitive of Reciprocal of x by Root of a x squared plus b x plus c"
] | [] | [
"Primitive of Power",
"Integration by Substitution",
"Primitive of Constant Multiple of Function"
] |
proofwiki-20883 | Primitive of Reciprocal of x by Root of x squared minus a squared/Arccosine Form | :$\ds \int \frac {\d x} {x \sqrt {x^2 - a^2} } = \frac 1 a \arccos \size {\frac a x} + C$
for $0 < a < \size x$. | We have that $\sqrt {x^2 - a^2}$ is defined only when $x^2 > a^2$, that is, either:
:$x > a$
or:
:$x < -a$
where it is assumed that $a > 0$.
Hence:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \sqrt {x^2 - a^2} }
| r = \frac 1 a \arcsec \size {\frac x a} + C
| c = Primitive of $\dfrac 1 {x \sqrt {x^2 - a^... | :$\ds \int \frac {\d x} {x \sqrt {x^2 - a^2} } = \frac 1 a \arccos \size {\frac a x} + C$
for $0 < a < \size x$. | We have that $\sqrt {x^2 - a^2}$ is defined only when $x^2 > a^2$, that is, either:
:$x > a$
or:
:$x < -a$
where it is assumed that $a > 0$.
Hence:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \sqrt {x^2 - a^2} }
| r = \frac 1 a \arcsec \size {\frac x a} + C
| c = [[Primitive of Reciprocal of x by Root... | Primitive of Reciprocal of x by Root of x squared minus a squared/Arccosine Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_x_squared_minus_a_squared/Arccosine_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_x_squared_minus_a_squared/Arccosine_Form | [
"Primitive of Reciprocal of x by Root of x squared minus a squared",
"Primitives involving Inverse Cosine Function"
] | [] | [
"Primitive of Reciprocal of x by Root of x squared minus a squared/Arcsecant Form",
"Arcsecant of Reciprocal equals Arccosine"
] |
proofwiki-20884 | Primitive of Reciprocal of Root of 2 a x minus x squared | :$\ds \int \frac {\d x} {\sqrt {2 a x - x^2} } = \arcsin \dfrac {x - a} a + C$ | Let $u := x - a$.
Then:
:$\dfrac {\d u} {\d x} = 1$
and:
:$x = u + a$
and we have:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sqrt {2 a x - x^2} }
| r = \int \frac {\d u} {\sqrt {2 a \paren {u + a} - \paren {u + a}^2} }
| c = Integration by Substitution
}}
{{eqn | r = \int \frac {\d u} {\sqrt {2 a u + 2 ... | :$\ds \int \frac {\d x} {\sqrt {2 a x - x^2} } = \arcsin \dfrac {x - a} a + C$ | Let $u := x - a$.
Then:
:$\dfrac {\d u} {\d x} = 1$
and:
:$x = u + a$
and we have:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sqrt {2 a x - x^2} }
| r = \int \frac {\d u} {\sqrt {2 a \paren {u + a} - \paren {u + a}^2} }
| c = [[Integration by Substitution]]
}}
{{eqn | r = \int \frac {\d u} {\sqrt {2 ... | Primitive of Reciprocal of Root of 2 a x minus x squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_2_a_x_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_2_a_x_minus_x_squared | [
"Primitives of Roots of Quadratic Functions"
] | [] | [
"Integration by Substitution",
"Primitive of Reciprocal of Root of a squared minus x squared/Arcsine Form"
] |
proofwiki-20885 | Primitive of Root of 2 a x minus x squared | :$\ds \int \sqrt {2 a x - x^2} \rd x = \frac {\paren {x - a} } 2 \sqrt {2 a x - x^2} + \frac {a^2} 2 \arcsin \frac {x - a} a + C$ | Let $u := x - a$.
Then:
:$\dfrac {\d u} {\d x} = 1$
and:
:$x = u + a$
Then:
{{begin-eqn}}
{{eqn | l = 2 a x - x^2
| r = 2 a \paren {u + a} - \paren {u + a}^2
| c =
}}
{{eqn | r = 2 a u + 2 a^2 - u^2 - 2 a u - a^2
| c =
}}
{{eqn | r = a^2 - u^2
| c =
}}
{{end-eqn}}
and we have:
{{begin-eqn}}
{... | :$\ds \int \sqrt {2 a x - x^2} \rd x = \frac {\paren {x - a} } 2 \sqrt {2 a x - x^2} + \frac {a^2} 2 \arcsin \frac {x - a} a + C$ | Let $u := x - a$.
Then:
:$\dfrac {\d u} {\d x} = 1$
and:
:$x = u + a$
Then:
{{begin-eqn}}
{{eqn | l = 2 a x - x^2
| r = 2 a \paren {u + a} - \paren {u + a}^2
| c =
}}
{{eqn | r = 2 a u + 2 a^2 - u^2 - 2 a u - a^2
| c =
}}
{{eqn | r = a^2 - u^2
| c =
}}
{{end-eqn}}
and we have:
{{begin-... | Primitive of Root of 2 a x minus x squared | https://proofwiki.org/wiki/Primitive_of_Root_of_2_a_x_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_Root_of_2_a_x_minus_x_squared | [
"Primitives of Roots of Quadratic Functions"
] | [] | [
"Primitive of Root of a squared minus x squared/Arcsine Form"
] |
proofwiki-20886 | Closed Ball in Normed Dual Space of Separable Normed Vector Space is Weak-* Metrizable | Let $X$ be a separable normed vector space.
Let $\sequence {x_n}_{n \in \N}$ be a countable everywhere dense subset of $X$.
Let $X^\ast$ be the normed dual space of $X$.
Let $\lambda > 0$.
Let $B^-_{X^\ast}$ be the closed unit ball of $X^\ast$.
Let $w^\ast$ be the weak-$\ast$ topology.
Define $d : X^\ast \times X^\a... | From the Banach-Alaoglu Theorem, we have that $\struct {B^-_{X^\ast}, w^\ast}$ is compact.
From Dilation of Compact Set in Topological Vector Space is Compact, $\struct {\lambda B^-_{X^\ast}, w^\ast}$ is compact.
From Weak-* Compact Set in Normed Dual Space of Separable Normed Vector Space is Weak-* Metrizable, $\struc... | Let $X$ be a [[Definition:Separable Space|separable]] [[Definition:Normed Vector Space|normed vector space]].
Let $\sequence {x_n}_{n \in \N}$ be a [[Definition:Countable Set|countable]] [[Definition:Everywhere Dense|everywhere dense subset]] of $X$.
Let $X^\ast$ be the [[Definition:Normed Dual Space|normed dual spac... | From the [[Banach-Alaoglu Theorem]], we have that $\struct {B^-_{X^\ast}, w^\ast}$ is [[Definition:Compact Topological Space|compact]].
From [[Dilation of Compact Set in Topological Vector Space is Compact]], $\struct {\lambda B^-_{X^\ast}, w^\ast}$ is [[Definition:Compact Topological Space|compact]].
From [[Weak-* C... | Closed Ball in Normed Dual Space of Separable Normed Vector Space is Weak-* Metrizable | https://proofwiki.org/wiki/Closed_Ball_in_Normed_Dual_Space_of_Separable_Normed_Vector_Space_is_Weak-*_Metrizable | https://proofwiki.org/wiki/Closed_Ball_in_Normed_Dual_Space_of_Separable_Normed_Vector_Space_is_Weak-*_Metrizable | [
"Weak-* Topologies",
"Metrizable Spaces"
] | [
"Definition:Separable Space",
"Definition:Normed Vector Space",
"Definition:Countable Set",
"Definition:Everywhere Dense",
"Definition:Normed Dual Space",
"Definition:Closed Unit Ball",
"Definition:Weak-* Topology",
"Definition:Metrizable Space",
"Definition:Restriction/Mapping"
] | [
"Banach-Alaoglu Theorem",
"Definition:Compact Topological Space",
"Dilation of Compact Set in Topological Vector Space is Compact",
"Definition:Compact Topological Space",
"Weak-* Compact Set in Normed Dual Space of Separable Normed Vector Space is Weak-* Metrizable",
"Definition:Metrizable Space",
"Def... |
proofwiki-20887 | Topological Closure in Coarser Topology is Larger | Let $X$ be a set.
Let $\tau_1$ and $\tau_2$ be topologies on $X$ such that:
:$\tau_1 \subseteq \tau_2$
That is, such that $\tau_1$ is coarser than $\tau_2$.
Let $S \subseteq X$.
Then we have:
:$\map {\cl_2} S \subseteq \map {\cl_1} S$
where $\cl_1$ and $\cl_2$ denote topological closure in $\struct {X, \tau_1}$ and ... | Let $\CC_1$ be the set of closed sets $C$ in the topological space $\struct {X, \tau_1}$ such that $S \subseteq C$.
Let $\CC_2$ be the set of closed sets $C$ in the topological space $\struct {X, \tau_2}$ such that $S \subseteq C$.
Let $C \in \CC_1$.
Then from Closed Set in Coarser Topology is Closed in Finer Topology:... | Let $X$ be a [[Definition:Set|set]].
Let $\tau_1$ and $\tau_2$ be [[Definition:Topology|topologies]] on $X$ such that:
:$\tau_1 \subseteq \tau_2$
That is, such that $\tau_1$ is [[Definition:Coarser Topology|coarser]] than $\tau_2$.
Let $S \subseteq X$.
Then we have:
:$\map {\cl_2} S \subseteq \map {\cl_1} S$
... | Let $\CC_1$ be the [[Definition:Set|set]] of [[Definition:Closed Set (Topology)|closed sets]] $C$ in the [[Definition:Topological Space|topological space]] $\struct {X, \tau_1}$ such that $S \subseteq C$.
Let $\CC_2$ be the [[Definition:Set|set]] of [[Definition:Closed Set (Topology)|closed sets]] $C$ in the [[Definit... | Topological Closure in Coarser Topology is Larger | https://proofwiki.org/wiki/Topological_Closure_in_Coarser_Topology_is_Larger | https://proofwiki.org/wiki/Topological_Closure_in_Coarser_Topology_is_Larger | [
"Set Closures"
] | [
"Definition:Set",
"Definition:Topology",
"Definition:Coarser Topology",
"Definition:Closure (Topology)"
] | [
"Definition:Set",
"Definition:Closed Set/Topology",
"Definition:Topological Space",
"Definition:Set",
"Definition:Closed Set/Topology",
"Definition:Topological Space",
"Closed Set in Coarser Topology is Closed in Finer Topology",
"Definition:Subset",
"Intersection is Decreasing",
"Definition:Closu... |
proofwiki-20888 | Closed Unit Ball in Normed Dual Space is Weak-* Closed | Let $X$ be a normed vector space.
Let $X^\ast$ be the normed dual space of $X$.
Let $B^-_{X^\ast}$ be the closed unit ball in $X^\ast$.
Then we have that $B^-_{X^\ast}$ is weak-$\ast$ closed. | From Weak-* Topology is Hausdorff, $\struct {X^\ast, w^\ast}$ is Hausdorff.
From the Banach-Alaoglu Theorem, $\struct {B^-_{X^\ast}, w^\ast}$ is compact.
From Compact Subspace of Hausdorff Space is Closed, it follows that $B^-_{X^\ast}$ is weak-$\ast$ closed.
{{qed}}
Category:Weak-* Topologies
Category:Normed Dual Spa... | Let $X$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $X^\ast$ be the [[Definition:Normed Dual Space|normed dual space]] of $X$.
Let $B^-_{X^\ast}$ be the [[Definition:Closed Unit Ball|closed unit ball]] in $X^\ast$.
Then we have that $B^-_{X^\ast}$ is [[Definition:Weak-* Closed|weak-$\ast$ cl... | From [[Weak-* Topology is Hausdorff]], $\struct {X^\ast, w^\ast}$ is [[Definition:Hausdorff Space|Hausdorff]].
From the [[Banach-Alaoglu Theorem]], $\struct {B^-_{X^\ast}, w^\ast}$ is [[Definition:Compact Topological Space|compact]].
From [[Compact Subspace of Hausdorff Space is Closed]], it follows that $B^-_{X^\as... | Closed Unit Ball in Normed Dual Space is Weak-* Closed | https://proofwiki.org/wiki/Closed_Unit_Ball_in_Normed_Dual_Space_is_Weak-*_Closed | https://proofwiki.org/wiki/Closed_Unit_Ball_in_Normed_Dual_Space_is_Weak-*_Closed | [
"Weak-* Topologies",
"Normed Dual Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Closed Unit Ball",
"Definition:Weak-* Closed"
] | [
"Weak-* Topology is Hausdorff",
"Definition:T2 Space",
"Banach-Alaoglu Theorem",
"Definition:Compact Topological Space",
"Compact Subspace of Hausdorff Space is Closed",
"Definition:Weak-* Closed",
"Category:Weak-* Topologies",
"Category:Normed Dual Spaces"
] |
proofwiki-20889 | Closed Unit Ball in Normed Dual Space of Separable Normed Vector Space is Weak-* Separable | Let $X$ be a separable normed vector space.
Let $X^\ast$ be the normed dual space of $X$.
Let $B^-_{X^\ast}$ be the closed unit ball of $X^\ast$.
Let $w^\ast$ be the weak-$\ast$ topology on $B^-_{X^\ast}$.
Then $\struct {B^-_{X^\ast}, w^\ast}$ is separable. | From the Banach-Alaoglu Theorem, $\struct {B^-_{X^\ast}, w^\ast}$ is compact.
From Closed Ball in Normed Dual Space of Separable Normed Vector Space is Weak-* Metrizable, $\struct {B^-_{X^\ast}, w^\ast}$ is metrizable.
From Compact Metric Space is Separable, we have that $\struct {B^-_{X^\ast}, w^\ast}$ is separable.... | Let $X$ be a [[Definition:Separable Space|separable]] [[Definition:Normed Vector Space|normed vector space]].
Let $X^\ast$ be the [[Definition:Normed Dual Space|normed dual space]] of $X$.
Let $B^-_{X^\ast}$ be the [[Definition:Closed Unit Ball|closed unit ball]] of $X^\ast$.
Let $w^\ast$ be the [[Definition:Weak-*... | From the [[Banach-Alaoglu Theorem]], $\struct {B^-_{X^\ast}, w^\ast}$ is [[Definition:Compact Topological Space|compact]].
From [[Closed Ball in Normed Dual Space of Separable Normed Vector Space is Weak-* Metrizable]], $\struct {B^-_{X^\ast}, w^\ast}$ is [[Definition:Metrizable Topology|metrizable]].
From [[Compac... | Closed Unit Ball in Normed Dual Space of Separable Normed Vector Space is Weak-* Separable | https://proofwiki.org/wiki/Closed_Unit_Ball_in_Normed_Dual_Space_of_Separable_Normed_Vector_Space_is_Weak-*_Separable | https://proofwiki.org/wiki/Closed_Unit_Ball_in_Normed_Dual_Space_of_Separable_Normed_Vector_Space_is_Weak-*_Separable | [
"Weak-* Topologies",
"Separable Spaces"
] | [
"Definition:Separable Space",
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Closed Unit Ball",
"Definition:Weak-* Topology",
"Definition:Separable Space"
] | [
"Banach-Alaoglu Theorem",
"Definition:Compact Topological Space",
"Closed Ball in Normed Dual Space of Separable Normed Vector Space is Weak-* Metrizable",
"Definition:Metrizable Space",
"Compact Metric Space is Separable",
"Definition:Separable Space",
"Category:Weak-* Topologies",
"Category:Separabl... |
proofwiki-20890 | Compactness and Sequential Compactness are Equivalent in Metric Spaces | Let $\struct {X, d}$ be a metric space.
Then $\struct {X, d}$ is compact {{iff}} it is sequentially compact. | === Necessary Condition ===
This is precisely Compact Subspace of Metric Space is Sequentially Compact in Itself.
{{qed|lemma}} | Let $\struct {X, d}$ be a [[Definition:Metric Space|metric space]].
Then $\struct {X, d}$ is [[Definition:Compact Topological Space|compact]] {{iff}} it is [[Definition:Sequentially Compact Space|sequentially compact]]. | === Necessary Condition ===
This is precisely [[Compact Subspace of Metric Space is Sequentially Compact in Itself]].
{{qed|lemma}} | Compactness and Sequential Compactness are Equivalent in Metric Spaces | https://proofwiki.org/wiki/Compactness_and_Sequential_Compactness_are_Equivalent_in_Metric_Spaces | https://proofwiki.org/wiki/Compactness_and_Sequential_Compactness_are_Equivalent_in_Metric_Spaces | [
"Sequentially Compact Spaces",
"Compact Topological Spaces",
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Compact Topological Space",
"Definition:Sequentially Compact Space"
] | [
"Compact Subspace of Metric Space is Sequentially Compact in Itself"
] |
proofwiki-20891 | Compact Metric Space is Separable | Let $\struct {X, d}$ be a compact metric space.
Then $\struct {X, d}$ is separable. | From Compactness and Sequential Compactness are Equivalent in Metric Spaces, $\struct {X, d}$ is sequentially compact.
From Sequentially Compact Metric Space is Separable, $\struct {X, d}$ is separable.
{{qed}}
Category:Separable Spaces
Category:Compact Metric Spaces
Category:Metric Spaces
0c0b8ghlibyso00ajtn701d817ks... | Let $\struct {X, d}$ be a [[Definition:Compact Metric Space|compact metric space]].
Then $\struct {X, d}$ is [[Definition:Separable Space|separable]]. | From [[Compactness and Sequential Compactness are Equivalent in Metric Spaces]], $\struct {X, d}$ is [[Definition:Sequentially Compact Space|sequentially compact]].
From [[Sequentially Compact Metric Space is Separable]], $\struct {X, d}$ is [[Definition:Separable Space|separable]].
{{qed}}
[[Category:Separable Spac... | Compact Metric Space is Separable | https://proofwiki.org/wiki/Compact_Metric_Space_is_Separable | https://proofwiki.org/wiki/Compact_Metric_Space_is_Separable | [
"Separable Spaces",
"Compact Metric Spaces",
"Metric Spaces"
] | [
"Definition:Compact Space/Metric Space",
"Definition:Separable Space"
] | [
"Compactness and Sequential Compactness are Equivalent in Metric Spaces",
"Definition:Sequentially Compact Space",
"Sequentially Compact Metric Space is Separable",
"Definition:Separable Space",
"Category:Separable Spaces",
"Category:Compact Metric Spaces",
"Category:Metric Spaces"
] |
proofwiki-20892 | Closed Unit Ball in Normed Dual Space of Separable Normed Vector Space is Weak-* Sequentially Compact | Let $X$ be a separable normed vector space.
Let $X^\ast$ be the normed dual space of $X$.
Let $B^-_{X^\ast}$ be the closed unit ball of $X^\ast$.
Let $w^\ast$ be the weak-$\ast$ topology on $B^-_{X^\ast}$.
Then $\struct {B^-_{X^\ast}, w^\ast}$ is sequentially compact. | From the Banach-Alaoglu Theorem, $\struct {B^-_{X^\ast}, w^\ast}$ is compact.
From Closed Ball in Normed Dual Space of Separable Normed Vector Space is Weak-* Metrizable, $\struct {B^-_{X^\ast}, w^\ast}$ is metrizable.
From Compactness and Sequential Compactness are Equivalent in Metric Spaces, $\struct {B^-_{X^\ast}... | Let $X$ be a [[Definition:Separable Space|separable]] [[Definition:Normed Vector Space|normed vector space]].
Let $X^\ast$ be the [[Definition:Normed Dual Space|normed dual space]] of $X$.
Let $B^-_{X^\ast}$ be the [[Definition:Closed Unit Ball|closed unit ball]] of $X^\ast$.
Let $w^\ast$ be the [[Definition:Weak-*... | From the [[Banach-Alaoglu Theorem]], $\struct {B^-_{X^\ast}, w^\ast}$ is [[Definition:Compact Topological Space|compact]].
From [[Closed Ball in Normed Dual Space of Separable Normed Vector Space is Weak-* Metrizable]], $\struct {B^-_{X^\ast}, w^\ast}$ is [[Definition:Metrizable Topology|metrizable]].
From [[Compac... | Closed Unit Ball in Normed Dual Space of Separable Normed Vector Space is Weak-* Sequentially Compact | https://proofwiki.org/wiki/Closed_Unit_Ball_in_Normed_Dual_Space_of_Separable_Normed_Vector_Space_is_Weak-*_Sequentially_Compact | https://proofwiki.org/wiki/Closed_Unit_Ball_in_Normed_Dual_Space_of_Separable_Normed_Vector_Space_is_Weak-*_Sequentially_Compact | [
"Sequentially Compact Spaces",
"Weak-* Topologies"
] | [
"Definition:Separable Space",
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Closed Unit Ball",
"Definition:Weak-* Topology",
"Definition:Sequentially Compact Space"
] | [
"Banach-Alaoglu Theorem",
"Definition:Compact Topological Space",
"Closed Ball in Normed Dual Space of Separable Normed Vector Space is Weak-* Metrizable",
"Definition:Metrizable Space",
"Compactness and Sequential Compactness are Equivalent in Metric Spaces",
"Definition:Sequentially Compact Space",
"C... |
proofwiki-20893 | Normed Dual Space of Separable Normed Vector Space is Weak-* Separable | Let $X$ be a separable normed vector space.
Let $X^\ast$ be the normed dual space of $X$.
Let $w^\ast$ be the weak-$\ast$ topology on $X^\ast$.
Then $\struct {X^\ast, w^\ast}$ is separable. | Let $B^-_{X^\ast}$ be the closed unit ball of $X^\ast$.
From Closed Unit Ball in Normed Dual Space of Separable Normed Vector Space is Weak-* Separable, $\struct {B^-_{X^\ast}, w^\ast}$ is separable space.
Let $S$ be a countable dense subset of $B^-_{X^\ast}$.
For $n \in \N$, we have:
:$\map {\cl_{w^\ast} } {n S} = n... | Let $X$ be a [[Definition:Separable Space|separable]] [[Definition:Normed Vector Space|normed vector space]].
Let $X^\ast$ be the [[Definition:Normed Dual Space|normed dual space]] of $X$.
Let $w^\ast$ be the [[Definition:Weak-* Topology|weak-$\ast$ topology]] on $X^\ast$.
Then $\struct {X^\ast, w^\ast}$ is [[Defi... | Let $B^-_{X^\ast}$ be the [[Definition:Closed Unit Ball|closed unit ball]] of $X^\ast$.
From [[Closed Unit Ball in Normed Dual Space of Separable Normed Vector Space is Weak-* Separable]], $\struct {B^-_{X^\ast}, w^\ast}$ is [[Definition:Separable Space|separable space]].
Let $S$ be a [[Definition:Countable Set|count... | Normed Dual Space of Separable Normed Vector Space is Weak-* Separable | https://proofwiki.org/wiki/Normed_Dual_Space_of_Separable_Normed_Vector_Space_is_Weak-*_Separable | https://proofwiki.org/wiki/Normed_Dual_Space_of_Separable_Normed_Vector_Space_is_Weak-*_Separable | [
"Normed Dual Spaces",
"Separable Spaces",
"Weak-* Topologies"
] | [
"Definition:Separable Space",
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Weak-* Topology",
"Definition:Separable Space"
] | [
"Definition:Closed Unit Ball",
"Closed Unit Ball in Normed Dual Space of Separable Normed Vector Space is Weak-* Separable",
"Definition:Separable Space",
"Definition:Countable Set",
"Definition:Everywhere Dense",
"Dilation of Closure of Set in Topological Vector Space is Closure of Dilation",
"Definiti... |
proofwiki-20894 | Infinite-Dimensional Banach Space has Uncountable Dimension | Let $\Bbb F \in \set {\R, \C}$.
Let $X$ be an infinite-dimensional Banach space over $\Bbb F$.
Let $B$ be a Hamel basis for $X$.
Then $B$ is uncountable.
That is, $\dim X > \aleph_0$.
{{explain|While the title talks about having an uncountable dimension, the result itself talks about the Hamel basis being uncountable... | Suppose that $B$ is countable.
Let:
:$B = \set {e_n : n \in \N}$
For each $n \in \N$, let:
:$F_n = \span \set {e_k : k \le n}$
Since $\dim F_n = n$, $F_n$ clearly has finite dimension.
We show that:
:$\ds X = \bigcup_{n \mathop = 1}^\infty F_n$
Let $x \in X$.
Then there exists $n_1, \ldots, n_k \in \N$ and $\alp... | Let $\Bbb F \in \set {\R, \C}$.
Let $X$ be an [[Definition:Infinite-Dimensional Vector Space|infinite-dimensional]] [[Definition:Banach Space|Banach space]] over $\Bbb F$.
Let $B$ be a [[Definition:Hamel Basis|Hamel basis]] for $X$.
Then $B$ is [[Definition:Uncountable Set|uncountable]].
That is, $\dim X > \alep... | Suppose that $B$ is [[Definition:Countable Set|countable]].
Let:
:$B = \set {e_n : n \in \N}$
For each $n \in \N$, let:
:$F_n = \span \set {e_k : k \le n}$
Since $\dim F_n = n$, $F_n$ clearly has [[Definition:Finite Dimensional Vector Space|finite dimension]].
We show that:
:$\ds X = \bigcup_{n \mathop = 1}... | Infinite-Dimensional Banach Space has Uncountable Dimension | https://proofwiki.org/wiki/Infinite-Dimensional_Banach_Space_has_Uncountable_Dimension | https://proofwiki.org/wiki/Infinite-Dimensional_Banach_Space_has_Uncountable_Dimension | [
"Banach Spaces"
] | [
"Definition:Infinite-Dimensional Vector Space",
"Definition:Banach Space",
"Definition:Basis of Vector Space",
"Definition:Uncountable/Set"
] | [
"Definition:Countable Set",
"Definition:Dimension of Vector Space/Finite",
"Finite Dimensional Subspace of Normed Vector Space is Closed",
"Definition:Closed Set/Normed Vector Space",
"Interior of Proper Subspace of Normed Vector Space is Empty",
"Definition:Empty Set",
"Definition:Interior (Topology)",... |
proofwiki-20895 | Normed Dual Space of Infinite-Dimensional Normed Vector Space is Infinite-Dimensional | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be an infinite dimensional normed vector space.
Let $X^\ast$ be the normed dual space of $X$.
Then $X^\ast$ is infinite dimensional. | {{MissingLinks}}
{{AimForCont}} $X^\ast$ is finite dimensional.
Then from Normed Dual Space of Finite-Dimensional Vector Space is Topologically Isomorphic to Original Space:
:$X^{\ast \ast} \cong X^{\ast}$
From Dimension of Image of Vector Space under Linear Transformation is Bounded Above by Dimension of Vector Space:... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be an [[Definition:Infinite Dimensional Vector Space|infinite dimensional]] [[Definition:Normed Vector Space|normed vector space]].
Let $X^\ast$ be the [[Definition:Normed Dual Space|normed dual space]] of $X$.
Then $X^\ast$ is [[Definition:Infin... | {{MissingLinks}}
{{AimForCont}} $X^\ast$ is [[Definition:Finite Dimensional Vector Space|finite dimensional]].
Then from [[Normed Dual Space of Finite-Dimensional Vector Space is Topologically Isomorphic to Original Space]]:
:$X^{\ast \ast} \cong X^{\ast}$
From [[Dimension of Image of Vector Space under Linear Trans... | Normed Dual Space of Infinite-Dimensional Normed Vector Space is Infinite-Dimensional | https://proofwiki.org/wiki/Normed_Dual_Space_of_Infinite-Dimensional_Normed_Vector_Space_is_Infinite-Dimensional | https://proofwiki.org/wiki/Normed_Dual_Space_of_Infinite-Dimensional_Normed_Vector_Space_is_Infinite-Dimensional | [
"Normed Vector Spaces",
"Normed Dual Spaces"
] | [
"Definition:Infinite Dimensional Vector Space",
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Infinite Dimensional Vector Space"
] | [
"Definition:Dimension of Vector Space/Finite",
"Normed Dual Space of Finite-Dimensional Vector Space is Topologically Isomorphic to Original Space",
"Dimension of Image of Vector Space under Linear Transformation is Bounded Above by Dimension of Vector Space/Corollary",
"Definition:Evaluation Linear Transform... |
proofwiki-20896 | Subset of Normed Vector Space is von Neumann-Bounded in Weak Topology iff Norm Bounded | Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space.
Let $w$ be the weak topology on $\struct {X, \norm {\, \cdot \,}_X}$.
Let $B \subseteq X$.
Then $B$ is bounded in $X$ {{iff}} it is von Neumann-bounded in $\struct {X, w}$. | Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\struct {X^{\ast \ast}, \norm {\, \cdot \,}_{X^{\ast \ast} } }$ be the second normed dual of $\struct {X, \norm {\, \cdot \,}_X}$. | Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $w$ be the [[Definition:Weak Topology on Topological Vector Space|weak topology]] on $\struct {X, \norm {\, \cdot \,}_X}$.
Let $B \subseteq X$.
Then $B$ is [[Definition:Bounded Subset of Normed Vector Space|... | Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\struct {X^{\ast \ast}, \norm {\, \cdot \,}_{X^{\ast \ast} } }$ be the [[Definition:Second Normed Dual|second normed dual]] of $\struct {X, \norm {\, \cdot \,}_X... | Subset of Normed Vector Space is von Neumann-Bounded in Weak Topology iff Norm Bounded | https://proofwiki.org/wiki/Subset_of_Normed_Vector_Space_is_von_Neumann-Bounded_in_Weak_Topology_iff_Norm_Bounded | https://proofwiki.org/wiki/Subset_of_Normed_Vector_Space_is_von_Neumann-Bounded_in_Weak_Topology_iff_Norm_Bounded | [
"Weak Topologies on Topological Vector Spaces",
"Von Neumann-Bounded Subsets of Topological Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Bounded Subset of Normed Vector Space",
"Definition:Von Neumann-Bounded Subset of Topological Vector Space"
] | [
"Definition:Normed Dual Space",
"Definition:Second Normed Dual"
] |
proofwiki-20897 | Union of Image of Convergent Sequence and Limit in Topological Space is Compact | Let $\struct {X, \tau}$ be a topological space.
Let $\sequence {x_n}_{n \in \N}$ be a convergent sequence such that:
:$x_n \to x$
Then:
:$\ds \set {x_n : n \in \N} \cup \set x$ is compact. | Let $\family {U_\alpha}_{\alpha \in A}$ be an open cover for $\ds \set {x_n : n \in \N} \cup \set x$.
Take $\beta \in A$ be such that $x \in U_\beta$.
Since $x_n \to x$, there exists $N \in \N$ such that $x_n \in U_\beta$ for $n \ge N$.
For each $n < N$, we can select $U_{\alpha_1}, \ldots, U_{\alpha_{N - 1} }$ such ... | Let $\struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\sequence {x_n}_{n \in \N}$ be a [[Definition:Convergent Sequence|convergent sequence]] such that:
:$x_n \to x$
Then:
:$\ds \set {x_n : n \in \N} \cup \set x$ is [[Definition:Compact Topological Space|compact]]. | Let $\family {U_\alpha}_{\alpha \in A}$ be an [[Definition:Open Cover|open cover]] for $\ds \set {x_n : n \in \N} \cup \set x$.
Take $\beta \in A$ be such that $x \in U_\beta$.
Since $x_n \to x$, there exists $N \in \N$ such that $x_n \in U_\beta$ for $n \ge N$.
For each $n < N$, we can select $U_{\alpha_1}, \ldot... | Union of Image of Convergent Sequence and Limit in Topological Space is Compact | https://proofwiki.org/wiki/Union_of_Image_of_Convergent_Sequence_and_Limit_in_Topological_Space_is_Compact | https://proofwiki.org/wiki/Union_of_Image_of_Convergent_Sequence_and_Limit_in_Topological_Space_is_Compact | [
"Compact Topological Spaces"
] | [
"Definition:Topological Space",
"Definition:Convergent Sequence",
"Definition:Compact Topological Space"
] | [
"Definition:Open Cover",
"Definition:Open Cover",
"Definition:Subcover/Finite",
"Definition:Compact Topological Space",
"Category:Compact Topological Spaces"
] |
proofwiki-20898 | Characterization of von Neumann-Boundedness in terms of Local Basis | Let $\struct {X, \tau}$ be a topological vector space.
Let $\mathcal B$ be a local basis for $\mathbf 0_X$ in $\struct {X, \tau}$.
Let $E \subseteq X$.
Then $E$ is von Neumann-bounded {{iff}}:
:for each $V \in \mathcal B$ there exists $s > 0$ such that $E \subseteq t V$ for each $t > s$. | === Necessary Condition ===
Suppose that $E$ is von Neumann-bounded.
Then:
:for each open neighbourhood $V$ of ${\mathbf 0}_X$, there exists $s > 0$ such that $E \subseteq t V$ for each $t > s$
Since $\mathcal B$ consists of open neighbourhoods of ${\mathbf 0}_X$, we in particular have:
:for each $V \in \mathcal B$ t... | Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]].
Let $\mathcal B$ be a [[Definition:Local Basis|local basis]] for $\mathbf 0_X$ in $\struct {X, \tau}$.
Let $E \subseteq X$.
Then $E$ is [[Definition:Von Neumann-Bounded Subset of Topological Vector Space|von Neumann-boun... | === Necessary Condition ===
Suppose that $E$ is [[Definition:Von Neumann-Bounded Subset of Topological Vector Space|von Neumann-bounded]].
Then:
:for each [[Definition:Open Neighborhood|open neighbourhood]] $V$ of ${\mathbf 0}_X$, there exists $s > 0$ such that $E \subseteq t V$ for each $t > s$
Since $\mathcal B$... | Characterization of von Neumann-Boundedness in terms of Local Basis | https://proofwiki.org/wiki/Characterization_of_von_Neumann-Boundedness_in_terms_of_Local_Basis | https://proofwiki.org/wiki/Characterization_of_von_Neumann-Boundedness_in_terms_of_Local_Basis | [
"Von Neumann-Bounded Subsets of Topological Vector Spaces",
"Local Bases"
] | [
"Definition:Topological Vector Space",
"Definition:Local Basis",
"Definition:Von Neumann-Bounded Subset of Topological Vector Space"
] | [
"Definition:Von Neumann-Bounded Subset of Topological Vector Space",
"Definition:Open Neighborhood",
"Definition:Open Neighborhood",
"Definition:Open Neighborhood",
"Definition:Von Neumann-Bounded Subset of Topological Vector Space"
] |
proofwiki-20899 | Characterization of Cauchy Sequence in Topological Vector Space in terms of Local Basis | Let $\struct {X, \tau}$ be a topological vector space.
Let $\BB$ be a local basis for $\mathbf 0_X$ in $\struct {X, \tau}$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence.
Then $\sequence {x_n}_{n \mathop \in \N}$ is Cauchy {{iff}}:
:for each $V \in \BB$ there exists $N \in \N$ such that $x_n - x_m \in V$ for ... | === Necessary Condition ===
Suppose that $\sequence {x_n}_{n \mathop \in \N}$ is Cauchy.
Then:
:for each open neighborhood $V$ of $\mathbf 0_X$ there exists $N \in \N$ such that:
::$x_n - x_m \in V$ for each $n, m \ge N$.
Since $\BB$ consists of open neighbourhoods of $\mathbf 0_X$, we in particular have:
:for each ... | Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]].
Let $\BB$ be a [[Definition:Local Basis|local basis]] for $\mathbf 0_X$ in $\struct {X, \tau}$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]].
Then $\sequence {x_n}_{n \mathop \in \N}$ is... | === Necessary Condition ===
Suppose that $\sequence {x_n}_{n \mathop \in \N}$ is [[Definition:Cauchy Sequence in Topological Vector Space|Cauchy]].
Then:
:for each [[Definition:Open Neighborhood|open neighborhood]] $V$ of $\mathbf 0_X$ there exists $N \in \N$ such that:
::$x_n - x_m \in V$ for each $n, m \ge N$. ... | Characterization of Cauchy Sequence in Topological Vector Space in terms of Local Basis | https://proofwiki.org/wiki/Characterization_of_Cauchy_Sequence_in_Topological_Vector_Space_in_terms_of_Local_Basis | https://proofwiki.org/wiki/Characterization_of_Cauchy_Sequence_in_Topological_Vector_Space_in_terms_of_Local_Basis | [
"Local Bases",
"Cauchy Sequences in Topological Vector Spaces"
] | [
"Definition:Topological Vector Space",
"Definition:Local Basis",
"Definition:Sequence",
"Definition:Cauchy Sequence/Topological Vector Space"
] | [
"Definition:Cauchy Sequence/Topological Vector Space",
"Definition:Open Neighborhood",
"Definition:Open Neighborhood",
"Definition:Open Neighborhood",
"Definition:Cauchy Sequence/Topological Vector Space"
] |
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