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 |
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proofwiki-14000 | Condition for Alexandroff Extension to be T1 | Let $T = \struct {S, \tau}$ be a non-empty topological space.
Let $p$ be a new element not in $S$.
Let $S^* := S \cup \set p$.
Let $T^* = \struct {S^*, \tau^*}$ be the Alexandroff extension on $S$.
Then $T^*$ is a $T_1$ space {{iff}} $T$ is a $T_1$ space. | === Necessary Condition ===
Let $T = \struct {S, \tau}$ be a $T_1$ space.
By definition, $T$ is a $T_1$ space {{iff}} all points of $S$ are closed in $T$.
We have that $S$ is open in $T$ by definition of a topology.
Thus by definition of the Alexandroff extension, $S$ is open in $T^*$.
So as $S = S^* \setminus \set p$ ... | Let $T = \struct {S, \tau}$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Topological Space|topological space]].
Let $p$ be a [[Definition:New Element|new element]] not in $S$.
Let $S^* := S \cup \set p$.
Let $T^* = \struct {S^*, \tau^*}$ be the [[Definition:Alexandroff Extension|Alexandroff extension]] o... | === Necessary Condition ===
Let $T = \struct {S, \tau}$ be a [[Definition:T1 Space|$T_1$ space]].
By definition, $T$ is a [[Definition:T1 Space|$T_1$ space]] {{iff}} all points of $S$ are [[Definition:Closed Point|closed]] in $T$.
We have that $S$ is [[Definition:Open Set (Topology)|open]] in $T$ by definition of a... | Condition for Alexandroff Extension to be T1 | https://proofwiki.org/wiki/Condition_for_Alexandroff_Extension_to_be_T1 | https://proofwiki.org/wiki/Condition_for_Alexandroff_Extension_to_be_T1 | [
"Alexandroff Extensions",
"Examples of T1 Spaces"
] | [
"Definition:Non-Empty Set",
"Definition:Topological Space",
"Definition:New Element",
"Definition:Alexandroff Extension",
"Definition:T1 Space",
"Definition:T1 Space"
] | [
"Definition:T1 Space",
"Definition:T1 Space",
"Definition:Closed Point",
"Definition:Open Set/Topology",
"Definition:Topology",
"Definition:Alexandroff Extension",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Closed Point",
"Defini... |
proofwiki-14001 | Condition for Alexandroff Extension to be T2 | Let $T = \struct {S, \tau}$ be a non-empty topological space.
Let $p$ be a new element not in $S$.
Let $S^* := S \cup \set p$.
Let $T^* =\struct {S^*, \tau^*}$ be the Alexandroff extension on $S$.
Then $T^*$ is a $T_2$ (Hausdorff) space {{iff}} $T$ is a locally compact Hausdorff space. | === Necessary Condition ===
Let $T = \struct {S, \tau}$ be a locally compact Hausdorff space.
Let $x, y \in S$.
Then as $T$ is a $T_2$ space, there exist two disjoint open sets $U, V \in \tau$ containing $x$ and $y$ respectively.
But by definition of the Alexandroff extension on $S$, $U$ and $V$ are also open sets of $... | Let $T = \struct {S, \tau}$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Topological Space|topological space]].
Let $p$ be a [[Definition:New Element|new element]] not in $S$.
Let $S^* := S \cup \set p$.
Let $T^* =\struct {S^*, \tau^*}$ be the [[Definition:Alexandroff Extension|Alexandroff extension]] on... | === Necessary Condition ===
Let $T = \struct {S, \tau}$ be a [[Definition:Locally Compact Hausdorff Space|locally compact Hausdorff space]].
Let $x, y \in S$.
Then as $T$ is a [[Definition:T2 Space|$T_2$ space]], there exist two [[Definition:Disjoint Sets|disjoint]] [[Definition:Open Set (Topology)|open sets]] $U, V... | Condition for Alexandroff Extension to be T2 | https://proofwiki.org/wiki/Condition_for_Alexandroff_Extension_to_be_T2 | https://proofwiki.org/wiki/Condition_for_Alexandroff_Extension_to_be_T2 | [
"Alexandroff Extensions",
"Examples of Hausdorff Spaces",
"Examples of Locally Compact Hausdorff Spaces"
] | [
"Definition:Non-Empty Set",
"Definition:Topological Space",
"Definition:New Element",
"Definition:Alexandroff Extension",
"Definition:T2 Space",
"Definition:Locally Compact Hausdorff Space"
] | [
"Definition:Locally Compact Hausdorff Space",
"Definition:T2 Space",
"Definition:Disjoint Sets",
"Definition:Open Set/Topology",
"Definition:Alexandroff Extension",
"Definition:Open Set/Topology",
"Definition:Locally Compact Hausdorff Space",
"Definition:Compact Topological Space/Subspace",
"Definit... |
proofwiki-14002 | Alexandroff Extension which is T2 is also T4 | Let $T = \struct {S, \tau}$ be a non-empty topological space.
Let $p$ be a new element not in $S$.
Let $S^* := S \cup \set p$.
Let $T^* = \struct {S^*, \tau^*}$ be the Alexandroff extension on $S$.
Let $T^*$ be a $T_2$ (Hausdorff) space.
Then $T^*$ is a $T_4$ space. | We have:
:Alexandroff Extension is Compact
:Compact Hausdorff Space is $T_4$.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Topological Space|topological space]].
Let $p$ be a [[Definition:New Element|new element]] not in $S$.
Let $S^* := S \cup \set p$.
Let $T^* = \struct {S^*, \tau^*}$ be the [[Definition:Alexandroff Extension|Alexandroff extension]] o... | We have:
:[[Alexandroff Extension is Compact]]
:[[Compact Hausdorff Space is T4|Compact Hausdorff Space is $T_4$]].
{{qed}} | Alexandroff Extension which is T2 is also T4 | https://proofwiki.org/wiki/Alexandroff_Extension_which_is_T2_is_also_T4 | https://proofwiki.org/wiki/Alexandroff_Extension_which_is_T2_is_also_T4 | [
"Alexandroff Extensions",
"Examples of Hausdorff Spaces",
"Examples of T4 Spaces"
] | [
"Definition:Non-Empty Set",
"Definition:Topological Space",
"Definition:New Element",
"Definition:Alexandroff Extension",
"Definition:T2 Space",
"Definition:T4 Space"
] | [
"Alexandroff Extension is Compact",
"Compact Hausdorff Space is T4"
] |
proofwiki-14003 | Alexandroff Extension of Rational Number Space is not T2 | Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.
Let $p$ be a new element not in $\Q$.
Let $\Q^* := \Q \cup \set p$.
Let $T^* = \struct {\Q^*, \tau^*}$ be the Alexandroff extension on $\struct {\Q, \tau_d}$.
Then $T^*$ is not a $T_2$ (Hausdorff) space. | From Condition for Alexandroff Extension to be $T_2$, $T^*$ is a $T_2$ space {{iff}} $\struct {\Q, \tau_d}$ is a locally compact Hausdorff Space.
But from Rational Number Space is not Locally Compact Hausdorff Space, $\struct {\Q, \tau_d}$ is not a locally compact Hausdorff Space.
Hence the result.
{{qed}} | Let $\struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Let $p$ be a [[Definition:New Element|new element]] not in $\Q$.
Let $\Q^* := \Q \cup \set p$.
Let $T^* = \struct {\Q^*, \tau^*}$ b... | From [[Condition for Alexandroff Extension to be T2|Condition for Alexandroff Extension to be $T_2$]], $T^*$ is a [[Definition:T2 Space|$T_2$ space]] {{iff}} $\struct {\Q, \tau_d}$ is a [[Definition:Locally Compact Hausdorff Space|locally compact Hausdorff Space]].
But from [[Rational Number Space is not Locally Compa... | Alexandroff Extension of Rational Number Space is not T2 | https://proofwiki.org/wiki/Alexandroff_Extension_of_Rational_Number_Space_is_not_T2 | https://proofwiki.org/wiki/Alexandroff_Extension_of_Rational_Number_Space_is_not_T2 | [
"Alexandroff Extensions",
"Rational Number Space",
"Examples of Hausdorff Spaces"
] | [
"Definition:Rational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:New Element",
"Definition:Alexandroff Extension",
"Definition:T2 Space"
] | [
"Condition for Alexandroff Extension to be T2",
"Definition:T2 Space",
"Definition:Locally Compact Hausdorff Space",
"Rational Number Space is not Locally Compact Hausdorff Space",
"Definition:Locally Compact Hausdorff Space"
] |
proofwiki-14004 | Alexandroff Extension of Rational Number Space is T1 | Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.
Let $p$ be a new element not in $\Q$.
Let $\Q^* := \Q \cup \set p$.
Let $T^* = \struct {\Q^*, \tau^*}$ be the Alexandroff extension on $\struct {\Q, \tau_d}$.
Then $T^*$ is a $T_1$ space. | From Condition for Alexandroff Extension to be $T_1$, $T^*$ is a $T_1$ space {{iff}} $\struct {\Q, \tau_d}$ is also a $T_1$ space.
From Rational Numbers form Metric Space, $\struct {\Q, d}$ is a metric space.
From Metric Space is $T_1$, $\struct {\Q, \tau_d}$ is a $T_1$ space.
Hence the result.
{{qed}} | Let $\struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Let $p$ be a [[Definition:New Element|new element]] not in $\Q$.
Let $\Q^* := \Q \cup \set p$.
Let $T^* = \struct {\Q^*, \tau^*}$ b... | From [[Condition for Alexandroff Extension to be T1|Condition for Alexandroff Extension to be $T_1$]], $T^*$ is a [[Definition:T1 Space|$T_1$ space]] {{iff}} $\struct {\Q, \tau_d}$ is also a [[Definition:T1 Space|$T_1$ space]].
From [[Rational Numbers form Metric Space]], $\struct {\Q, d}$ is a [[Definition:Metric Spa... | Alexandroff Extension of Rational Number Space is T1 | https://proofwiki.org/wiki/Alexandroff_Extension_of_Rational_Number_Space_is_T1 | https://proofwiki.org/wiki/Alexandroff_Extension_of_Rational_Number_Space_is_T1 | [
"Alexandroff Extensions",
"Rational Number Space",
"Examples of T1 Spaces"
] | [
"Definition:Rational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:New Element",
"Definition:Alexandroff Extension",
"Definition:T1 Space"
] | [
"Condition for Alexandroff Extension to be T1",
"Definition:T1 Space",
"Definition:T1 Space",
"Rational Numbers form Metric Space",
"Definition:Metric Space",
"Metric Space is T1",
"Definition:T1 Space"
] |
proofwiki-14005 | Separation Properties of Alexandroff Extension of Rational Number Space | Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.
Let $p$ be a new element not in $\Q$.
Let $\Q^* := \Q \cup \set p$.
Let $T^* = \struct {\Q^*, \tau^*}$ be the Alexandroff extension on $\struct {\Q, \tau_d}$.
Then $T^*$ satisfies no Tychonoff separation axioms higher than a ... | From Alexandroff Extension of Rational Number Space is $T_1$, $T^*$ is a $T_1$ space.
From Alexandroff Extension of Rational Number Space is not T2, $T^*$ is not a $T_2$ (Hausdorff) space.
From $T_{2 \frac 1 2}$ Space is $T_2$, $T^*$ is not a $T_{2 \frac 1 2}$ space.
{{Recall|Semiregular Space|semiregular space}}
{{:De... | Let $\struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Let $p$ be a [[Definition:New Element|new element]] not in $\Q$.
Let $\Q^* := \Q \cup \set p$.
Let $T^* = \struct {\Q^*, \tau^*}$ b... | From [[Alexandroff Extension of Rational Number Space is T1|Alexandroff Extension of Rational Number Space is $T_1$]], $T^*$ is a [[Definition:T1 Space|$T_1$ space]].
From [[Alexandroff Extension of Rational Number Space is not T2]], $T^*$ is not a [[Definition:T2 Space|$T_2$ (Hausdorff) space]].
From [[T2.5 Space is... | Separation Properties of Alexandroff Extension of Rational Number Space | https://proofwiki.org/wiki/Separation_Properties_of_Alexandroff_Extension_of_Rational_Number_Space | https://proofwiki.org/wiki/Separation_Properties_of_Alexandroff_Extension_of_Rational_Number_Space | [
"Alexandroff Extensions",
"Rational Number Space",
"Examples of Separation Axioms"
] | [
"Definition:Rational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:New Element",
"Definition:Alexandroff Extension",
"Definition:Tychonoff Separation Axioms",
"Definition:T1 Space"
] | [
"Alexandroff Extension of Rational Number Space is T1",
"Definition:T1 Space",
"Alexandroff Extension of Rational Number Space is not T2",
"Definition:T2 Space",
"T2.5 Space is T2",
"Definition:T2.5 Space",
"Definition:T2 Space",
"Definition:Semiregular Space",
"Regular Space is T2.5",
"Definition... |
proofwiki-14006 | Particular Point of Alexandroff Extension of Rational Number Space is Dispersion Point | Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.
Let $p$ be a new element not in $\Q$.
Let $\Q^* := \Q \cup \set p$.
Let $T^* = \struct {\Q^*, \tau^*}$ be the Alexandroff extension on $\left({\Q, \tau_d}\right)$.
Then $p$ is a dispersion point of $T^*$. | By definition, $p$ is a dispersion point of $T^*$ {{iff}}:
:$\Q^*$ is a connected set in $T^*$
:$\Q^* \setminus \set p$ is totally disconnected in $T^*$.
From Alexandroff Extension of Rational Number Space is Connected, $\Q^*$ is a connected set in $T^*$.
It remains to be shown that $\Q^* \setminus \set p$ is totally d... | Let $\struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Let $p$ be a [[Definition:New Element|new element]] not in $\Q$.
Let $\Q^* := \Q \cup \set p$.
Let $T^* = \struct {\Q^*, \tau^*}$ b... | By definition, $p$ is a [[Definition:Dispersion Point|dispersion point]] of $T^*$ {{iff}}:
:$\Q^*$ is a [[Definition:Connected Set (Topology)|connected set]] in $T^*$
:$\Q^* \setminus \set p$ is [[Definition:Totally Disconnected Space|totally disconnected]] in $T^*$.
From [[Alexandroff Extension of Rational Number Spa... | Particular Point of Alexandroff Extension of Rational Number Space is Dispersion Point | https://proofwiki.org/wiki/Particular_Point_of_Alexandroff_Extension_of_Rational_Number_Space_is_Dispersion_Point | https://proofwiki.org/wiki/Particular_Point_of_Alexandroff_Extension_of_Rational_Number_Space_is_Dispersion_Point | [
"Alexandroff Extensions",
"Rational Number Space",
"Examples of Dispersion Points"
] | [
"Definition:Rational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:New Element",
"Definition:Alexandroff Extension",
"Definition:Dispersion Point"
] | [
"Definition:Dispersion Point",
"Definition:Connected Set (Topology)",
"Definition:Totally Disconnected Space",
"Alexandroff Extension of Rational Number Space is Connected",
"Definition:Connected Set (Topology)",
"Definition:Totally Disconnected Space",
"Definition:Rational Number Space",
"Rational Nu... |
proofwiki-14007 | Alexandroff Extension of Rational Number Space is Connected | Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.
Let $p$ be a new element not in $\Q$.
Let $\Q^* := \Q \cup \set p$.
Let $T^* = \struct {\Q^*, \tau^*}$ be the Alexandroff extension on $\struct {\Q, \tau_d}$.
Then $T^*$ is a connected space. | By definition $T^*$ is a connected space of $T^*$ {{iff}} it admits no separation.
{{AimForCont}} $T^*$ does admit a separation.
That is, there exist open sets $A, B \in \tau^*$ such that $A, B \ne \O$, $A \cup B = \Q^*$ and $A \cap B = \O$.
That is, both $A$ and $B = \relcomp {\Q^*} A$ are open in $T^*$.
{{WLOG}}, Let... | Let $\struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Let $p$ be a [[Definition:New Element|new element]] not in $\Q$.
Let $\Q^* := \Q \cup \set p$.
Let $T^* = \struct {\Q^*, \tau^*}$ b... | By definition $T^*$ is a [[Definition:Connected Topological Space|connected space]] of $T^*$ {{iff}} it admits no [[Definition:Separation (Topology)|separation]].
{{AimForCont}} $T^*$ does admit a [[Definition:Separation (Topology)|separation]].
That is, there exist [[Definition:Open Set (Topology)|open sets]] $A, B... | Alexandroff Extension of Rational Number Space is Connected | https://proofwiki.org/wiki/Alexandroff_Extension_of_Rational_Number_Space_is_Connected | https://proofwiki.org/wiki/Alexandroff_Extension_of_Rational_Number_Space_is_Connected | [
"Alexandroff Extensions",
"Rational Number Space",
"Examples of Connected Topological Spaces"
] | [
"Definition:Rational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:New Element",
"Definition:Alexandroff Extension",
"Definition:Connected Topological Space"
] | [
"Definition:Connected Topological Space",
"Definition:Separation (Topology)",
"Definition:Separation (Topology)",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Compact Topological Space/Subspace",
"Compact Set of Rational Numbers is Nowhe... |
proofwiki-14008 | Alexandroff Extension of Rational Number Space is Biconnected | Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.
Let $p$ be a new element not in $\Q$.
Let $\Q^* := \Q \cup \set p$.
Let $T^* = \struct {\Q^*, \tau^*}$ be the Alexandroff extension on $\struct {\Q, \tau_d}$.
Then $T^*$ is a biconnected space. | From Particular Point of Alexandroff Extension of Rational Number Space is Dispersion Point, $p$ is a dispersion point of $T^*$.
The result follows from Set with Dispersion Point is Biconnected.
{{qed}} | Let $\struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Let $p$ be a [[Definition:New Element|new element]] not in $\Q$.
Let $\Q^* := \Q \cup \set p$.
Let $T^* = \struct {\Q^*, \tau^*}$ b... | From [[Particular Point of Alexandroff Extension of Rational Number Space is Dispersion Point]], $p$ is a [[Definition:Dispersion Point|dispersion point]] of $T^*$.
The result follows from [[Set with Dispersion Point is Biconnected]].
{{qed}} | Alexandroff Extension of Rational Number Space is Biconnected | https://proofwiki.org/wiki/Alexandroff_Extension_of_Rational_Number_Space_is_Biconnected | https://proofwiki.org/wiki/Alexandroff_Extension_of_Rational_Number_Space_is_Biconnected | [
"Alexandroff Extensions",
"Rational Number Space",
"Examples of Biconnected Sets"
] | [
"Definition:Rational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:New Element",
"Definition:Alexandroff Extension",
"Definition:Biconnected Set"
] | [
"Particular Point of Alexandroff Extension of Rational Number Space is Dispersion Point",
"Definition:Dispersion Point",
"Set with Dispersion Point is Biconnected"
] |
proofwiki-14009 | Alexandroff Extension of Rational Number Space is Sequentially Compact | Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.
Let $p$ be a new element not in $\Q$.
Let $\Q^* := \Q \cup \set p$.
Let $T^* = \struct {\Q^*, \tau^*}$ be the Alexandroff extension on $\struct {\Q, \tau_d}$.
Then $T^*$ is a sequentially compact space. | The strategy here is to demonstrate that every sequence in $T^*$ is either contained in a compact subspace of $T^*$, or must contain a subsequence which converges to $p$.
{{ProofWanted}} | Let $\struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Let $p$ be a [[Definition:New Element|new element]] not in $\Q$.
Let $\Q^* := \Q \cup \set p$.
Let $T^* = \struct {\Q^*, \tau^*}$ b... | The strategy here is to demonstrate that every [[Definition:Sequence|sequence]] in $T^*$ is either contained in a [[Definition:Compact Topological Subspace|compact subspace]] of $T^*$, or must contain a [[Definition:Subsequence|subsequence]] which [[Definition:Convergent Sequence (Topology)|converges]] to $p$.
{{Proof... | Alexandroff Extension of Rational Number Space is Sequentially Compact | https://proofwiki.org/wiki/Alexandroff_Extension_of_Rational_Number_Space_is_Sequentially_Compact | https://proofwiki.org/wiki/Alexandroff_Extension_of_Rational_Number_Space_is_Sequentially_Compact | [
"Alexandroff Extensions",
"Rational Number Space",
"Examples of Sequentially Compact Spaces"
] | [
"Definition:Rational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:New Element",
"Definition:Alexandroff Extension",
"Definition:Sequentially Compact Space"
] | [
"Definition:Sequence",
"Definition:Compact Topological Space/Subspace",
"Definition:Subsequence",
"Definition:Convergent Sequence/Topology"
] |
proofwiki-14010 | Hilbert Sequence Space is Complete Metric Space | Let $A$ be the set of all real sequences $\left\langle{x_i}\right\rangle$ such that the series $\ds \sum_{i \mathop = 0}^\infty x_i^2$ is convergent.
Let $\ell^2 = \struct {A, d_2}$ be the Hilbert sequence space on $\R$.
Then $\ell^2$ is a complete metric space. | We have that Hilbert Sequence Space is Metric Space.
It remains to be shown that it is complete.
Recall that from Real Number Line is Complete Metric Space, $\R$ is a complete metric space.
Let $x^1, x^2, x^3, \ldots$ be a Cauchy sequence $\ell^2$.
Then for each $i \in \N_{>0}$, we have that $\sequence { {x_i}^j}_{j \m... | Let $A$ be the [[Definition:Set|set]] of all [[Definition:Real Sequence|real sequences]] $\left\langle{x_i}\right\rangle$ such that the [[Definition:Series of Numbers|series]] $\ds \sum_{i \mathop = 0}^\infty x_i^2$ is [[Definition:Convergent Series of Numbers|convergent]].
Let $\ell^2 = \struct {A, d_2}$ be the [[Def... | We have that [[Hilbert Sequence Space is Metric Space]].
It remains to be shown that it is [[Definition:Complete Metric Space|complete]].
Recall that from [[Real Number Line is Complete Metric Space]], $\R$ is a [[Definition:Complete Metric Space|complete metric space]].
Let $x^1, x^2, x^3, \ldots$ be a [[Definitio... | Hilbert Sequence Space is Complete Metric Space | https://proofwiki.org/wiki/Hilbert_Sequence_Space_is_Complete_Metric_Space | https://proofwiki.org/wiki/Hilbert_Sequence_Space_is_Complete_Metric_Space | [
"Hilbert Sequence Space",
"Examples of Complete Metric Spaces"
] | [
"Definition:Set",
"Definition:Real Sequence",
"Definition:Series/Number Field",
"Definition:Convergent Series/Number Field",
"Definition:Hilbert Sequence Space",
"Definition:Complete Metric Space"
] | [
"Hilbert Sequence Space is Metric Space",
"Definition:Complete Metric Space",
"Real Number Line is Complete Metric Space",
"Definition:Complete Metric Space",
"Definition:Cauchy Sequence/Metric Space",
"Definition:Cauchy Sequence/Metric Space",
"Definition:Complete Metric Space",
"Definition:Convergen... |
proofwiki-14011 | Hilbert Sequence Space is Separable | Let $A$ be the set of all real sequences $\sequence {x_i}$ such that the series $\ds \sum_{i \mathop \ge 0} x_i^2$ is convergent.
Let $\ell^2 = \struct {A, d_2}$ be the Hilbert sequence space on $\R$.
Then $\ell^2$ is a separable space. | Consider the set $H$ of all points of $\ell^2$ which have finitely many rational coordinates and all the rest zero.
$H$ is countable, since
:Rational Numbers are Countably Infinite
:Cartesian Product of Countable Sets is Countable
:Countable Union of Countable Sets is Countable
It remains to show that $H$ is everywhere... | Let $A$ be the [[Definition:Set|set]] of all [[Definition:Real Sequence|real sequences]] $\sequence {x_i}$ such that the [[Definition:Series of Numbers|series]] $\ds \sum_{i \mathop \ge 0} x_i^2$ is [[Definition:Convergent Series of Numbers|convergent]].
Let $\ell^2 = \struct {A, d_2}$ be the [[Definition:Hilbert Sequ... | Consider the [[Definition:Set|set]] $H$ of all [[Definition:Point of Set|points]] of $\ell^2$ which have [[Definition:Finite Set|finitely many]] [[Definition:Rational Number|rational]] [[Definition:Coordinate of Ordered Tuple|coordinates]] and all the rest [[Definition:Zero (Number)|zero]].
$H$ is [[Definition:Countab... | Hilbert Sequence Space is Separable | https://proofwiki.org/wiki/Hilbert_Sequence_Space_is_Separable | https://proofwiki.org/wiki/Hilbert_Sequence_Space_is_Separable | [
"Hilbert Sequence Space",
"Examples of Separable Spaces"
] | [
"Definition:Set",
"Definition:Real Sequence",
"Definition:Series/Number Field",
"Definition:Convergent Series/Number Field",
"Definition:Hilbert Sequence Space",
"Definition:Separable Space"
] | [
"Definition:Set",
"Definition:Element",
"Definition:Finite Set",
"Definition:Rational Number",
"Definition:Cartesian Product/Coordinate",
"Definition:Zero (Number)",
"Definition:Countable Set",
"Rational Numbers are Countably Infinite",
"Cartesian Product of Countable Sets is Countable",
"Countabl... |
proofwiki-14012 | Hilbert Sequence Space is Second-Countable | Let $\ell^2$ be the Hilbert sequence space on $\R$.
Then $\ell^2$ is a second-countable space. | From Hilbert Sequence Space is Separable, $\ell^2$ is a separable space.
We also have that Hilbert Sequence Space is Metric Space.
The result follows from Separable Metric Space is Second-Countable.
{{qed}} | Let $\ell^2$ be the [[Definition:Hilbert Sequence Space|Hilbert sequence space on $\R$]].
Then $\ell^2$ is a [[Definition:Second-Countable Space|second-countable space]]. | From [[Hilbert Sequence Space is Separable]], $\ell^2$ is a [[Definition:Separable Space|separable space]].
We also have that [[Hilbert Sequence Space is Metric Space]].
The result follows from [[Separable Metric Space is Second-Countable]].
{{qed}} | Hilbert Sequence Space is Second-Countable | https://proofwiki.org/wiki/Hilbert_Sequence_Space_is_Second-Countable | https://proofwiki.org/wiki/Hilbert_Sequence_Space_is_Second-Countable | [
"Hilbert Sequence Space",
"Examples of Second-Countable Spaces"
] | [
"Definition:Hilbert Sequence Space",
"Definition:Second-Countable Space"
] | [
"Hilbert Sequence Space is Separable",
"Definition:Separable Space",
"Hilbert Sequence Space is Metric Space",
"Separable Metric Space is Second-Countable"
] |
proofwiki-14013 | Hilbert Sequence Space is Lindelöf | Let $\ell^2$ be the Hilbert sequence space on $\R$.
Then $\ell^2$ is a Lindelöf space. | From Hilbert Sequence Space is Second-Countable, $\ell^2$ is a second-countable space.
The result follows from Second-Countable Space is Lindelöf.
{{qed}} | Let $\ell^2$ be the [[Definition:Hilbert Sequence Space|Hilbert sequence space on $\R$]].
Then $\ell^2$ is a [[Definition:Lindelöf Space|Lindelöf space]]. | From [[Hilbert Sequence Space is Second-Countable]], $\ell^2$ is a [[Definition:Second-Countable Space|second-countable space]].
The result follows from [[Second-Countable Space is Lindelöf]].
{{qed}} | Hilbert Sequence Space is Lindelöf | https://proofwiki.org/wiki/Hilbert_Sequence_Space_is_Lindelöf | https://proofwiki.org/wiki/Hilbert_Sequence_Space_is_Lindelöf | [
"Hilbert Sequence Space",
"Examples of Lindelöf Spaces"
] | [
"Definition:Hilbert Sequence Space",
"Definition:Lindelöf Space"
] | [
"Hilbert Sequence Space is Second-Countable",
"Definition:Second-Countable Space",
"Second-Countable Space is Lindelöf"
] |
proofwiki-14014 | Hilbert Sequence Space is not Locally Compact Hausdorff Space | Let $A$ be the set of all real sequences $\sequence {x_i}$ such that the series $\ds \sum_{i \mathop \ge 0} {x_i}^2$ is convergent.
Let $\ell^2 = \struct {A, d_2}$ be the Hilbert sequence space on $\R$.
Then $\ell^2$ is not a locally compact Hausdorff space. | From Hilbert Sequence Space is Metric Space, $\ell^2$ is a metric space.
From Metric Space is $T_2$, $\ell^2$ is a Hausdorff space.
Let $x = \sequence {x_i} \in A$ be a point of $\ell^2$.
From Point in Hilbert Sequence Space has no Compact Neighborhood, $x$ has no compact neighborhood.
Hence the result by definition of... | Let $A$ be the [[Definition:Set|set]] of all [[Definition:Real Sequence|real sequences]] $\sequence {x_i}$ such that the [[Definition:Series of Numbers|series]] $\ds \sum_{i \mathop \ge 0} {x_i}^2$ is [[Definition:Convergent Series of Numbers|convergent]].
Let $\ell^2 = \struct {A, d_2}$ be the [[Definition:Hilbert Se... | From [[Hilbert Sequence Space is Metric Space]], $\ell^2$ is a [[Definition:Metric Space|metric space]].
From [[Metric Space is T2|Metric Space is $T_2$]], $\ell^2$ is a [[Definition:Hausdorff Space|Hausdorff space]].
Let $x = \sequence {x_i} \in A$ be a point of $\ell^2$.
From [[Point in Hilbert Sequence Space has... | Hilbert Sequence Space is not Locally Compact Hausdorff Space | https://proofwiki.org/wiki/Hilbert_Sequence_Space_is_not_Locally_Compact_Hausdorff_Space | https://proofwiki.org/wiki/Hilbert_Sequence_Space_is_not_Locally_Compact_Hausdorff_Space | [
"Hilbert Sequence Space",
"Examples of Locally Compact Hausdorff Spaces"
] | [
"Definition:Set",
"Definition:Real Sequence",
"Definition:Series/Number Field",
"Definition:Convergent Series/Number Field",
"Definition:Hilbert Sequence Space",
"Definition:Locally Compact Hausdorff Space"
] | [
"Hilbert Sequence Space is Metric Space",
"Definition:Metric Space",
"Metric Space is T2",
"Definition:T2 Space",
"Point in Hilbert Sequence Space has no Compact Neighborhood",
"Definition:Compact Topological Space/Subspace",
"Definition:Neighborhood (Topology)/Point",
"Definition:Locally Compact Haus... |
proofwiki-14015 | Compact Subset of Hilbert Sequence Space is Closed | Let $A$ be the set of all real sequences $\sequence {x_i}$ such that the series $\ds \sum_{i \mathop \ge 0} {x_i}^2$ is convergent.
Let $\ell^2 = \struct {A, d_2}$ be the Hilbert sequence space on $\R$.
Let $H$ be a compact subset of $\ell^2$.
Then $H$ is closed in $\ell^2$. | From Hilbert Sequence Space is Metric Space, $\ell^2$ is a metric space.
From Metric Space is $T_2$, $\ell^2$ is a Hausdorff space.
The result follows from Compact Subspace of Hausdorff Space is Closed.
{{qed}} | Let $A$ be the [[Definition:Set|set]] of all [[Definition:Real Sequence|real sequences]] $\sequence {x_i}$ such that the [[Definition:Series of Numbers|series]] $\ds \sum_{i \mathop \ge 0} {x_i}^2$ is [[Definition:Convergent Series of Numbers|convergent]].
Let $\ell^2 = \struct {A, d_2}$ be the [[Definition:Hilbert Se... | From [[Hilbert Sequence Space is Metric Space]], $\ell^2$ is a [[Definition:Metric Space|metric space]].
From [[Metric Space is T2|Metric Space is $T_2$]], $\ell^2$ is a [[Definition:Hausdorff Space|Hausdorff space]].
The result follows from [[Compact Subspace of Hausdorff Space is Closed]].
{{qed}} | Compact Subset of Hilbert Sequence Space is Closed | https://proofwiki.org/wiki/Compact_Subset_of_Hilbert_Sequence_Space_is_Closed | https://proofwiki.org/wiki/Compact_Subset_of_Hilbert_Sequence_Space_is_Closed | [
"Hilbert Sequence Space",
"Examples of Compact Topological Spaces",
"Examples of Closed Sets"
] | [
"Definition:Set",
"Definition:Real Sequence",
"Definition:Series/Number Field",
"Definition:Convergent Series/Number Field",
"Definition:Hilbert Sequence Space",
"Definition:Compact Topological Space/Subspace",
"Definition:Closed Set/Topology"
] | [
"Hilbert Sequence Space is Metric Space",
"Definition:Metric Space",
"Metric Space is T2",
"Definition:T2 Space",
"Compact Subspace of Hausdorff Space is Closed"
] |
proofwiki-14016 | Subset of Hilbert Sequence Space with Non-Empty Interior is not Compact | Let $A$ be the set of all real sequences $\sequence {x_i}$ such that the series $\ds \sum_{i \mathop \ge 0} {x_i}^2$ is convergent.
Let $\ell^2 = \struct {A, d_2}$ be the Hilbert sequence space on $\R$.
Let $H$ be a subset of $\ell^2$ whose interior is non-empty.
Then $H$ is not compact in $\ell^2$. | Let $x \in H^\circ$, where $H^\circ$ denotes the interior of $H$.
By definition, $H^\circ$ is an open set of $\ell^2$ containing $x$.
Again by definition, $H$ is a neighborhood of $x$.
But from Point in Hilbert Sequence Space has no Compact Neighborhood, $x$ has no compact neighborhood in $\ell^2$.
Thus $H$ cannot be c... | Let $A$ be the [[Definition:Set|set]] of all [[Definition:Real Sequence|real sequences]] $\sequence {x_i}$ such that the [[Definition:Series of Numbers|series]] $\ds \sum_{i \mathop \ge 0} {x_i}^2$ is [[Definition:Convergent Series of Numbers|convergent]].
Let $\ell^2 = \struct {A, d_2}$ be the [[Definition:Hilbert Se... | Let $x \in H^\circ$, where $H^\circ$ denotes the [[Definition:Interior (Topology)|interior]] of $H$.
By definition, $H^\circ$ is an [[Definition:Open Set (Topology)|open set]] of $\ell^2$ containing $x$.
Again by definition, $H$ is a [[Definition:Neighborhood of Point|neighborhood]] of $x$.
But from [[Point in Hilbe... | Subset of Hilbert Sequence Space with Non-Empty Interior is not Compact | https://proofwiki.org/wiki/Subset_of_Hilbert_Sequence_Space_with_Non-Empty_Interior_is_not_Compact | https://proofwiki.org/wiki/Subset_of_Hilbert_Sequence_Space_with_Non-Empty_Interior_is_not_Compact | [
"Hilbert Sequence Space",
"Set Interiors",
"Examples of Compact Topological Spaces"
] | [
"Definition:Set",
"Definition:Real Sequence",
"Definition:Series/Number Field",
"Definition:Convergent Series/Number Field",
"Definition:Hilbert Sequence Space",
"Definition:Subset",
"Definition:Interior (Topology)",
"Definition:Non-Empty Set",
"Definition:Compact Topological Space/Subspace"
] | [
"Definition:Interior (Topology)",
"Definition:Open Set/Topology",
"Definition:Neighborhood (Topology)/Point",
"Point in Hilbert Sequence Space has no Compact Neighborhood",
"Definition:Compact Topological Space/Subspace",
"Definition:Neighborhood (Topology)/Point",
"Definition:Compact Topological Space/... |
proofwiki-14017 | Point in Hilbert Sequence Space has no Compact Neighborhood | Let $A$ be the set of all real sequences $\sequence {x_i}$ such that the series $\ds \sum_{i \mathop \ge 0} {x_i}^2$ is convergent.
Let $\ell^2 = \struct {A, d_2}$ be the Hilbert sequence space on $\R$.
Then no point of $\ell^2$ has a compact neighborhood. | From Hilbert Sequence Space is Metric Space, $\ell^2$ is a metric space.
Let $x = \sequence {x_i} \in A$ be a point of $\ell^2$.
Consider the closed $\epsilon$-ball of $x$ in $\ell^2$:
:$\map { {B_\epsilon}^-} x := \set {y \in A: \map {d_2} {x, y} \le \epsilon}$
for some $\epsilon \in \R_{>0}$.
Consider the point:
:$\s... | Let $A$ be the [[Definition:Set|set]] of all [[Definition:Real Sequence|real sequences]] $\sequence {x_i}$ such that the [[Definition:Series of Numbers|series]] $\ds \sum_{i \mathop \ge 0} {x_i}^2$ is [[Definition:Convergent Series of Numbers|convergent]].
Let $\ell^2 = \struct {A, d_2}$ be the [[Definition:Hilbert Se... | From [[Hilbert Sequence Space is Metric Space]], $\ell^2$ is a [[Definition:Metric Space|metric space]].
Let $x = \sequence {x_i} \in A$ be a [[Definition:Point of Set|point]] of $\ell^2$.
Consider the [[Definition:Closed Ball|closed $\epsilon$-ball]] of $x$ in $\ell^2$:
:$\map { {B_\epsilon}^-} x := \set {y \in A: \... | Point in Hilbert Sequence Space has no Compact Neighborhood | https://proofwiki.org/wiki/Point_in_Hilbert_Sequence_Space_has_no_Compact_Neighborhood | https://proofwiki.org/wiki/Point_in_Hilbert_Sequence_Space_has_no_Compact_Neighborhood | [
"Hilbert Sequence Space"
] | [
"Definition:Set",
"Definition:Real Sequence",
"Definition:Series/Number Field",
"Definition:Convergent Series/Number Field",
"Definition:Hilbert Sequence Space",
"Definition:Compact Topological Space/Subspace",
"Definition:Neighborhood (Topology)/Point"
] | [
"Hilbert Sequence Space is Metric Space",
"Definition:Metric Space",
"Definition:Element",
"Definition:Closed Ball",
"Definition:Element",
"Definition:Convergent Sequence/Metric Space",
"Definition:Subsequence",
"Definition:Compact Topological Space/Subspace",
"Definition:Compact Topological Space/S... |
proofwiki-14018 | Compact Subset of Hilbert Sequence Space is Nowhere Dense | Let $A$ be the set of all real sequences $\sequence {x_i}$ such that the series $\ds \sum_{i \mathop \ge 0} {x_i}^2$ is convergent.
Let $\ell^2 = \struct {A, d_2}$ be the Hilbert sequence space on $\R$.
Let $H$ be a compact subset of $\ell^2$.
Then $H$ is nowhere dense in $\ell^2$. | By Compact Subset of Hilbert Sequence Space is Closed, $H$ is a closed set of $\ell^2$.
From Set is Closed iff Equals Topological Closure:
:$H^- = H$
where $H^-$ denotes the closure of $H$.
From Subset of Hilbert Sequence Space with Non-Empty Interior is not Compact:
:$H^\circ = \O$
where $H^\circ$ denotes the interior... | Let $A$ be the [[Definition:Set|set]] of all [[Definition:Real Sequence|real sequences]] $\sequence {x_i}$ such that the [[Definition:Series of Numbers|series]] $\ds \sum_{i \mathop \ge 0} {x_i}^2$ is [[Definition:Convergent Series of Numbers|convergent]].
Let $\ell^2 = \struct {A, d_2}$ be the [[Definition:Hilbert Se... | By [[Compact Subset of Hilbert Sequence Space is Closed]], $H$ is a [[Definition:Closed Set (Topology)|closed set]] of $\ell^2$.
From [[Set is Closed iff Equals Topological Closure]]:
:$H^- = H$
where $H^-$ denotes the [[Definition:Closure (Topology)|closure]] of $H$.
From [[Subset of Hilbert Sequence Space with Non-... | Compact Subset of Hilbert Sequence Space is Nowhere Dense | https://proofwiki.org/wiki/Compact_Subset_of_Hilbert_Sequence_Space_is_Nowhere_Dense | https://proofwiki.org/wiki/Compact_Subset_of_Hilbert_Sequence_Space_is_Nowhere_Dense | [
"Hilbert Sequence Space",
"Examples of Nowhere Dense"
] | [
"Definition:Set",
"Definition:Real Sequence",
"Definition:Series/Number Field",
"Definition:Convergent Series/Number Field",
"Definition:Hilbert Sequence Space",
"Definition:Compact Topological Space/Subspace",
"Definition:Nowhere Dense"
] | [
"Compact Subset of Hilbert Sequence Space is Closed",
"Definition:Closed Set/Topology",
"Set is Closed iff Equals Topological Closure",
"Definition:Closure (Topology)",
"Subset of Hilbert Sequence Space with Non-Empty Interior is not Compact",
"Definition:Interior (Topology)",
"Definition:Nowhere Dense"... |
proofwiki-14019 | Hilbert Sequence Space is not Sigma-Compact | Let $A$ be the set of all real sequences $\sequence {x_i}$ such that the series $\ds \sum_{i \mathop \ge 0} x_i^2$ is convergent.
Let $\ell^2 = \struct {A, d_2}$ be the Hilbert sequence space on $\R$.
Then $\ell^2$ is not $\sigma$-compact. | By Compact Subset of Hilbert Sequence Space is Nowhere Dense, a compact subset of $\ell^2$ is nowhere dense in $\ell^2$.
We have that Hilbert Sequence Space is Complete Metric Space.
From Complete Metric Space is Non-Meager, $\ell^2$ is non-meager.
It follows that $\ell^2$ is not $\sigma$-compact.
{{explain|How?}}
{{qe... | Let $A$ be the [[Definition:Set|set]] of all [[Definition:Real Sequence|real sequences]] $\sequence {x_i}$ such that the [[Definition:Series of Numbers|series]] $\ds \sum_{i \mathop \ge 0} x_i^2$ is [[Definition:Convergent Series of Numbers|convergent]].
Let $\ell^2 = \struct {A, d_2}$ be the [[Definition:Hilbert Sequ... | By [[Compact Subset of Hilbert Sequence Space is Nowhere Dense]], a [[Definition:Compact Topological Subspace|compact subset]] of $\ell^2$ is [[Definition:Nowhere Dense|nowhere dense]] in $\ell^2$.
We have that [[Hilbert Sequence Space is Complete Metric Space]].
From [[Complete Metric Space is Non-Meager]], $\ell^2$... | Hilbert Sequence Space is not Sigma-Compact | https://proofwiki.org/wiki/Hilbert_Sequence_Space_is_not_Sigma-Compact | https://proofwiki.org/wiki/Hilbert_Sequence_Space_is_not_Sigma-Compact | [
"Hilbert Sequence Space",
"Examples of Sigma-Compact Spaces"
] | [
"Definition:Set",
"Definition:Real Sequence",
"Definition:Series/Number Field",
"Definition:Convergent Series/Number Field",
"Definition:Hilbert Sequence Space",
"Definition:Sigma-Compact Space"
] | [
"Compact Subset of Hilbert Sequence Space is Nowhere Dense",
"Definition:Compact Topological Space/Subspace",
"Definition:Nowhere Dense",
"Hilbert Sequence Space is Complete Metric Space",
"Complete Metric Space is Non-Meager",
"Definition:Meager Space/Non-Meager",
"Definition:Sigma-Compact Space"
] |
proofwiki-14020 | Hilbert Sequence Space is Injectively Path-Connected | Let $A$ be the set of all real sequences $\sequence {x_i}$ such that the series $\ds \sum_{i \mathop \ge 0} {x_i}^2$ is convergent.
Let $\ell^2 = \struct {A, d_2}$ be the Hilbert sequence space on $\R$.
Then $\ell^2$ is injectively path-connected. | Let $x = \sequence {x_i}$ and $y = \sequence {y_i}$.
Consider the mapping $f: \closedint 0 1 \to \ell^2$ defined as:
:$\forall t \in \closedint 0 1: \map f t = t x + \paren {1 - t} y = \sequence {t x_i + \paren {1 - t} y_i}$
{{begin-eqn}}
{{eqn | l = \sum_{i \mathop \ge 0} \paren {t x_i + \paren {1 - t} y_i}^2
| ... | Let $A$ be the [[Definition:Set|set]] of all [[Definition:Real Sequence|real sequences]] $\sequence {x_i}$ such that the [[Definition:Series of Numbers|series]] $\ds \sum_{i \mathop \ge 0} {x_i}^2$ is [[Definition:Convergent Series of Numbers|convergent]].
Let $\ell^2 = \struct {A, d_2}$ be the [[Definition:Hilbert Se... | Let $x = \sequence {x_i}$ and $y = \sequence {y_i}$.
Consider the [[Definition:Mapping|mapping]] $f: \closedint 0 1 \to \ell^2$ defined as:
:$\forall t \in \closedint 0 1: \map f t = t x + \paren {1 - t} y = \sequence {t x_i + \paren {1 - t} y_i}$
{{begin-eqn}}
{{eqn | l = \sum_{i \mathop \ge 0} \paren {t x_i + \par... | Hilbert Sequence Space is Injectively Path-Connected | https://proofwiki.org/wiki/Hilbert_Sequence_Space_is_Injectively_Path-Connected | https://proofwiki.org/wiki/Hilbert_Sequence_Space_is_Injectively_Path-Connected | [
"Hilbert Sequence Space",
"Examples of Injectively Path-Connected Spaces"
] | [
"Definition:Set",
"Definition:Real Sequence",
"Definition:Series/Number Field",
"Definition:Convergent Series/Number Field",
"Definition:Hilbert Sequence Space",
"Definition:Injectively Path-Connected/Topological Space"
] | [
"Definition:Mapping",
"Definition:Convergent Series/Number Field",
"Definition:Injective Path"
] |
proofwiki-14021 | Fréchet Product Space is Metric Space | Let $\struct {\R^\omega, d}$ be the '''Fréchet product space on $\R^\omega$'''.
Then $\struct {\R^\omega, d}$ is a metric space. | It is to be demonstrated that $d$ satisfies all the metric space axioms.
Recall from the definition of the '''Fréchet product space''' that the distance function $d: \R^\omega \times \R^\omega \to \R$ is defined on $\R^\omega$ as:
:$\forall x, y \in \R^\omega: \map d {x, y} = \ds \sum_{i \mathop \in \N} \dfrac {2^{-i} ... | Let $\struct {\R^\omega, d}$ be the '''[[Definition:Fréchet Product Space|Fréchet product space]] on $\R^\omega$'''.
Then $\struct {\R^\omega, d}$ is a [[Definition:Metric Space|metric space]]. | It is to be demonstrated that $d$ satisfies all the [[Axiom:Metric Space Axioms|metric space axioms]].
Recall from the definition of the '''[[Definition:Fréchet Product Space|Fréchet product space]]''' that the [[Definition:Distance Function|distance function]] $d: \R^\omega \times \R^\omega \to \R$ is defined on $\R^... | Fréchet Product Space is Metric Space | https://proofwiki.org/wiki/Fréchet_Product_Space_is_Metric_Space | https://proofwiki.org/wiki/Fréchet_Product_Space_is_Metric_Space | [
"Fréchet Product Metric"
] | [
"Definition:Fréchet Product Space",
"Definition:Metric Space"
] | [
"Axiom:Metric Space Axioms",
"Definition:Fréchet Product Space",
"Definition:Distance Function",
"Definition:Element",
"Fréchet Product Metric is Absolutely Convergent",
"Ratio Test",
"Definition:Absolutely Convergent Series",
"Fréchet Product Metric is Absolutely Convergent",
"Axiom:Metric Space Ax... |
proofwiki-14022 | Fréchet Product Space is Complete Metric Space | Let $\struct {\R^\omega, d}$ be the '''Fréchet product space on $\R^\omega$'''.
Then $\struct {\R^\omega, d}$ is a complete metric space. | From Fréchet Product Space is Metric Space, $\struct {\R^\omega, d}$ is a metric space.
It remains to be demonstrated that $\struct {\R^\omega, d}$ is complete.
Let $\sequence {x^{\paren n} }_{n \mathop \in \N}$ be an arbitrary Cauchy sequence in $\R^\omega$, where as an exception we denote the index of the Cauchy sequ... | Let $\struct {\R^\omega, d}$ be the '''[[Definition:Fréchet Product Space|Fréchet product space]] on $\R^\omega$'''.
Then $\struct {\R^\omega, d}$ is a [[Definition:Complete Metric Space|complete metric space]]. | From [[Fréchet Product Space is Metric Space]], $\struct {\R^\omega, d}$ is a [[Definition:Metric Space|metric space]].
It remains to be demonstrated that $\struct {\R^\omega, d}$ is [[Definition:Complete Metric Space|complete]].
Let $\sequence {x^{\paren n} }_{n \mathop \in \N}$ be an arbitrary [[Definition:Cauchy S... | Fréchet Product Space is Complete Metric Space | https://proofwiki.org/wiki/Fréchet_Product_Space_is_Complete_Metric_Space | https://proofwiki.org/wiki/Fréchet_Product_Space_is_Complete_Metric_Space | [
"Fréchet Product Space is Complete Metric Space",
"Fréchet Product Metric",
"Complete Metric Spaces"
] | [
"Definition:Fréchet Product Space",
"Definition:Complete Metric Space"
] | [
"Fréchet Product Space is Metric Space",
"Definition:Metric Space",
"Definition:Complete Metric Space",
"Definition:Cauchy Sequence/Metric Space",
"Definition:Term of Sequence/Index",
"Definition:Cauchy Sequence/Metric Space",
"Definition:Real Sequence",
"Definition:Fréchet Product Space"
] |
proofwiki-14023 | Separable Metric Space is Homeomorphic to Subspace of Fréchet Metric Space | Let $M = \struct {A, d}$ be a metric space whose induced topology is separable.
Then $M$ is homeomorphic to a subspace of the Fréchet product space $\struct {\R^\omega, d}$ on the countable-dimensional real Cartesian space $\R^\omega$. | Let $f: M \to \R^\omega$ be the mapping defined as:
:$\forall x \in M: \map f x = \sequence {\map d {x, x_i} }$
where $\set {x_i}$ is a countable dense subset of $A$.
It remains to be shown that $f$ is a homeomorphism.
{{ProofWanted}} | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]] whose [[Definition:Topology Induced by Metric|induced topology]] is [[Definition:Separable Space|separable]].
Then $M$ is [[Definition:Homeomorphic Topological Spaces|homeomorphic]] to a [[Definition:Topological Subspace|subspace]] of the [[Definit... | Let $f: M \to \R^\omega$ be the [[Definition:Mapping|mapping]] defined as:
:$\forall x \in M: \map f x = \sequence {\map d {x, x_i} }$
where $\set {x_i}$ is a [[Definition:Countable Set|countable]] [[Definition:Everywhere Dense|dense]] [[Definition:Subset|subset]] of $A$.
It remains to be shown that $f$ is a [[Defini... | Separable Metric Space is Homeomorphic to Subspace of Fréchet Metric Space | https://proofwiki.org/wiki/Separable_Metric_Space_is_Homeomorphic_to_Subspace_of_Fréchet_Metric_Space | https://proofwiki.org/wiki/Separable_Metric_Space_is_Homeomorphic_to_Subspace_of_Fréchet_Metric_Space | [
"Fréchet Product Metric",
"Examples of Separable Spaces"
] | [
"Definition:Metric Space",
"Definition:Topology Induced by Metric",
"Definition:Separable Space",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Topological Subspace",
"Definition:Fréchet Product Space",
"Definition:Cartesian Product/Cartesian Space/Real Cartesian Space/Countable"
] | [
"Definition:Mapping",
"Definition:Countable Set",
"Definition:Everywhere Dense",
"Definition:Subset",
"Definition:Homeomorphism/Topological Spaces"
] |
proofwiki-14024 | Hilbert Cube is Metric Space | Let $M = \struct {I^\omega, d_2}$ be the Hilbert cube.
Then $M$ is a metric space. | As defined, $M$ is a subspace of the Hilbert sequence space $\ell^2$.
We have that Hilbert Sequence Space is Metric Space.
The result follows from Subspace of Metric Space is Metric Space.
{{qed}} | Let $M = \struct {I^\omega, d_2}$ be the [[Definition:Hilbert Cube|Hilbert cube]].
Then $M$ is a [[Definition:Metric Space|metric space]]. | As defined, $M$ is a [[Definition:Metric Subspace|subspace]] of the [[Definition:Hilbert Sequence Space|Hilbert sequence space]] $\ell^2$.
We have that [[Hilbert Sequence Space is Metric Space]].
The result follows from [[Subspace of Metric Space is Metric Space]].
{{qed}} | Hilbert Cube is Metric Space | https://proofwiki.org/wiki/Hilbert_Cube_is_Metric_Space | https://proofwiki.org/wiki/Hilbert_Cube_is_Metric_Space | [
"Hilbert Cube",
"Examples of Metric Spaces"
] | [
"Definition:Hilbert Cube",
"Definition:Metric Space"
] | [
"Definition:Metric Subspace",
"Definition:Hilbert Sequence Space",
"Hilbert Sequence Space is Metric Space",
"Subspace of Metric Space is Metric Space"
] |
proofwiki-14025 | Hilbert Cube is Homeomorphic to Countable Infinite Product of Real Number Unit Intervals | Let $M_1 = \struct {I^\omega, d_2}$ be the Hilbert cube:
:$M_1 = \ds \prod_{k \mathop \in \N} \closedint 0 {\dfrac 1 k}$
under the same metric as that of the Hilbert sequence space:
:$\ds \forall x = \sequence {x_i}, y = \sequence {y_i} \in I^\omega: \map {d_2} {x, y} := \paren {\sum_{k \mathop \ge 0} \paren {x_k - y_k... | Let $x = \sequence {x_i}$ be an arbitrary element of $M_1$.
Let $f: M_1 \to M_2$ be the mapping defined as:
:$\forall x \in M_1: \map f x = \tuple {x_1, 2 x_2, 3 x_3, \ldots}$
Then $f$ is seen to be a bijection.
It remains to be shown that an open set in $M_1$ is mapped to an open set in $M_2$ by $f$.
{{ProofWanted}} | Let $M_1 = \struct {I^\omega, d_2}$ be the [[Definition:Hilbert Cube|Hilbert cube]]:
:$M_1 = \ds \prod_{k \mathop \in \N} \closedint 0 {\dfrac 1 k}$
under the same [[Definition:Metric|metric]] as that of the [[Definition:Hilbert Sequence Space|Hilbert sequence space]]:
:$\ds \forall x = \sequence {x_i}, y = \sequence... | Let $x = \sequence {x_i}$ be an arbitrary [[Definition:Element|element]] of $M_1$.
Let $f: M_1 \to M_2$ be the [[Definition:Mapping|mapping]] defined as:
:$\forall x \in M_1: \map f x = \tuple {x_1, 2 x_2, 3 x_3, \ldots}$
Then $f$ is seen to be a [[Definition:Bijection|bijection]].
It remains to be shown that an [[D... | Hilbert Cube is Homeomorphic to Countable Infinite Product of Real Number Unit Intervals | https://proofwiki.org/wiki/Hilbert_Cube_is_Homeomorphic_to_Countable_Infinite_Product_of_Real_Number_Unit_Intervals | https://proofwiki.org/wiki/Hilbert_Cube_is_Homeomorphic_to_Countable_Infinite_Product_of_Real_Number_Unit_Intervals | [
"Hilbert Cube"
] | [
"Definition:Hilbert Cube",
"Definition:Metric Space/Metric",
"Definition:Hilbert Sequence Space",
"Definition:Metric Space",
"Definition:Product Topology",
"Definition:Homeomorphism/Metric Spaces"
] | [
"Definition:Element",
"Definition:Mapping",
"Definition:Bijection",
"Definition:Open Set/Metric Space",
"Definition:Open Set/Metric Space"
] |
proofwiki-14026 | Hilbert Cube is Completely Normal | Let $M = \struct {I^\omega, d_2}$ be the Hilbert cube.
Then $M$ is a completely normal space. | We have that Hilbert Cube is Metric Space.
The result follows from Metric Space is Completely Normal.
{{qed}} | Let $M = \struct {I^\omega, d_2}$ be the [[Definition:Hilbert Cube|Hilbert cube]].
Then $M$ is a [[Definition:Completely Normal Space|completely normal space]]. | We have that [[Hilbert Cube is Metric Space]].
The result follows from [[Metric Space is Completely Normal]].
{{qed}} | Hilbert Cube is Completely Normal | https://proofwiki.org/wiki/Hilbert_Cube_is_Completely_Normal | https://proofwiki.org/wiki/Hilbert_Cube_is_Completely_Normal | [
"Hilbert Cube",
"Examples of Completely Normal Spaces"
] | [
"Definition:Hilbert Cube",
"Definition:Completely Normal Space"
] | [
"Hilbert Cube is Metric Space",
"Metric Space is Completely Normal"
] |
proofwiki-14027 | Hilbert Cube is Separable | Let $M = \struct {I^\omega, d_2}$ be the Hilbert cube.
Then $M$ is a separable space. | Consider the set $H$ of all points of $M$ which have finitely many rational coordinates and all the rest zero.
Then $H$ forms a countable subset of $A$ which is (everywhere) dense.
{{finish|Demonstrate that it is (everywhere) dense.}}
The result follows by definition of separable space.
{{qed}} | Let $M = \struct {I^\omega, d_2}$ be the [[Definition:Hilbert Cube|Hilbert cube]].
Then $M$ is a [[Definition:Separable Space|separable space]]. | Consider the [[Definition:Set|set]] $H$ of all points of $M$ which have [[Definition:Finite Set|finitely many]] [[Definition:Rational Number|rational]] [[Definition:Coordinate of Ordered Tuple|coordinates]] and all the rest [[Definition:Zero (Number)|zero]].
Then $H$ forms a [[Definition:Countable Set|countable]] [[De... | Hilbert Cube is Separable | https://proofwiki.org/wiki/Hilbert_Cube_is_Separable | https://proofwiki.org/wiki/Hilbert_Cube_is_Separable | [
"Hilbert Cube",
"Examples of Separable Spaces"
] | [
"Definition:Hilbert Cube",
"Definition:Separable Space"
] | [
"Definition:Set",
"Definition:Finite Set",
"Definition:Rational Number",
"Definition:Cartesian Product/Coordinate",
"Definition:Zero (Number)",
"Definition:Countable Set",
"Definition:Subset",
"Definition:Everywhere Dense",
"Definition:Everywhere Dense",
"Definition:Separable Space"
] |
proofwiki-14028 | Hilbert Cube is Second-Countable | Let $M = \struct {I^\omega, d_2}$ be the Hilbert cube.
Then $M$ is a second-countable space. | From Hilbert Cube is Separable, $M$ is a separable space.
We also have that Hilbert Cube is Metric Space.
The result follows from Separable Metric Space is Second-Countable.
{{qed}} | Let $M = \struct {I^\omega, d_2}$ be the [[Definition:Hilbert Cube|Hilbert cube]].
Then $M$ is a [[Definition:Second-Countable Space|second-countable space]]. | From [[Hilbert Cube is Separable]], $M$ is a [[Definition:Separable Space|separable space]].
We also have that [[Hilbert Cube is Metric Space]].
The result follows from [[Separable Metric Space is Second-Countable]].
{{qed}} | Hilbert Cube is Second-Countable | https://proofwiki.org/wiki/Hilbert_Cube_is_Second-Countable | https://proofwiki.org/wiki/Hilbert_Cube_is_Second-Countable | [
"Hilbert Cube",
"Examples of Second-Countable Spaces"
] | [
"Definition:Hilbert Cube",
"Definition:Second-Countable Space"
] | [
"Hilbert Cube is Separable",
"Definition:Separable Space",
"Hilbert Cube is Metric Space",
"Separable Metric Space is Second-Countable"
] |
proofwiki-14029 | Hilbert Cube is Compact | Let $M = \struct {I^\omega, d_2}$ be the Hilbert cube.
Then $M$ is a compact space. | Let $M'$ be the metric space defined as:
:$M' = \ds \prod_{k \mathop \in \N} \closedint 0 1$
under the product topology.
By definition, $\closedint 0 1$ is the closed unit interval under the usual (Euclidean) topology.
From Hilbert Cube is Homeomorphic to Countable Infinite Product of Real Number Unit Intervals, $M$ is... | Let $M = \struct {I^\omega, d_2}$ be the [[Definition:Hilbert Cube|Hilbert cube]].
Then $M$ is a [[Definition:Compact Topological Space|compact space]]. | Let $M'$ be the [[Definition:Metric Space|metric space]] defined as:
:$M' = \ds \prod_{k \mathop \in \N} \closedint 0 1$
under the [[Definition:Product Topology|product topology]].
By definition, $\closedint 0 1$ is the [[Definition:Closed Unit Interval|closed unit interval]] under the [[Definition:Euclidean Topology ... | Hilbert Cube is Compact | https://proofwiki.org/wiki/Hilbert_Cube_is_Compact | https://proofwiki.org/wiki/Hilbert_Cube_is_Compact | [
"Hilbert Cube is Compact",
"Hilbert Cube",
"Examples of Compact Topological Spaces"
] | [
"Definition:Hilbert Cube",
"Definition:Compact Topological Space"
] | [
"Definition:Metric Space",
"Definition:Product Topology",
"Definition:Real Interval/Unit Interval/Closed",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Hilbert Cube is Homeomorphic to Countable Infinite Product of Real Number Unit Intervals",
"Definition:Homeomorphism/Metric Spaces",... |
proofwiki-14030 | Hilbert Cube is Injectively Path-Connected | Let $M = \struct {I^\omega, d_2}$ be the Hilbert cube.
Then $M$ is an injectively path-connected space. | Let $x = \sequence {x_i}$ and $y = \sequence {y_i}$.
Consider the mapping $f: \closedint 0 1 \to I^\omega$ defined as:
:$\forall t \in \closedint 0 1: \map f t = t x + \paren {1 - t} y = \sequence {t x_i + \paren {1 - t} y_i}$
{{begin-eqn}}
{{eqn | l = \sum_{i \mathop \ge 0} \paren {t x_i + \paren {1 - t} y_i}^2
... | Let $M = \struct {I^\omega, d_2}$ be the [[Definition:Hilbert Cube|Hilbert cube]].
Then $M$ is an [[Definition:Injectively Path-Connected Space|injectively path-connected space]]. | Let $x = \sequence {x_i}$ and $y = \sequence {y_i}$.
Consider the [[Definition:Mapping|mapping]] $f: \closedint 0 1 \to I^\omega$ defined as:
:$\forall t \in \closedint 0 1: \map f t = t x + \paren {1 - t} y = \sequence {t x_i + \paren {1 - t} y_i}$
{{begin-eqn}}
{{eqn | l = \sum_{i \mathop \ge 0} \paren {t x_i + \p... | Hilbert Cube is Injectively Path-Connected | https://proofwiki.org/wiki/Hilbert_Cube_is_Injectively_Path-Connected | https://proofwiki.org/wiki/Hilbert_Cube_is_Injectively_Path-Connected | [
"Hilbert Cube",
"Examples of Injectively Path-Connected Spaces"
] | [
"Definition:Hilbert Cube",
"Definition:Injectively Path-Connected/Topological Space"
] | [
"Definition:Mapping",
"Definition:Convergent Series/Number Field",
"Definition:Injective Path"
] |
proofwiki-14031 | Interval of Totally Ordered Set is Order-Convex | Let $\struct {S, \preccurlyeq}$ be a totally ordered set.
Let $I \subseteq S$ be an interval in $S$.
Then $I$ is order-convex. | There are a number of cases to investigate.
=== Open Interval ===
Let $I = \openint a b$ be an open interval:
:$I = \set {x \in S: a \prec x \prec b}$
Let $s, t, x \in I$ such that $s \prec x \prec t$.
Then by definition:
:$a \prec s \prec x$
and:
:$x \prec t \prec b$
and so:
:$a \prec x \prec b$
and $x \in I$.
Thus we... | Let $\struct {S, \preccurlyeq}$ be a [[Definition:Totally Ordered Set|totally ordered set]].
Let $I \subseteq S$ be an [[Definition:Interval of Ordered Set|interval]] in $S$.
Then $I$ is [[Definition:Order-Convex Set|order-convex]]. | There are a number of cases to investigate.
=== Open Interval ===
Let $I = \openint a b$ be an [[Definition:Open Interval|open interval]]:
:$I = \set {x \in S: a \prec x \prec b}$
Let $s, t, x \in I$ such that $s \prec x \prec t$.
Then by definition:
:$a \prec s \prec x$
and:
:$x \prec t \prec b$
and so:
:$a \pre... | Interval of Totally Ordered Set is Order-Convex/Proof 1 | https://proofwiki.org/wiki/Interval_of_Totally_Ordered_Set_is_Order-Convex | https://proofwiki.org/wiki/Interval_of_Totally_Ordered_Set_is_Order-Convex/Proof_1 | [
"Interval of Totally Ordered Set is Order-Convex",
"Intervals",
"Order-Convex Sets"
] | [
"Definition:Totally Ordered Set",
"Definition:Interval/Ordered Set",
"Definition:Order-Convex Set"
] | [
"Definition:Interval/Ordered Set/Open",
"Definition:Order-Convex Set",
"Definition:Interval/Ordered Set/Left Half-Open",
"Definition:Order-Convex Set",
"Definition:Interval/Ordered Set/Right Half-Open",
"Definition:Order-Convex Set",
"Definition:Interval/Ordered Set/Closed",
"Definition:Order-Convex S... |
proofwiki-14032 | Interval of Totally Ordered Set is Order-Convex | Let $\struct {S, \preccurlyeq}$ be a totally ordered set.
Let $I \subseteq S$ be an interval in $S$.
Then $I$ is order-convex. | An interval can be represented as the intersection of two rays.
{{explain|Obvious though it is, the above needs to be stated as a theorem in its own right.}}
Thus by Ray is Order-Convex and Intersection of Order-Convex Sets is Order-Convex, $I$ is order-convex.
{{qed}} | Let $\struct {S, \preccurlyeq}$ be a [[Definition:Totally Ordered Set|totally ordered set]].
Let $I \subseteq S$ be an [[Definition:Interval of Ordered Set|interval]] in $S$.
Then $I$ is [[Definition:Order-Convex Set|order-convex]]. | An [[Definition:Interval of Ordered Set|interval]] can be represented as the [[Definition:Set Intersection|intersection]] of two [[Definition:Ray (Order Theory)|rays]].
{{explain|Obvious though it is, the above needs to be stated as a theorem in its own right.}}
Thus by [[Ray is Order-Convex]] and [[Intersection of O... | Interval of Totally Ordered Set is Order-Convex/Proof 2 | https://proofwiki.org/wiki/Interval_of_Totally_Ordered_Set_is_Order-Convex | https://proofwiki.org/wiki/Interval_of_Totally_Ordered_Set_is_Order-Convex/Proof_2 | [
"Interval of Totally Ordered Set is Order-Convex",
"Intervals",
"Order-Convex Sets"
] | [
"Definition:Totally Ordered Set",
"Definition:Interval/Ordered Set",
"Definition:Order-Convex Set"
] | [
"Definition:Interval/Ordered Set",
"Definition:Set Intersection",
"Definition:Ray (Order Theory)",
"Ray is Order-Convex",
"Intersection of Order-Convex Sets is Order-Convex",
"Definition:Order-Convex Set"
] |
proofwiki-14033 | Order-Convex Subset of Ordered Set is not necessarily Interval | Let $\struct {S, \preccurlyeq}$ be an ordered set.
Let $C$ be an order-convex subset of $S$.
Then it is not necessarily the case that $C$ is an interval of $S$. | Consider the open ray of $S$:
:$R = \set {x \in S: a \prec x}$
for some $a \in S$.
From Ray is Order-Convex, $R$ is a order-convex subset of $S$.
But $R$ is not an interval of $S$.
{{qed}} | Let $\struct {S, \preccurlyeq}$ be an [[Definition:Ordered Set|ordered set]].
Let $C$ be an [[Definition:Order-Convex Set|order-convex]] [[Definition:Subset|subset]] of $S$.
Then it is not necessarily the case that $C$ is an [[Definition:Interval of Ordered Set|interval]] of $S$. | Consider the [[Definition:Open Ray|open ray]] of $S$:
:$R = \set {x \in S: a \prec x}$
for some $a \in S$.
From [[Ray is Order-Convex]], $R$ is a [[Definition:Order-Convex Set|order-convex]] [[Definition:Subset|subset]] of $S$.
But $R$ is not an [[Definition:Interval of Ordered Set|interval]] of $S$.
{{qed}} | Order-Convex Subset of Ordered Set is not necessarily Interval | https://proofwiki.org/wiki/Order-Convex_Subset_of_Ordered_Set_is_not_necessarily_Interval | https://proofwiki.org/wiki/Order-Convex_Subset_of_Ordered_Set_is_not_necessarily_Interval | [
"Order-Convex Sets",
"Intervals"
] | [
"Definition:Ordered Set",
"Definition:Order-Convex Set",
"Definition:Subset",
"Definition:Interval/Ordered Set"
] | [
"Definition:Ray (Order Theory)/Open",
"Ray is Order-Convex",
"Definition:Order-Convex Set",
"Definition:Subset",
"Definition:Interval/Ordered Set"
] |
proofwiki-14034 | Union of Non-Disjoint Order-Convex Sets is Order-Convex | Let $\struct {S, \preccurlyeq}$ be an ordered set.
Let $\CC$ be a set of order-convex sets of $S$ such that their intersection is non-empty:
:$\bigcap \CC \ne \O$
Then the union $\bigcup \CC$ is also order-convex. | Let $x, y, z \in S$ be arbitrary elements of $S$ such that $x \prec y \prec z$.
Let $x, z \in \bigcup \CC$.
First let $x, z \in C$ where $C \in \CC$.
Then as $C$ is order-convex, $y \in C$.
Hence, by definition of union, $y \in \bigcup \CC$.
Now let $x \in C_1, z \in C_2$ where $C_1, C_2 \in \CC$.
We have that $\bigcap... | Let $\struct {S, \preccurlyeq}$ be an [[Definition:Ordered Set|ordered set]].
Let $\CC$ be a [[Definition:Set of Sets|set]] of [[Definition:Order-Convex Set|order-convex sets]] of $S$ such that their [[Definition:Intersection of Set of Sets|intersection]] is [[Definition:Non-Empty Set|non-empty]]:
:$\bigcap \CC \ne \O... | Let $x, y, z \in S$ be [[Definition:Arbitrary|arbitrary]] [[Definition:Element|elements]] of $S$ such that $x \prec y \prec z$.
Let $x, z \in \bigcup \CC$.
First let $x, z \in C$ where $C \in \CC$.
Then as $C$ is [[Definition:Order-Convex Set|order-convex]], $y \in C$.
Hence, by definition of [[Definition:Set Unio... | Union of Non-Disjoint Order-Convex Sets is Order-Convex | https://proofwiki.org/wiki/Union_of_Non-Disjoint_Order-Convex_Sets_is_Order-Convex | https://proofwiki.org/wiki/Union_of_Non-Disjoint_Order-Convex_Sets_is_Order-Convex | [
"Order-Convex Sets",
"Set Union"
] | [
"Definition:Ordered Set",
"Definition:Set of Sets",
"Definition:Order-Convex Set",
"Definition:Set Intersection/Set of Sets",
"Definition:Non-Empty Set",
"Definition:Set Union/Set of Sets",
"Definition:Order-Convex Set"
] | [
"Definition:Arbitrary",
"Definition:Element",
"Definition:Order-Convex Set",
"Definition:Set Union",
"Definition:Order-Convex Set",
"Definition:Order-Convex Set",
"Definition:Order-Convex Set"
] |
proofwiki-14035 | Separated Subsets of Linearly Ordered Space under Order Topology | Let $T = \struct {S, \preceq, \tau}$ be a linearly ordered space.
Let $A$ and $B$ be separated sets of $T$.
Let $A^*$ and $B^*$ be defined as:
{{begin-eqn}}
{{eqn | l = A^*
| r = \bigcup \set {\closedint a b: a, b \in A, \closedint a b \cap B^- = \O}
}}
{{eqn | l = B^*
| r = \bigcup \set {\closedint a b: a,... | === Lemma ===
{{:Separated Subsets of Linearly Ordered Space under Order Topology/Lemma}}{{qed|lemma}}
So, from the {{Lemma|Separated Subsets of Linearly Ordered Space under Order Topology}}:
:$A \subseteq A^*$
:$B \subseteq B^*$
:$A^* \cap B^* = \O$
Let $p \notin A^* \cup A^-$.
Thus $p \notin A^*$ and $p \notin A^-$.
... | Let $T = \struct {S, \preceq, \tau}$ be a [[Definition:Linearly Ordered Space|linearly ordered space]].
Let $A$ and $B$ be [[Definition:Separated Sets|separated sets]] of $T$.
Let $A^*$ and $B^*$ be defined as:
{{begin-eqn}}
{{eqn | l = A^*
| r = \bigcup \set {\closedint a b: a, b \in A, \closedint a b \cap B^-... | === [[Separated Subsets of Linearly Ordered Space under Order Topology/Lemma|Lemma]] ===
{{:Separated Subsets of Linearly Ordered Space under Order Topology/Lemma}}{{qed|lemma}}
So, from the {{Lemma|Separated Subsets of Linearly Ordered Space under Order Topology}}:
:$A \subseteq A^*$
:$B \subseteq B^*$
:$A^* \cap... | Separated Subsets of Linearly Ordered Space under Order Topology | https://proofwiki.org/wiki/Separated_Subsets_of_Linearly_Ordered_Space_under_Order_Topology | https://proofwiki.org/wiki/Separated_Subsets_of_Linearly_Ordered_Space_under_Order_Topology | [
"Separated Subsets of Linearly Ordered Space under Order Topology",
"Linearly Ordered Spaces",
"Examples of Separated Sets"
] | [
"Definition:Linearly Ordered Space",
"Definition:Separated Sets",
"Definition:Closure (Topology)",
"Definition:Separated Sets"
] | [
"Separated Subsets of Linearly Ordered Space under Order Topology/Lemma",
"Definition:Interval/Ordered Set/Open",
"Definition:Disjoint Sets",
"Definition:Set Intersection",
"Definition:Set Intersection"
] |
proofwiki-14036 | Generators of Special Linear Group of Order 2 over Integers | Let:
:$ S = \begin{pmatrix}
0 & - 1 \\
1 & 0
\end{pmatrix}$
and:
:$T = \begin{pmatrix}
1 & 1 \\
0 & 1
\end{pmatrix}$
Then $S$ and $T$ are generators for the special linear group of order $2$ over $\Z$. | Let:
:$ g = \begin{pmatrix}
a & b \\
c & d
\end{pmatrix}$
be an element of $\SL {2, \Z}$.
Observe that:
:$T^n = \begin{pmatrix}
1 & n \\
0 & 1
\end{pmatrix}$
so that:
:$T^n g = \begin{pmatrix}
a + nc & b + nd \\
c & d
\end{pmatrix}$
Also note that:
:$S^2 = -I$
and:
:$S g = \begin{pmatrix}
-c & -d \\
a & b
\end{pmatri... | Let:
:$ S = \begin{pmatrix}
0 & - 1 \\
1 & 0
\end{pmatrix}$
and:
:$T = \begin{pmatrix}
1 & 1 \\
0 & 1
\end{pmatrix}$
Then $S$ and $T$ are [[Definition:Generator of Group|generators]] for the [[Definition:Special Linear Group|special linear group]] of order $2$ over $\Z$. | Let:
:$ g = \begin{pmatrix}
a & b \\
c & d
\end{pmatrix}$
be an element of $\SL {2, \Z}$.
Observe that:
:$T^n = \begin{pmatrix}
1 & n \\
0 & 1
\end{pmatrix}$
so that:
:$T^n g = \begin{pmatrix}
a + nc & b + nd \\
c & d
\end{pmatrix}$
Also note that:
:$S^2 = -I$
and:
:$S g = \begin{pmatrix}
-c & -d \\
a & b
\end... | Generators of Special Linear Group of Order 2 over Integers | https://proofwiki.org/wiki/Generators_of_Special_Linear_Group_of_Order_2_over_Integers | https://proofwiki.org/wiki/Generators_of_Special_Linear_Group_of_Order_2_over_Integers | [
" Special Linear Group"
] | [
"Definition:Generator of Group",
"Definition:Special Linear Group"
] | [
"Division Theorem",
"Category: Special Linear Group"
] |
proofwiki-14037 | Partition of Linearly Ordered Space by Convex Components is Linearly Ordered Set | Let $T = \struct {S, \preceq, \tau}$ be a linearly ordered space.
Let $A$ and $B$ be separated sets of $T$.
Let $A^*$ and $B^*$ be defined as:
{{begin-eqn}}
{{eqn | l = A^*
| r = \bigcup \set {\closedint a b: a, b \in A, \closedint a b \cap B^- = \O}
}}
{{eqn | l = B^*
| r = \bigcup \set {\closedint a b: a,... | {{ProofWanted|I lack both the skill and the patience to disentangle this material.}} | Let $T = \struct {S, \preceq, \tau}$ be a [[Definition:Linearly Ordered Space|linearly ordered space]].
Let $A$ and $B$ be [[Definition:Separated Sets|separated sets]] of $T$.
Let $A^*$ and $B^*$ be defined as:
{{begin-eqn}}
{{eqn | l = A^*
| r = \bigcup \set {\closedint a b: a, b \in A, \closedint a b \cap B^... | {{ProofWanted|I lack both the skill and the patience to disentangle this material.}} | Partition of Linearly Ordered Space by Convex Components is Linearly Ordered Set | https://proofwiki.org/wiki/Partition_of_Linearly_Ordered_Space_by_Convex_Components_is_Linearly_Ordered_Set | https://proofwiki.org/wiki/Partition_of_Linearly_Ordered_Space_by_Convex_Components_is_Linearly_Ordered_Set | [
"Linearly Ordered Spaces"
] | [
"Definition:Linearly Ordered Space",
"Definition:Separated Sets",
"Definition:Closure (Topology)",
"Definition:Set Union",
"Definition:Convex Component",
"Definition:Relative Complement",
"Definition:Set",
"Definition:Total Ordering",
"Definition:Strictly Totally Ordered Set"
] | [] |
proofwiki-14038 | Successor Sets of Linearly Ordered Set Induced by Convex Component Partition | Let $T = \struct {S, \preceq, \tau}$ be a linearly ordered space.
Let $A$ and $B$ be separated sets of $T$.
Let $A^*$ and $B^*$ be defined as:
{{begin-eqn}}
{{eqn | l = A^*
| r = \bigcup \set {\closedint a b: a, b \in A, \closedint a b \cap B^- = \O}
}}
{{eqn | l = B^*
| r = \bigcup \set {\closedint a b: a,... | Let $A_\alpha \cap {S_\alpha}^- \ne \O$.
Then $A_\alpha \cap {S_\alpha}^-$ contains exactly $1$ point, say $p$.
This belongs to the complement in $S$ of the closed set $\paren {B^*}^-$.
Hence there exists a neighborhood $\openint x y$ of $p$ which is disjoint from $\paren {B^*}^-$.
Then:
:$\openint x y \cap S_\alpha \n... | Let $T = \struct {S, \preceq, \tau}$ be a [[Definition:Linearly Ordered Space|linearly ordered space]].
Let $A$ and $B$ be [[Definition:Separated Sets|separated sets]] of $T$.
Let $A^*$ and $B^*$ be defined as:
{{begin-eqn}}
{{eqn | l = A^*
| r = \bigcup \set {\closedint a b: a, b \in A, \closedint a b \cap B^... | Let $A_\alpha \cap {S_\alpha}^- \ne \O$.
Then $A_\alpha \cap {S_\alpha}^-$ contains exactly $1$ point, say $p$.
This belongs to the [[Definition:Relative Complement|complement in $S$]] of the [[Definition:Closed Set (Topology)|closed set]] $\paren {B^*}^-$.
Hence there exists a [[Definition:Neighborhood of Point|nei... | Successor Sets of Linearly Ordered Set Induced by Convex Component Partition | https://proofwiki.org/wiki/Successor_Sets_of_Linearly_Ordered_Set_Induced_by_Convex_Component_Partition | https://proofwiki.org/wiki/Successor_Sets_of_Linearly_Ordered_Set_Induced_by_Convex_Component_Partition | [
"Linearly Ordered Spaces"
] | [
"Definition:Linearly Ordered Space",
"Definition:Separated Sets",
"Definition:Closure (Topology)",
"Definition:Set Union",
"Definition:Convex Component",
"Definition:Relative Complement",
"Definition:Strictly Totally Ordered Set",
"Partition of Linearly Ordered Space by Convex Components is Linearly O... | [
"Definition:Relative Complement",
"Definition:Closed Set/Topology",
"Definition:Neighborhood (Topology)/Point",
"Definition:Disjoint Sets",
"Definition:Disjoint Sets"
] |
proofwiki-14039 | Linearly Ordered Space is T5 | Let $T = \struct {S, \preceq, \tau}$ be a linearly ordered space.
Then $T$ is a $T_5$ space. | Let $A$ and $B$ be separated sets of $T$.
Let $A^*$ and $B^*$ be defined as:
{{begin-eqn}}
{{eqn | l = A^*
| r = \bigcup \set {\closedint a b: a, b \in A, \closedint a b \cap B^- = \O}
}}
{{eqn | l = B^*
| r = \bigcup \set {\closedint a b: a, b \in B, \closedint a b \cap A^- = \O}
}}
{{end-eqn}}
where $A^-$... | Let $T = \struct {S, \preceq, \tau}$ be a [[Definition:Linearly Ordered Space|linearly ordered space]].
Then $T$ is a [[Definition:T5 Space|$T_5$ space]]. | Let $A$ and $B$ be [[Definition:Separated Sets|separated sets]] of $T$.
Let $A^*$ and $B^*$ be defined as:
{{begin-eqn}}
{{eqn | l = A^*
| r = \bigcup \set {\closedint a b: a, b \in A, \closedint a b \cap B^- = \O}
}}
{{eqn | l = B^*
| r = \bigcup \set {\closedint a b: a, b \in B, \closedint a b \cap A^- ... | Linearly Ordered Space is T5 | https://proofwiki.org/wiki/Linearly_Ordered_Space_is_T5 | https://proofwiki.org/wiki/Linearly_Ordered_Space_is_T5 | [
"Linearly Ordered Spaces",
"Examples of T5 Spaces"
] | [
"Definition:Linearly Ordered Space",
"Definition:T5 Space"
] | [
"Definition:Separated Sets",
"Definition:Closure (Topology)",
"Definition:Set Union",
"Definition:Convex Component",
"Definition:Relative Complement",
"Definition:Strictly Totally Ordered Set",
"Partition of Linearly Ordered Space by Convex Components is Linearly Ordered Set",
"Definition:Set",
"Def... |
proofwiki-14040 | Linearly Ordered Space is T1 | Let $T = \struct {S, \preceq, \tau}$ be a linearly ordered space.
Then $T$ is a $T_1$ space. | Let $p \in S$.
By definition of linearly ordered space, the rays:
{{begin-eqn}}
{{eqn | l = R_1
| o = :=
| r = \set {x \in S: x \prec p}
}}
{{eqn | l = R_2
| o = :=
| r = \set {x \in S: p \prec x}
}}
{{end-eqn}}
are open in $T$.
Thus their union:
:$R_1 \cup R_2 = \set {x \in S: x \prec p \lor p ... | Let $T = \struct {S, \preceq, \tau}$ be a [[Definition:Linearly Ordered Space|linearly ordered space]].
Then $T$ is a [[Definition:T1 Space|$T_1$ space]]. | Let $p \in S$.
By definition of [[Definition:Linearly Ordered Space|linearly ordered space]], the [[Definition:Ray (Order Theory)|rays]]:
{{begin-eqn}}
{{eqn | l = R_1
| o = :=
| r = \set {x \in S: x \prec p}
}}
{{eqn | l = R_2
| o = :=
| r = \set {x \in S: p \prec x}
}}
{{end-eqn}}
are [[Defi... | Linearly Ordered Space is T1 | https://proofwiki.org/wiki/Linearly_Ordered_Space_is_T1 | https://proofwiki.org/wiki/Linearly_Ordered_Space_is_T1 | [
"Linearly Ordered Spaces",
"Examples of T1 Spaces"
] | [
"Definition:Linearly Ordered Space",
"Definition:T1 Space"
] | [
"Definition:Linearly Ordered Space",
"Definition:Ray (Order Theory)",
"Definition:Open Set/Topology",
"Definition:Set Union",
"Definition:Open Set/Topology",
"Definition:Relative Complement",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Closed Point",
"Definition... |
proofwiki-14041 | Linearly Ordered Space is Completely Normal | Let $T = \struct {S, \preceq, \tau}$ be a linearly ordered space.
Then $T$ is a completely normal space. | By Linearly Ordered Space is $T_1$, $T$ is a $T_1$ space.
By Linearly Ordered Space is $T_5$, $T$ is a $T_5$ space.
Hence the result, by definition of completely normal space.
{{qed}} | Let $T = \struct {S, \preceq, \tau}$ be a [[Definition:Linearly Ordered Space|linearly ordered space]].
Then $T$ is a [[Definition:Completely Normal Space|completely normal space]]. | By [[Linearly Ordered Space is T1|Linearly Ordered Space is $T_1$]], $T$ is a [[Definition:T1 Space|$T_1$ space]].
By [[Linearly Ordered Space is T5|Linearly Ordered Space is $T_5$]], $T$ is a [[Definition:T5 Space|$T_5$ space]].
Hence the result, by definition of [[Definition:Completely Normal Space|completely norma... | Linearly Ordered Space is Completely Normal | https://proofwiki.org/wiki/Linearly_Ordered_Space_is_Completely_Normal | https://proofwiki.org/wiki/Linearly_Ordered_Space_is_Completely_Normal | [
"Linearly Ordered Spaces",
"Examples of Completely Normal Spaces"
] | [
"Definition:Linearly Ordered Space",
"Definition:Completely Normal Space"
] | [
"Linearly Ordered Space is T1",
"Definition:T1 Space",
"Linearly Ordered Space is T5",
"Definition:T5 Space",
"Definition:Completely Normal Space"
] |
proofwiki-14042 | Linearly Ordered Space is Compact iff Complete | Let $T = \struct {S, \preceq, \tau}$ be a linearly ordered space.
Then $T$ is a compact space {{iff}} it is complete. | === Necessary Condition ===
Let $T$ be a compact space.
Let $A \subseteq S$.
{{AimForCont}} $A$ has no supremum.
Consider the sets:
{{begin-eqn}}
{{eqn | l = P_\alpha
| r = \set {x \in S: x \prec \alpha}
| c = for $\alpha \in A$
}}
{{eqn | l = B_\beta
| r = \set {x \in S: \beta \prec x}
| c = fo... | Let $T = \struct {S, \preceq, \tau}$ be a [[Definition:Linearly Ordered Space|linearly ordered space]].
Then $T$ is a [[Definition:Compact Topological Space|compact space]] {{iff}} it is [[Definition:Complete Ordered Set|complete]]. | === Necessary Condition ===
Let $T$ be a [[Definition:Compact Topological Space|compact space]].
Let $A \subseteq S$.
{{AimForCont}} $A$ has no [[Definition:Supremum of Set|supremum]].
Consider the [[Definition:Set|sets]]:
{{begin-eqn}}
{{eqn | l = P_\alpha
| r = \set {x \in S: x \prec \alpha}
| c = fo... | Linearly Ordered Space is Compact iff Complete | https://proofwiki.org/wiki/Linearly_Ordered_Space_is_Compact_iff_Complete | https://proofwiki.org/wiki/Linearly_Ordered_Space_is_Compact_iff_Complete | [
"Linearly Ordered Spaces",
"Examples of Compact Topological Spaces",
"Examples of Complete Lattices"
] | [
"Definition:Linearly Ordered Space",
"Definition:Compact Topological Space",
"Definition:Complete Lattice"
] | [
"Definition:Compact Topological Space",
"Definition:Supremum of Set",
"Definition:Set",
"Definition:Upper Bound of Set",
"Definition:Cover of Set",
"Definition:Subcover/Finite",
"Definition:Compact Topological Space",
"Definition:Infimum of Set",
"Definition:Supremum of Set",
"Definition:Infimum o... |
proofwiki-14043 | Every Point except Endpoint in Connected Linearly Ordered Space is Cut Point | Let $T = \struct {S, \preceq, \tau}$ be a linearly ordered space.
Let $A \subseteq S$ be a connected space.
Let $p \in A$ be a point of $A$ which is not an endpoint of $A$.
Then $p$ is a cut point of $A$. | We have that $A \setminus \set p$ is separated by $\set {x \in A: x \prec p}$ and $\set {x \in A: p \prec x}$.
If $p$ is an endpoint of $A$, then either:
:$\set {x \in A: x \prec p} = \O$
or:
:$\set {x \in A: p \prec x} = \O$
{{qed}} | Let $T = \struct {S, \preceq, \tau}$ be a [[Definition:Linearly Ordered Space|linearly ordered space]].
Let $A \subseteq S$ be a [[Definition:Connected Topological Space|connected space]].
Let $p \in A$ be a point of $A$ which is not an [[Definition:Endpoint of Interval|endpoint]] of $A$.
Then $p$ is a [[Definition... | We have that $A \setminus \set p$ is [[Definition:Separation (Topology)|separated]] by $\set {x \in A: x \prec p}$ and $\set {x \in A: p \prec x}$.
If $p$ is an [[Definition:Endpoint of Interval|endpoint]] of $A$, then either:
:$\set {x \in A: x \prec p} = \O$
or:
:$\set {x \in A: p \prec x} = \O$
{{qed}} | Every Point except Endpoint in Connected Linearly Ordered Space is Cut Point | https://proofwiki.org/wiki/Every_Point_except_Endpoint_in_Connected_Linearly_Ordered_Space_is_Cut_Point | https://proofwiki.org/wiki/Every_Point_except_Endpoint_in_Connected_Linearly_Ordered_Space_is_Cut_Point | [
"Linearly Ordered Spaces",
"Examples of Connected Topological Spaces",
"Examples of Cut Points"
] | [
"Definition:Linearly Ordered Space",
"Definition:Connected Topological Space",
"Definition:Interval/Ordered Set/Endpoint",
"Definition:Cut Point"
] | [
"Definition:Separation (Topology)",
"Definition:Interval/Ordered Set/Endpoint"
] |
proofwiki-14044 | Omega is Closed in Uncountable Closed Ordinal Space but not G-Delta Set | Let $\Omega$ denote the first uncountable ordinal.
Let $\closedint 0 \Omega$ denote the closed ordinal space on $\Omega$.
Then $\set \Omega$ is a closed set of $\closedint 0 \Omega$ but not a $G_\delta$ set. | The complement relative to $\closedint 0 \Omega$ of $\set \Omega$ is $\hointr 0 \Omega$, which is open in $\closedint 0 \Omega$.
Hence, by definition, $\set \Omega$ is a closed set of $\closedint 0 \Omega$.
Let $G_i$ be a countable set of open sets of $\closedint 0 \Omega$ which contain $\Omega$.
Then we can find a set... | Let $\Omega$ denote the first [[Definition:Uncountable Ordinal|uncountable ordinal]].
Let $\closedint 0 \Omega$ denote the [[Definition:Uncountable Closed Ordinal Space|closed ordinal space]] on $\Omega$.
Then $\set \Omega$ is a [[Definition:Closed Set (Topology)|closed set]] of $\closedint 0 \Omega$ but not a [[Def... | The [[Definition:Relative Complement|complement relative to $\closedint 0 \Omega$]] of $\set \Omega$ is $\hointr 0 \Omega$, which is [[Definition:Open Set (Topology)|open]] in $\closedint 0 \Omega$.
Hence, by definition, $\set \Omega$ is a [[Definition:Closed Set (Topology)|closed set]] of $\closedint 0 \Omega$.
Let ... | Omega is Closed in Uncountable Closed Ordinal Space but not G-Delta Set | https://proofwiki.org/wiki/Omega_is_Closed_in_Uncountable_Closed_Ordinal_Space_but_not_G-Delta_Set | https://proofwiki.org/wiki/Omega_is_Closed_in_Uncountable_Closed_Ordinal_Space_but_not_G-Delta_Set | [
"Uncountable Closed Ordinal Spaces",
"Examples of G-Delta Sets"
] | [
"Definition:Uncountable Ordinal",
"Definition:Ordinal Space/Closed/Uncountable",
"Definition:Closed Set/Topology",
"Definition:G-Delta Set"
] | [
"Definition:Relative Complement",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Countable Set",
"Definition:Open Set/Topology",
"Definition:Set",
"Definition:Basis (Topology)",
"Definition:Countable Set",
"Countable Union of Countable Sets is Countable",
"Definition... |
proofwiki-14045 | Uncountable Closed Ordinal Space is not First-Countable | Let $\Omega$ denote the first uncountable ordinal.
Let $\closedint 0 \Omega$ denote the closed ordinal space on $\Omega$.
Then $\closedint 0 \Omega$ is not a first-countable space. | From Omega is Closed in Uncountable Closed Ordinal Space but not G-Delta Set, $\set \Omega$ cannot be expressed as a countable intersection of open sets of $\closedint 0 \Omega$.
Thus, by definition, $\Omega$ does not have a countable local basis.
Hence the result by definition of first-countable space.
{{qed}} | Let $\Omega$ denote the first [[Definition:Uncountable Ordinal|uncountable ordinal]].
Let $\closedint 0 \Omega$ denote the [[Definition:Uncountable Closed Ordinal Space|closed ordinal space]] on $\Omega$.
Then $\closedint 0 \Omega$ is not a [[Definition:First-Countable Space|first-countable space]]. | From [[Omega is Closed in Uncountable Closed Ordinal Space but not G-Delta Set]], $\set \Omega$ cannot be expressed as a [[Definition:Countable Intersection|countable intersection]] of [[Definition:Open Set (Topology)|open sets]] of $\closedint 0 \Omega$.
Thus, by definition, $\Omega$ does not have a [[Definition:Coun... | Uncountable Closed Ordinal Space is not First-Countable | https://proofwiki.org/wiki/Uncountable_Closed_Ordinal_Space_is_not_First-Countable | https://proofwiki.org/wiki/Uncountable_Closed_Ordinal_Space_is_not_First-Countable | [
"Uncountable Closed Ordinal Spaces",
"Examples of First-Countable Spaces"
] | [
"Definition:Uncountable Ordinal",
"Definition:Ordinal Space/Closed/Uncountable",
"Definition:First-Countable Space"
] | [
"Omega is Closed in Uncountable Closed Ordinal Space but not G-Delta Set",
"Definition:Set Intersection/Countable Intersection",
"Definition:Open Set/Topology",
"Definition:Countable Set",
"Definition:Local Basis",
"Definition:First-Countable Space"
] |
proofwiki-14046 | Uncountable Open Ordinal Space is not Separable | Let $\Omega$ denote the first uncountable ordinal.
Let $\hointr 0 \Omega$ denote the open ordinal space on $\Omega$.
Then $\hointr 0 \Omega$ is not a separable space. | Because $\Omega$ is the first uncountable ordinal, any ordinal which strictly precedes $\Omega$ is countable.
Let $H \subseteq \hointr 0 \Omega$ be a countable subset of $\hointr 0 \Omega$.
Let $\sigma$ be the supremum of $H$.
As $H$ by definition strictly precedes $\Omega$, $H$ itself is countable.
Thus $\sigma$ stric... | Let $\Omega$ denote the first [[Definition:Uncountable Ordinal|uncountable ordinal]].
Let $\hointr 0 \Omega$ denote the [[Definition:Uncountable Open Ordinal Space|open ordinal space]] on $\Omega$.
Then $\hointr 0 \Omega$ is not a [[Definition:Separable Space|separable space]]. | Because $\Omega$ is the first [[Definition:Uncountable Ordinal|uncountable ordinal]], any [[Definition:Ordinal|ordinal]] which [[Definition:Strictly Precede|strictly precedes]] $\Omega$ is [[Definition:Countable Set|countable]].
Let $H \subseteq \hointr 0 \Omega$ be a [[Definition:Countable Set|countable]] [[Definitio... | Uncountable Open Ordinal Space is not Separable | https://proofwiki.org/wiki/Uncountable_Open_Ordinal_Space_is_not_Separable | https://proofwiki.org/wiki/Uncountable_Open_Ordinal_Space_is_not_Separable | [
"Uncountable Open Ordinal Spaces",
"Examples of Separable Spaces"
] | [
"Definition:Uncountable Ordinal",
"Definition:Ordinal Space/Open/Uncountable",
"Definition:Separable Space"
] | [
"Definition:Uncountable Ordinal",
"Definition:Ordinal",
"Definition:Strictly Precede",
"Definition:Countable Set",
"Definition:Countable Set",
"Definition:Subset",
"Definition:Supremum of Set",
"Definition:Strictly Precede",
"Definition:Countable Set",
"Definition:Strictly Precede",
"Definition:... |
proofwiki-14047 | Uncountable Closed Ordinal Space is not Separable | Let $\Omega$ denote the first uncountable ordinal.
Let $\closedint 0 \Omega$ denote the closed ordinal space on $\Omega$.
Then $\closedint 0 \Omega$ is not a separable space. | Let $H \subseteq \closedint 0 \Omega$ be a countable subset of $\closedint 0 \Omega$.
From Uncountable Open Ordinal Space is not Separable, there exists an open interval $\openint \sigma \Omega$ in the complement of $H^-$ in $\hointr 0 \Omega$, and so also in $\closedint 0 \Omega$.
Thus the closure of $H$ in $\closedin... | Let $\Omega$ denote the first [[Definition:Uncountable Ordinal|uncountable ordinal]].
Let $\closedint 0 \Omega$ denote the [[Definition:Uncountable Closed Ordinal Space|closed ordinal space]] on $\Omega$.
Then $\closedint 0 \Omega$ is not a [[Definition:Separable Space|separable space]]. | Let $H \subseteq \closedint 0 \Omega$ be a [[Definition:Countable Set|countable]] [[Definition:Subset|subset]] of $\closedint 0 \Omega$.
From [[Uncountable Open Ordinal Space is not Separable]], there exists an [[Definition:Open Interval|open interval]] $\openint \sigma \Omega$ in the [[Definition:Relative Complement|... | Uncountable Closed Ordinal Space is not Separable | https://proofwiki.org/wiki/Uncountable_Closed_Ordinal_Space_is_not_Separable | https://proofwiki.org/wiki/Uncountable_Closed_Ordinal_Space_is_not_Separable | [
"Uncountable Closed Ordinal Spaces",
"Examples of Separable Spaces"
] | [
"Definition:Uncountable Ordinal",
"Definition:Ordinal Space/Closed/Uncountable",
"Definition:Separable Space"
] | [
"Definition:Countable Set",
"Definition:Subset",
"Uncountable Open Ordinal Space is not Separable",
"Definition:Interval/Ordered Set/Open",
"Definition:Relative Complement",
"Definition:Closure (Topology)",
"Definition:Everywhere Dense",
"Definition:Separable Space"
] |
proofwiki-14048 | Uncountable Open Ordinal Space is First-Countable | Let $\Omega$ denote the first uncountable ordinal.
Let $\hointr 0 \Omega$ denote the open ordinal space on $\Omega$.
Then $\hointr 0 \Omega$ is a first-countable space. | {{ProofWanted|It is to be shown that the only point of $\closedint 0 \Omega$ that does not have a countable local basis is $\Omega$. So all the others have, and that means all the points in $\hointr 0 \Omega$ have a countable local basis.}} | Let $\Omega$ denote the first [[Definition:Uncountable Ordinal|uncountable ordinal]].
Let $\hointr 0 \Omega$ denote the [[Definition:Uncountable Open Ordinal Space|open ordinal space]] on $\Omega$.
Then $\hointr 0 \Omega$ is a [[Definition:First-Countable Space|first-countable space]]. | {{ProofWanted|It is to be shown that the only point of $\closedint 0 \Omega$ that does not have a countable local basis is $\Omega$. So all the others have, and that means all the points in $\hointr 0 \Omega$ have a countable local basis.}} | Uncountable Open Ordinal Space is First-Countable | https://proofwiki.org/wiki/Uncountable_Open_Ordinal_Space_is_First-Countable | https://proofwiki.org/wiki/Uncountable_Open_Ordinal_Space_is_First-Countable | [
"Uncountable Open Ordinal Spaces",
"Examples of First-Countable Spaces"
] | [
"Definition:Uncountable Ordinal",
"Definition:Ordinal Space/Open/Uncountable",
"Definition:First-Countable Space"
] | [] |
proofwiki-14049 | Ordinal Space is Completely Normal | Let $\Gamma$ denote a limit ordinal.
Let $\hointr 0 \Gamma$ denote the open ordinal space on $\Gamma$.
Let $\closedint 0 \Gamma$ denote the closed ordinal space on $\Gamma$.
Then $\hointr 0 \Gamma$ and $\closedint 0 \Gamma$ are both completely normal. | By definition, $\hointr 0 \Gamma$ and $\closedint 0 \Gamma$ are both linearly ordered spaces.
The result follows from Linearly Ordered Space is Completely Normal.
{{qed}} | Let $\Gamma$ denote a [[Definition:Limit Ordinal|limit ordinal]].
Let $\hointr 0 \Gamma$ denote the [[Definition:Open Ordinal Space|open ordinal space]] on $\Gamma$.
Let $\closedint 0 \Gamma$ denote the [[Definition:Closed Ordinal Space|closed ordinal space]] on $\Gamma$.
Then $\hointr 0 \Gamma$ and $\closedint 0 \... | By definition, $\hointr 0 \Gamma$ and $\closedint 0 \Gamma$ are both [[Definition:Linearly Ordered Space|linearly ordered spaces]].
The result follows from [[Linearly Ordered Space is Completely Normal]].
{{qed}} | Ordinal Space is Completely Normal | https://proofwiki.org/wiki/Ordinal_Space_is_Completely_Normal | https://proofwiki.org/wiki/Ordinal_Space_is_Completely_Normal | [
"Ordinal Spaces",
"Examples of Completely Normal Spaces"
] | [
"Definition:Limit Ordinal",
"Definition:Ordinal Space/Open",
"Definition:Ordinal Space/Closed",
"Definition:Completely Normal Space"
] | [
"Definition:Linearly Ordered Space",
"Linearly Ordered Space is Completely Normal"
] |
proofwiki-14050 | Uncountable Closed Ordinal Space is not Perfectly Normal | Let $\Omega$ denote the first uncountable ordinal.
Let $\closedint 0 \Omega$ denote the closed ordinal space on $\Omega$.
Then $\closedint 0 \Omega$ is not a perfectly normal space. | From Omega is Closed in Uncountable Closed Ordinal Space but not $G_\delta$ Set, $\set \Omega$ is not a $G_\delta$ set.
From Ordinal Space is Completely Normal, $\closedint 0 \Omega$ is a $T_1$ space.
Thus by definition $\set \Omega$ is closed in $\closedint 0 \Omega$.
Thus we have that $\set \Omega$ is a closed set of... | Let $\Omega$ denote the first [[Definition:Uncountable Ordinal|uncountable ordinal]].
Let $\closedint 0 \Omega$ denote the [[Definition:Uncountable Closed Ordinal Space|closed ordinal space]] on $\Omega$.
Then $\closedint 0 \Omega$ is not a [[Definition:Perfectly Normal Space|perfectly normal space]]. | From [[Omega is Closed in Uncountable Closed Ordinal Space but not G-Delta Set|Omega is Closed in Uncountable Closed Ordinal Space but not $G_\delta$ Set]], $\set \Omega$ is not a [[Definition:G-Delta Set|$G_\delta$ set]].
From [[Ordinal Space is Completely Normal]], $\closedint 0 \Omega$ is a [[Definition:T1 Space|$T... | Uncountable Closed Ordinal Space is not Perfectly Normal | https://proofwiki.org/wiki/Uncountable_Closed_Ordinal_Space_is_not_Perfectly_Normal | https://proofwiki.org/wiki/Uncountable_Closed_Ordinal_Space_is_not_Perfectly_Normal | [
"Uncountable Closed Ordinal Spaces",
"Examples of Perfectly Normal Spaces"
] | [
"Definition:Uncountable Ordinal",
"Definition:Ordinal Space/Closed/Uncountable",
"Definition:Perfectly Normal Space"
] | [
"Omega is Closed in Uncountable Closed Ordinal Space but not G-Delta Set",
"Definition:G-Delta Set",
"Ordinal Space is Completely Normal",
"Definition:T1 Space",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:G-Delta Set",
"Definition:Perfectly Normal Space"
] |
proofwiki-14051 | Uncountable Closed Ordinal Space is not Second-Countable | Let $\Omega$ denote the first uncountable ordinal.
Let $\closedint 0 \Omega$ denote the closed ordinal space on $\Omega$.
Then $\closedint 0 \Omega$ is not a second-countable space. | {{ProofWanted|Show it does not have a countable basis}} | Let $\Omega$ denote the first [[Definition:Uncountable Ordinal|uncountable ordinal]].
Let $\closedint 0 \Omega$ denote the [[Definition:Uncountable Closed Ordinal Space|closed ordinal space]] on $\Omega$.
Then $\closedint 0 \Omega$ is not a [[Definition:Second-Countable Space|second-countable space]]. | {{ProofWanted|Show it does not have a countable basis}} | Uncountable Closed Ordinal Space is not Second-Countable | https://proofwiki.org/wiki/Uncountable_Closed_Ordinal_Space_is_not_Second-Countable | https://proofwiki.org/wiki/Uncountable_Closed_Ordinal_Space_is_not_Second-Countable | [
"Uncountable Closed Ordinal Spaces",
"Examples of Second-Countable Spaces"
] | [
"Definition:Uncountable Ordinal",
"Definition:Ordinal Space/Closed/Uncountable",
"Definition:Second-Countable Space"
] | [] |
proofwiki-14052 | Uncountable Open Ordinal Space is not Second-Countable | Let $\Omega$ denote the first uncountable ordinal.
Let $\hointr 0 \Omega$ denote the open ordinal space on $\Omega$.
Then $\hointr 0 \Omega$ is not a second-countable space. | {{ProofWanted|Show it does not have a countable basis}} | Let $\Omega$ denote the first [[Definition:Uncountable Ordinal|uncountable ordinal]].
Let $\hointr 0 \Omega$ denote the [[Definition:Uncountable Open Ordinal Space|open ordinal space]] on $\Omega$.
Then $\hointr 0 \Omega$ is not a [[Definition:Second-Countable Space|second-countable space]]. | {{ProofWanted|Show it does not have a countable basis}} | Uncountable Open Ordinal Space is not Second-Countable | https://proofwiki.org/wiki/Uncountable_Open_Ordinal_Space_is_not_Second-Countable | https://proofwiki.org/wiki/Uncountable_Open_Ordinal_Space_is_not_Second-Countable | [
"Uncountable Open Ordinal Spaces",
"Examples of Second-Countable Spaces"
] | [
"Definition:Uncountable Ordinal",
"Definition:Ordinal Space/Open/Uncountable",
"Definition:Second-Countable Space"
] | [] |
proofwiki-14053 | Ring Element is Unit iff Unit in Integral Extension | let $A$ be a commutative ring with unity.
Let $a \in A$.
Let $B$ be an integral ring extension of $A$.
{{TFAE}}
:$(1): \quad a$ is a unit of $A$
:$(2): \quad a$ is a unit of $B$ | === 1 implies 2 ===
Follows from Ring Homomorphism Preserves Units.
{{qed|lemma}} | let $A$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $a \in A$.
Let $B$ be an [[Definition:Integral Ring Extension|integral ring extension]] of $A$.
{{TFAE}}
:$(1): \quad a$ is a [[Definition:Unit of Ring|unit]] of $A$
:$(2): \quad a$ is a [[Definition:Unit of Ring|unit]] of $B$ | === 1 implies 2 ===
Follows from [[Ring Homomorphism Preserves Units]].
{{qed|lemma}} | Ring Element is Unit iff Unit in Integral Extension | https://proofwiki.org/wiki/Ring_Element_is_Unit_iff_Unit_in_Integral_Extension | https://proofwiki.org/wiki/Ring_Element_is_Unit_iff_Unit_in_Integral_Extension | [
"Integral Ring Extensions"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Integral Ring Extension",
"Definition:Unit of Ring",
"Definition:Unit of Ring"
] | [
"Ring Homomorphism Preserves Units"
] |
proofwiki-14054 | Separable Elements Form Field | Let $E / F$ be an algebraic field extension.
Let $K$ be the relative separable closure of $F$ in $E$.
Then $K$ is an intermediate field of $E / F$. | We need to show that $K$ is a field.
By Transitivity of Separable Field Extensions, an algebraic extension generated by a family of separable elements is separable.
{{proof wanted|proof of Theorem 4.5 in Lang's Algebra}} | Let $E / F$ be an [[Definition:Algebraic Field Extension|algebraic field extension]].
Let $K$ be the [[Definition:Relative Separable Closure|relative separable closure]] of $F$ in $E$.
Then $K$ is an [[Definition:Intermediate Field|intermediate field]] of $E / F$. | We need to show that $K$ is a [[Definition:Field (Abstract Algebra)|field]].
By [[Transitivity of Separable Field Extensions]], an algebraic extension generated by a family of separable elements is separable.
{{proof wanted|proof of Theorem 4.5 in Lang's Algebra}} | Separable Elements Form Field | https://proofwiki.org/wiki/Separable_Elements_Form_Field | https://proofwiki.org/wiki/Separable_Elements_Form_Field | [
"Separable Field Extensions"
] | [
"Definition:Algebraic Field Extension",
"Definition:Relative Separable Closure",
"Definition:Intermediate Field"
] | [
"Definition:Field (Abstract Algebra)",
"Transitivity of Separable Field Extensions"
] |
proofwiki-14055 | Decomposition of Field Extension as Separable Extension followed by Purely Inseparable | Let $E / F$ be an algebraic field extension.
Then the relative separable closure $K = F^{sep}$ in $E$ is the unique intermediate field with the following properties:
:$K / F$ is separable
:$E / K$ is purely inseparable. | Let $K = F^{sep}$ be the set of elements of $E$ which are separable over $F$.
This is a subextension by Separable Elements Form Field.
Then elements of $E \setminus K$ are not separable over $F$, since all elements that are separable over $F$ are in $K$.
Then elements of $E \setminus K$ are not separable over $K$, sinc... | Let $E / F$ be an [[Definition:Algebraic Field Extension|algebraic field extension]].
Then the [[Definition:Relative Separable Closure|relative separable closure]] $K = F^{sep}$ in $E$ is the [[Definition:Unique|unique]] [[Definition:Intermediate Field|intermediate field]] with the following properties:
:$K / F$ is [... | Let $K = F^{sep}$ be the set of elements of $E$ which are [[Definition:Separable Element|separable]] over $F$.
This is a subextension by [[Separable Elements Form Field]].
Then elements of $E \setminus K$ are not [[Definition:Separable Element|separable]] over $F$, since all elements that are [[Definition:Separable E... | Decomposition of Field Extension as Separable Extension followed by Purely Inseparable | https://proofwiki.org/wiki/Decomposition_of_Field_Extension_as_Separable_Extension_followed_by_Purely_Inseparable | https://proofwiki.org/wiki/Decomposition_of_Field_Extension_as_Separable_Extension_followed_by_Purely_Inseparable | [
"Field Extensions"
] | [
"Definition:Algebraic Field Extension",
"Definition:Relative Separable Closure",
"Definition:Unique",
"Definition:Intermediate Field",
"Definition:Separable Extension",
"Definition:Purely Inseparable Field Extension"
] | [
"Definition:Separable Element",
"Separable Elements Form Field",
"Definition:Separable Element",
"Definition:Separable Element",
"Definition:Separable Element",
"Definition:Separable Element",
"Definition:Separable Element",
"Transitivity of Separable Field Extensions",
"Definition:Purely Inseparabl... |
proofwiki-14056 | Subextensions of Separable Field Extension are Separable | Let $E / K / F$ be a tower of fields.
Let $E / F$ be separable.
Then $E / K$ and $K / F$ are separable. | === Upper extension ===
We prove that $E / K$ is separable.
Let $\alpha \in E$.
Let $f$ be its minimal polynomial over $F$.
Let $g$ be its minimal polynomial over $K$.
Then {{hypothesis}}, $f$ is separable.
On the other hand:
:$f \in K \sqbrk x$
and:
:$\map f \alpha = 0$
Hence by definition $g$ divides $f$.
{{ExtractTh... | Let $E / K / F$ be a [[Definition:Tower of Fields|tower of fields]].
Let $E / F$ be [[Definition:Separable Field Extension|separable]].
Then $E / K$ and $K / F$ are [[Definition:Separable Field Extension|separable]]. | === Upper extension ===
We prove that $E / K$ is [[Definition:Separable Field Extension|separable]].
Let $\alpha \in E$.
Let $f$ be its [[Definition:Minimal Polynomial|minimal polynomial]] over $F$.
Let $g$ be its [[Definition:Minimal Polynomial|minimal polynomial]] over $K$.
Then {{hypothesis}}, $f$ is [[Definiti... | Subextensions of Separable Field Extension are Separable | https://proofwiki.org/wiki/Subextensions_of_Separable_Field_Extension_are_Separable | https://proofwiki.org/wiki/Subextensions_of_Separable_Field_Extension_are_Separable | [
"Separable Field Extensions"
] | [
"Definition:Tower of Fields",
"Definition:Separable Extension",
"Definition:Separable Extension"
] | [
"Definition:Separable Extension",
"Definition:Minimal Polynomial",
"Definition:Minimal Polynomial",
"Definition:Separable Polynomial",
"Definition:Divisor of Polynomial",
"Divisor of Separable Polynomial is Separable",
"Definition:Algebraic Closure",
"Definition:Algebraic Closure",
"Definition:Separ... |
proofwiki-14057 | Finite Orbit under Group of Automorphisms of Field implies Separable over Fixed Field | Let $E$ be a field.
Let $G \le \Aut E$ be a subgroup of its automorphism group.
Let $F = \map {\operatorname {Fix}_E} G$ be its fixed field.
Let $\alpha \in E$ have a finite orbit under $G$.
Then $\alpha$ is separable over $F$. | Let $\Lambda$ be the orbit of $\alpha$ under $G$.
By:
:Minimal Polynomial of Element with Finite Orbit under Group of Automorphisms over Fixed Field in terms of Orbit
the product:
:$\map p x = \ds \prod_{\beta \in \Lambda} \paren {x - \beta}$
is the minimal polynomial of $\alpha$ over $F$.
By Product of Distinct Monic ... | Let $E$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $G \le \Aut E$ be a [[Definition:Subgroup|subgroup]] of its [[Definition:Automorphism Group of Field|automorphism group]].
Let $F = \map {\operatorname {Fix}_E} G$ be its [[Definition:Fixed Field|fixed field]].
Let $\alpha \in E$ have a [[Definition:Fin... | Let $\Lambda$ be the [[Definition:Orbit under Group of Permutations|orbit]] of $\alpha$ under $G$.
By:
:[[Minimal Polynomial of Element with Finite Orbit under Group of Automorphisms over Fixed Field in terms of Orbit]]
the [[Definition:Product over Finite Set|product]]:
:$\map p x = \ds \prod_{\beta \in \Lambda} \par... | Finite Orbit under Group of Automorphisms of Field implies Separable over Fixed Field | https://proofwiki.org/wiki/Finite_Orbit_under_Group_of_Automorphisms_of_Field_implies_Separable_over_Fixed_Field | https://proofwiki.org/wiki/Finite_Orbit_under_Group_of_Automorphisms_of_Field_implies_Separable_over_Fixed_Field | [
"Field Extensions"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Subgroup",
"Definition:Automorphism Group of Field",
"Definition:Fixed Field",
"Definition:Finite Set",
"Definition:Orbit under Group of Permutations",
"Definition:Separable Element"
] | [
"Definition:Orbit under Group of Permutations",
"Minimal Polynomial of Element with Finite Orbit under Group of Automorphisms over Fixed Field in terms of Orbit",
"Definition:Product over Finite Set",
"Definition:Minimal Polynomial",
"Product of Distinct Monic Linear Polynomials is Separable",
"Definition... |
proofwiki-14058 | Countable Closed Ordinal Space is Second-Countable | Let $\Omega$ denote the first uncountable ordinal.
Let $\Gamma$ be a limit ordinal which strictly precedes $\Omega$.
Let $\closedint 0 \Gamma$ denote the closed ordinal space on $\Gamma$.
Then $\closedint 0 \Gamma$ is a second-countable space. | From Basis for Open Ordinal Topology, the set $\BB$ of subsets of $\closedint 0 \Gamma$ of the form:
:$\openint \alpha {\beta + 1} = \hointl \alpha \beta = \set {x \in \hointr 0 \Gamma: \alpha < x < \beta + 1}$
for $\alpha, \beta \in \hointr 0 \Gamma$, forms a basis for $\closedint 0 \Gamma$.
As $\Gamma$ strictly prece... | Let $\Omega$ denote the first [[Definition:Uncountable Ordinal|uncountable ordinal]].
Let $\Gamma$ be a [[Definition:Limit Ordinal|limit ordinal]] which [[Definition:Strictly Precede|strictly precedes]] $\Omega$.
Let $\closedint 0 \Gamma$ denote the [[Definition:Countable Closed Ordinal Space|closed ordinal space]] o... | From [[Basis for Open Ordinal Topology]], the [[Definition:Set of Sets|set]] $\BB$ of [[Definition:Subset|subsets]] of $\closedint 0 \Gamma$ of the form:
:$\openint \alpha {\beta + 1} = \hointl \alpha \beta = \set {x \in \hointr 0 \Gamma: \alpha < x < \beta + 1}$
for $\alpha, \beta \in \hointr 0 \Gamma$, forms a [[Defi... | Countable Closed Ordinal Space is Second-Countable | https://proofwiki.org/wiki/Countable_Closed_Ordinal_Space_is_Second-Countable | https://proofwiki.org/wiki/Countable_Closed_Ordinal_Space_is_Second-Countable | [
"Countable Closed Ordinal Spaces",
"Examples of Second-Countable Spaces"
] | [
"Definition:Uncountable Ordinal",
"Definition:Limit Ordinal",
"Definition:Strictly Precede",
"Definition:Ordinal Space/Closed/Countable",
"Definition:Second-Countable Space"
] | [
"Basis for Open Ordinal Topology",
"Definition:Set of Sets",
"Definition:Subset",
"Definition:Basis (Topology)",
"Definition:Strictly Precede",
"Definition:Countable Set",
"Definition:Countable Set",
"Definition:Basis (Topology)",
"Countable Union of Countable Sets is Countable",
"Definition:Count... |
proofwiki-14059 | Countable Open Ordinal Space is Second-Countable | Let $\Omega$ denote the first uncountable ordinal.
Let $\Gamma$ be a limit ordinal which strictly precedes $\Omega$.
Let $\hointr 0 \Gamma$ denote the open ordinal space on $\Gamma$.
Then $\hointr 0 \Gamma$ is a second-countable space. | From Basis for Open Ordinal Topology, the set $\BB$ of subsets of $\hointr 0 \Gamma$ of the form:
:$\openint \alpha {\beta + 1} = \hointl \alpha \beta = \set {x \in \hointr 0 \Gamma: \alpha < x < \beta + 1}$
for $\alpha, \beta \in \hointr 0 \Gamma$, forms a basis for $\hointr 0 \Gamma$.
As $\Gamma$ strictly precedes $\... | Let $\Omega$ denote the first [[Definition:Uncountable Ordinal|uncountable ordinal]].
Let $\Gamma$ be a [[Definition:Limit Ordinal|limit ordinal]] which [[Definition:Strictly Precede|strictly precedes]] $\Omega$.
Let $\hointr 0 \Gamma$ denote the [[Definition:Countable Open Ordinal Space|open ordinal space]] on $\Gam... | From [[Basis for Open Ordinal Topology]], the [[Definition:Set of Sets|set]] $\BB$ of [[Definition:Subset|subsets]] of $\hointr 0 \Gamma$ of the form:
:$\openint \alpha {\beta + 1} = \hointl \alpha \beta = \set {x \in \hointr 0 \Gamma: \alpha < x < \beta + 1}$
for $\alpha, \beta \in \hointr 0 \Gamma$, forms a [[Definit... | Countable Open Ordinal Space is Second-Countable | https://proofwiki.org/wiki/Countable_Open_Ordinal_Space_is_Second-Countable | https://proofwiki.org/wiki/Countable_Open_Ordinal_Space_is_Second-Countable | [
"Countable Open Ordinal Spaces",
"Examples of Second-Countable Spaces"
] | [
"Definition:Uncountable Ordinal",
"Definition:Limit Ordinal",
"Definition:Strictly Precede",
"Definition:Ordinal Space/Open/Countable",
"Definition:Second-Countable Space"
] | [
"Basis for Open Ordinal Topology",
"Definition:Set of Sets",
"Definition:Subset",
"Definition:Basis (Topology)",
"Definition:Strictly Precede",
"Definition:Countable Set",
"Definition:Countable Set",
"Definition:Basis (Topology)",
"Countable Union of Countable Sets is Countable",
"Definition:Count... |
proofwiki-14060 | Minimal Polynomial of Element with Finite Orbit under Group of Automorphisms over Fixed Field in terms of Orbit | Let $E$ be a field.
Let $G \le \Aut E$ be a subgroup of its automorphism group.
Let $F = \map {\operatorname {Fix_E} } G$ be its fixed field.
Let $\alpha \in E$ have a finite orbit under $G$.
Then $\alpha$ is algebraic over $F$ and the product of polynomials
:$\ds \map p x = \prod_{\beta \mathop \in \Lambda} \paren {x ... | By Product over Finite Set with Zero Factor, we have $\map p \alpha = 0$.
By definition, $p \in E \sqbrk x$. | Let $E$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $G \le \Aut E$ be a [[Definition:Subgroup|subgroup]] of its [[Definition:Automorphism Group of Field|automorphism group]].
Let $F = \map {\operatorname {Fix_E} } G$ be its [[Definition:Fixed Field|fixed field]].
Let $\alpha \in E$ have a [[Definition:Fi... | By [[Product over Finite Set with Zero Factor]], we have $\map p \alpha = 0$.
By definition, $p \in E \sqbrk x$. | Minimal Polynomial of Element with Finite Orbit under Group of Automorphisms over Fixed Field in terms of Orbit | https://proofwiki.org/wiki/Minimal_Polynomial_of_Element_with_Finite_Orbit_under_Group_of_Automorphisms_over_Fixed_Field_in_terms_of_Orbit | https://proofwiki.org/wiki/Minimal_Polynomial_of_Element_with_Finite_Orbit_under_Group_of_Automorphisms_over_Fixed_Field_in_terms_of_Orbit | [
"Field Extensions"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Subgroup",
"Definition:Automorphism Group of Field",
"Definition:Fixed Field",
"Definition:Finite Set",
"Definition:Orbit under Group of Permutations",
"Definition:Algebraic Element of Field Extension",
"Definition:Product over Finite Set",
"Definit... | [
"Product over Finite Set with Zero Factor"
] |
proofwiki-14061 | Countable Closed Ordinal Space is Metrizable | Let $\Omega$ denote the first uncountable ordinal.
Let $\Gamma$ be a limit ordinal which strictly precedes $\Omega$.
Let $\closedint 0 \Gamma$ denote the closed ordinal space on $\Gamma$.
Then $\closedint 0 \Gamma$ is a metrizable space. | From Countable Closed Ordinal Space is Second-Countable, $\closedint 0 \Gamma$ has a basis which is $\sigma$-locally finite.
From Ordinal Space is Regular, $\closedint 0 \Gamma$ is a regular space.
The result follows from Metrizable iff Regular and has Sigma-Locally Finite Basis.
{{qed}} | Let $\Omega$ denote the first [[Definition:Uncountable Ordinal|uncountable ordinal]].
Let $\Gamma$ be a [[Definition:Limit Ordinal|limit ordinal]] which [[Definition:Strictly Precede|strictly precedes]] $\Omega$.
Let $\closedint 0 \Gamma$ denote the [[Definition:Countable Closed Ordinal Space|closed ordinal space]] o... | From [[Countable Closed Ordinal Space is Second-Countable]], $\closedint 0 \Gamma$ has a [[Definition:Basis (Topology)|basis]] which is [[Definition:Sigma-Locally Finite Basis|$\sigma$-locally finite]].
From [[Ordinal Space is Regular]], $\closedint 0 \Gamma$ is a [[Definition:Regular Space|regular space]].
The resul... | Countable Closed Ordinal Space is Metrizable | https://proofwiki.org/wiki/Countable_Closed_Ordinal_Space_is_Metrizable | https://proofwiki.org/wiki/Countable_Closed_Ordinal_Space_is_Metrizable | [
"Countable Closed Ordinal Spaces",
"Countable Ordinal Spaces",
"Examples of Metrizable Spaces"
] | [
"Definition:Uncountable Ordinal",
"Definition:Limit Ordinal",
"Definition:Strictly Precede",
"Definition:Ordinal Space/Closed/Countable",
"Definition:Metrizable Space"
] | [
"Countable Closed Ordinal Space is Second-Countable",
"Definition:Basis (Topology)",
"Definition:Sigma-Locally Finite Basis",
"Ordinal Space is Regular",
"Definition:Regular Space",
"Nagata-Smirnov Metrization Theorem"
] |
proofwiki-14062 | Countable Open Ordinal Space is Metrizable | Let $\Omega$ denote the first uncountable ordinal.
Let $\Gamma$ be a limit ordinal which strictly precedes $\Omega$.
Let $\hointr 0 \Gamma$ denote the open ordinal space on $\Gamma$.
Then $\hointr 0 \Gamma$ is a metrizable space. | From Countable Open Ordinal Space is Second-Countable, $\hointr 0 \Gamma$ has a basis which is $\sigma$-locally finite.
From Ordinal Space is Regular, $\hointr 0 \Gamma$ is a regular space.
The result follows from Metrizable iff Regular and has Sigma-Locally Finite Basis.
{{qed}} | Let $\Omega$ denote the first [[Definition:Uncountable Ordinal|uncountable ordinal]].
Let $\Gamma$ be a [[Definition:Limit Ordinal|limit ordinal]] which [[Definition:Strictly Precede|strictly precedes]] $\Omega$.
Let $\hointr 0 \Gamma$ denote the [[Definition:Countable Open Ordinal Space|open ordinal space]] on $\Gam... | From [[Countable Open Ordinal Space is Second-Countable]], $\hointr 0 \Gamma$ has a [[Definition:Basis (Topology)|basis]] which is [[Definition:Sigma-Locally Finite Basis|$\sigma$-locally finite]].
From [[Ordinal Space is Regular]], $\hointr 0 \Gamma$ is a [[Definition:Regular Space|regular space]].
The result follow... | Countable Open Ordinal Space is Metrizable | https://proofwiki.org/wiki/Countable_Open_Ordinal_Space_is_Metrizable | https://proofwiki.org/wiki/Countable_Open_Ordinal_Space_is_Metrizable | [
"Countable Open Ordinal Spaces",
"Countable Ordinal Spaces",
"Examples of Metrizable Spaces"
] | [
"Definition:Uncountable Ordinal",
"Definition:Limit Ordinal",
"Definition:Strictly Precede",
"Definition:Ordinal Space/Open/Countable",
"Definition:Metrizable Space"
] | [
"Countable Open Ordinal Space is Second-Countable",
"Definition:Basis (Topology)",
"Definition:Sigma-Locally Finite Basis",
"Ordinal Space is Regular",
"Definition:Regular Space",
"Nagata-Smirnov Metrization Theorem"
] |
proofwiki-14063 | Closed Ordinal Space is Complete Order Space | Let $\Gamma$ be a limit ordinal.
Let $\closedint 0 \Gamma$ denote the closed ordinal space on $\Gamma$.
Then $\closedint 0 \Gamma$ is a complete order space. | Let $H$ be a subset of an ordinal space.
Then $H$ has an infimum: its first element.
Let $H$ be a subset of $\closedint 0 \Gamma$.
Then $H$ has a supremum.
Therefore $\closedint 0 \Gamma$ is a complete order space.
{{qed}} | Let $\Gamma$ be a [[Definition:Limit Ordinal|limit ordinal]].
Let $\closedint 0 \Gamma$ denote the [[Definition:Closed Ordinal Space|closed ordinal space]] on $\Gamma$.
Then $\closedint 0 \Gamma$ is a [[Definition:Complete Order Topology|complete order space]]. | Let $H$ be a [[Definition:Subset|subset]] of an [[Definition:Ordinal Space|ordinal space]].
Then $H$ has an [[Definition:Infimum of Set|infimum]]: its first [[Definition:Element|element]].
Let $H$ be a [[Definition:Subset|subset]] of $\closedint 0 \Gamma$.
Then $H$ has a [[Definition:Supremum|supremum]].
Therefore ... | Closed Ordinal Space is Complete Order Space | https://proofwiki.org/wiki/Closed_Ordinal_Space_is_Complete_Order_Space | https://proofwiki.org/wiki/Closed_Ordinal_Space_is_Complete_Order_Space | [
"Closed Ordinal Spaces",
"Examples of Complete Order Topologies"
] | [
"Definition:Limit Ordinal",
"Definition:Ordinal Space/Closed",
"Definition:Complete Order Topology"
] | [
"Definition:Subset",
"Definition:Ordinal Space",
"Definition:Infimum of Set",
"Definition:Element",
"Definition:Subset",
"Definition:Supremum",
"Definition:Complete Order Topology"
] |
proofwiki-14064 | Closed Ordinal Space is Compact | Let $\Gamma$ be a limit ordinal.
Let $\closedint 0 \Gamma$ denote the closed ordinal space on $\Gamma$.
Then $\closedint 0 \Gamma$ is a compact space. | By definition, $\closedint 0 \Gamma$ is a linearly ordered space.
The result follows from Linearly Ordered Space is Compact iff Complete.
{{qed}} | Let $\Gamma$ be a [[Definition:Limit Ordinal|limit ordinal]].
Let $\closedint 0 \Gamma$ denote the [[Definition:Closed Ordinal Space|closed ordinal space]] on $\Gamma$.
Then $\closedint 0 \Gamma$ is a [[Definition:Compact Topological Space|compact space]]. | By definition, $\closedint 0 \Gamma$ is a [[Definition:Linearly Ordered Space|linearly ordered space]].
The result follows from [[Linearly Ordered Space is Compact iff Complete]].
{{qed}} | Closed Ordinal Space is Compact | https://proofwiki.org/wiki/Closed_Ordinal_Space_is_Compact | https://proofwiki.org/wiki/Closed_Ordinal_Space_is_Compact | [
"Closed Ordinal Spaces",
"Examples of Compact Topological Spaces"
] | [
"Definition:Limit Ordinal",
"Definition:Ordinal Space/Closed",
"Definition:Compact Topological Space"
] | [
"Definition:Linearly Ordered Space",
"Linearly Ordered Space is Compact iff Complete"
] |
proofwiki-14065 | Ordinal Space is Strongly Locally Compact | Let $T$ denote an ordinal space on a limit ordinal $\Gamma$.
Then $T$ is a strongly locally compact space. | {{ProofWanted|Demonstrated by showing that the closure of each basis neighborhood is compact.}} | Let $T$ denote an [[Definition:Ordinal Space|ordinal space]] on a [[Definition:Limit Ordinal|limit ordinal]] $\Gamma$.
Then $T$ is a [[Definition:Strongly Locally Compact Space|strongly locally compact space]]. | {{ProofWanted|Demonstrated by showing that the closure of each basis neighborhood is compact.}} | Ordinal Space is Strongly Locally Compact | https://proofwiki.org/wiki/Ordinal_Space_is_Strongly_Locally_Compact | https://proofwiki.org/wiki/Ordinal_Space_is_Strongly_Locally_Compact | [
"Ordinal Spaces",
"Examples of Strongly Locally Compact Spaces"
] | [
"Definition:Ordinal Space",
"Definition:Limit Ordinal",
"Definition:Strongly Locally Compact Space"
] | [] |
proofwiki-14066 | Open Ordinal Space is not Compact Space | Let $\Gamma$ be a limit ordinal.
Let $\hointr 0 \Gamma$ denote the open ordinal space on $\Gamma$.
Consider the compact subspace $\hointr 0 \Gamma$.
Then $\hointr 0 \Gamma$ is not compact in $\closedint 0 \Gamma$. | Consider the set:
:$\set {\hointr 0 \Gamma: \alpha < \Gamma}$
This is an open cover of $\hointr 0 \Gamma$.
But because $\Gamma$ is a limit ordinal, it has no finite subcover.
Hence the result by definition of compact.
{{qed}} | Let $\Gamma$ be a [[Definition:Limit Ordinal|limit ordinal]].
Let $\hointr 0 \Gamma$ denote the [[Definition:Open Ordinal Space|open ordinal space]] on $\Gamma$.
Consider the [[Definition:Compact Topological Subspace|compact subspace]] $\hointr 0 \Gamma$.
Then $\hointr 0 \Gamma$ is not [[Definition:Compact Topologi... | Consider the [[Definition:Set|set]]:
:$\set {\hointr 0 \Gamma: \alpha < \Gamma}$
This is an [[Definition:Open Cover|open cover]] of $\hointr 0 \Gamma$.
But because $\Gamma$ is a [[Definition:Limit Ordinal|limit ordinal]], it has no [[Definition:Finite Subcover|finite subcover]].
Hence the result by definition of [[... | Open Ordinal Space is not Compact Space | https://proofwiki.org/wiki/Open_Ordinal_Space_is_not_Compact_Space | https://proofwiki.org/wiki/Open_Ordinal_Space_is_not_Compact_Space | [
"Open Ordinal Spaces",
"Examples of Compact Topological Spaces"
] | [
"Definition:Limit Ordinal",
"Definition:Ordinal Space/Open",
"Definition:Compact Topological Space/Subspace",
"Definition:Compact Topological Space"
] | [
"Definition:Set",
"Definition:Open Cover",
"Definition:Limit Ordinal",
"Definition:Subcover/Finite",
"Definition:Compact Topological Space"
] |
proofwiki-14067 | Integral Transform is Mapping | Let $\map F p$ be an integral transform:
:$\map F p = \ds \int_a^b \map f x \map K {p, x} \rd x$
Let $T$ be the integral operator associated with $\map F p$.
Then $T$ is a mapping from the domain of $T$ to its image.
That is, for every $\map f x$ there exists a unique $\map F p$. | Let $p$ be fixed.
In this context, $\map f x \map K {p, x}$ is the pointwise product of the functions $\map f x$ and $\map K {p, x}$.
From Pointwise Operation is Well-Defined, it follows that $\map f x \map K {p, x}$ is a real function on $x$.
We have that both $\map f x$ and $\map K {p, x}$ are integrable.
It follows ... | Let $\map F p$ be an [[Definition:Integral Transform|integral transform]]:
:$\map F p = \ds \int_a^b \map f x \map K {p, x} \rd x$
Let $T$ be the [[Definition:Integral Transform/Operator|integral operator]] associated with $\map F p$.
Then $T$ is a [[Definition:Mapping|mapping]] from the [[Definition:Domain of Mapp... | Let $p$ be fixed.
In this context, $\map f x \map K {p, x}$ is the [[Definition:Pointwise Multiplication of Real-Valued Functions|pointwise product]] of the [[Definition:Real Function|functions]] $\map f x$ and $\map K {p, x}$.
From [[Pointwise Operation is Well-Defined]], it follows that $\map f x \map K {p, x}$ is ... | Integral Transform is Mapping | https://proofwiki.org/wiki/Integral_Transform_is_Mapping | https://proofwiki.org/wiki/Integral_Transform_is_Mapping | [
"Integral Transforms"
] | [
"Definition:Integral Transform",
"Definition:Integral Transform/Operator",
"Definition:Mapping",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Unique"
] | [
"Definition:Pointwise Multiplication of Real-Valued Functions",
"Definition:Real Function",
"Pointwise Operation is Well-Defined",
"Definition:Real Function",
"Definition:Integrable Function",
"Pointwise Product of Integrable Real Functions is Integrable",
"Definition:Integrable Function",
"Definite I... |
proofwiki-14068 | Integral Operator is Linear | Let $T$ be an integral operator.
Let $f$ and $g$ be integrable real functions on a domain appropriate to $T$.
Then $T$ is a linear operator:
:$\forall \alpha, \beta \in \R: \map T {\alpha f + \beta g} = \alpha \map T f + \beta \map T g$ | Let $T$ be expressed in its full form as an integral fransform:
:$\map T f := \ds \int_a^b \map f x \map K {p, x} \rd x$
for some integrable function $\map K {p, x}$.
Then:
{{begin-eqn}}
{{eqn | l = \map T {\alpha f + \beta g}
| r = \int_a^b \paren {\alpha \map f x + \beta \map g x} \map K {p, x} \rd x
| c ... | Let $T$ be an [[Definition:Integral Operator|integral operator]].
Let $f$ and $g$ be [[Definition:Integrable Function|integrable]] [[Definition:Real Function|real functions]] on a [[Definition:Domain of Real Function|domain]] appropriate to $T$.
Then $T$ is a [[Definition:Linear Operator|linear operator]]:
:$\forall... | Let $T$ be expressed in its full form as an [[Definition:Integral Transform|integral fransform]]:
:$\map T f := \ds \int_a^b \map f x \map K {p, x} \rd x$
for some [[Definition:Integrable Function|integrable]] [[Definition:Real-Valued Function|function]] $\map K {p, x}$.
Then:
{{begin-eqn}}
{{eqn | l = \map T {\alph... | Integral Operator is Linear | https://proofwiki.org/wiki/Integral_Operator_is_Linear | https://proofwiki.org/wiki/Integral_Operator_is_Linear | [
"Integral Transforms",
"Integral Operator is Linear"
] | [
"Definition:Integral Operator",
"Definition:Integrable Function",
"Definition:Real Function",
"Definition:Real Function/Domain",
"Definition:Linear Operator"
] | [
"Definition:Integral Transform",
"Definition:Integrable Function",
"Definition:Real-Valued Function",
"Real Multiplication Distributes over Addition",
"Linear Combination of Integrals/Definite",
"Category:Integral Transforms",
"Category:Integral Operator is Linear"
] |
proofwiki-14069 | Integral Operator is Linear/Corollary 1 | :$\forall \alpha, \beta \in \R: \map T {f + g} = \map T f + \map T g$ | From Integral Operator is Linear:
:$\forall \alpha, \beta \in \R: \map T {\alpha f + \beta g} = \alpha \map T f + \beta \map T g$
The result follows by setting $\alpha = \beta = 1$.
{{Qed}} | :$\forall \alpha, \beta \in \R: \map T {f + g} = \map T f + \map T g$ | From [[Integral Operator is Linear]]:
:$\forall \alpha, \beta \in \R: \map T {\alpha f + \beta g} = \alpha \map T f + \beta \map T g$
The result follows by setting $\alpha = \beta = 1$.
{{Qed}} | Integral Operator is Linear/Corollary 1 | https://proofwiki.org/wiki/Integral_Operator_is_Linear/Corollary_1 | https://proofwiki.org/wiki/Integral_Operator_is_Linear/Corollary_1 | [
"Integral Operator is Linear"
] | [] | [
"Integral Operator is Linear"
] |
proofwiki-14070 | Integral Operator is Linear/Corollary 2 | :$\forall \alpha \in \R: \map T {\alpha f} = \alpha \map T f$ | From Integral Operator is Linear:
:$\forall \alpha, \beta \in \R: \map T {\alpha f + \beta g} = \alpha \map T f + \beta \map T g$
The result follows by setting $\beta = 0$.
{{Qed}} | :$\forall \alpha \in \R: \map T {\alpha f} = \alpha \map T f$ | From [[Integral Operator is Linear]]:
:$\forall \alpha, \beta \in \R: \map T {\alpha f + \beta g} = \alpha \map T f + \beta \map T g$
The result follows by setting $\beta = 0$.
{{Qed}} | Integral Operator is Linear/Corollary 2 | https://proofwiki.org/wiki/Integral_Operator_is_Linear/Corollary_2 | https://proofwiki.org/wiki/Integral_Operator_is_Linear/Corollary_2 | [
"Integral Operator is Linear"
] | [] | [
"Integral Operator is Linear"
] |
proofwiki-14071 | Inverse Integral Operator is Linear if Unique | Let $T$ be an integral operator.
Let $f$ be an integrable real function on a domain appropriate to $T$.
Let $F = \map T f$ and $G = \map T g$.
Let $T$ have a unique inverse $T^{-1}$.
Then $T^{-1}$ is a linear operator:
:$\forall p, q \in \R: \map {T^{-1} } {p F + q G} = p \map {T^{-1} } F + q \map {T^{-1} } G$ | Let:
:$x_1 = \map {T^{-1} } F$
:$x_2 = \map {T^{-1} } G$
Thus:
:$F = \map T {x_1}$
:$G = \map T {x_2}$
Then for all $p, q \in \R$:
{{begin-eqn}}
{{eqn | l = \map T {p x_1 + q x_2}
| r = p \map T {x_1} + q \map T {x_2}
| c = Integral Operator is Linear
}}
{{eqn | r = p F + q G
| c =
}}
{{end-eqn}}
and... | Let $T$ be an [[Definition:Integral Operator|integral operator]].
Let $f$ be an [[Definition:Integrable Function|integrable]] [[Definition:Real Function|real function]] on a [[Definition:Domain of Real Function|domain]] appropriate to $T$.
Let $F = \map T f$ and $G = \map T g$.
Let $T$ have a [[Definition:Unique|uni... | Let:
:$x_1 = \map {T^{-1} } F$
:$x_2 = \map {T^{-1} } G$
Thus:
:$F = \map T {x_1}$
:$G = \map T {x_2}$
Then for all $p, q \in \R$:
{{begin-eqn}}
{{eqn | l = \map T {p x_1 + q x_2}
| r = p \map T {x_1} + q \map T {x_2}
| c = [[Integral Operator is Linear]]
}}
{{eqn | r = p F + q G
| c =
}}
{{end-e... | Inverse Integral Operator is Linear if Unique | https://proofwiki.org/wiki/Inverse_Integral_Operator_is_Linear_if_Unique | https://proofwiki.org/wiki/Inverse_Integral_Operator_is_Linear_if_Unique | [
"Integral Transforms"
] | [
"Definition:Integral Operator",
"Definition:Integrable Function",
"Definition:Real Function",
"Definition:Real Function/Domain",
"Definition:Unique",
"Definition:Inverse Integral Operator",
"Definition:Linear Operator"
] | [
"Integral Operator is Linear",
"Definition:Linear Operator"
] |
proofwiki-14072 | Trigonometric Series is Convergent if Sum of Absolute Values of Coefficients is Convergent | Let $\map S x$ be a trigonometric series:
:$\map S x = \dfrac {a_0} 2 + \ds \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$
Let the series:
:$\ds \sum_{n \mathop = 1}^\infty \paren {\size {a_n} + \size {b_n} }$
be convergent.
Then $\map S x$ is a convergent series for each $x \in \R$. | For all $n \in \N_{\ge 1}$ and $x \in \R$, we have:
{{begin-eqn}}
{{eqn | o = \le
| l = \size {a_n \cos n x + b_n \sin n x}
| r = \size {a_n \cos n x} + \size {b_n \sin n x}
| c = Triangle Inequality for Real Numbers
}}
{{eqn | r = \size {a_n} \size {\cos n x} + \size {b_n} \size {\sin n x}
| c... | Let $\map S x$ be a [[Definition:Trigonometric Series|trigonometric series]]:
:$\map S x = \dfrac {a_0} 2 + \ds \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$
Let the [[Definition:Real Series|series]]:
:$\ds \sum_{n \mathop = 1}^\infty \paren {\size {a_n} + \size {b_n} }$
be [[Definition:Convergent... | For all $n \in \N_{\ge 1}$ and $x \in \R$, we have:
{{begin-eqn}}
{{eqn | o = \le
| l = \size {a_n \cos n x + b_n \sin n x}
| r = \size {a_n \cos n x} + \size {b_n \sin n x}
| c = [[Triangle Inequality for Real Numbers]]
}}
{{eqn | r = \size {a_n} \size {\cos n x} + \size {b_n} \size {\sin n x}
... | Trigonometric Series is Convergent if Sum of Absolute Values of Coefficients is Convergent | https://proofwiki.org/wiki/Trigonometric_Series_is_Convergent_if_Sum_of_Absolute_Values_of_Coefficients_is_Convergent | https://proofwiki.org/wiki/Trigonometric_Series_is_Convergent_if_Sum_of_Absolute_Values_of_Coefficients_is_Convergent | [
"Trigonometric Series",
"Convergence"
] | [
"Definition:Trigonometric Series",
"Definition:Series/Real",
"Definition:Convergent Series/Number Field",
"Definition:Convergent Series/Number Field"
] | [
"Triangle Inequality/Real Numbers",
"Absolute Value Function is Completely Multiplicative",
"Real Cosine Function is Bounded",
"Real Sine Function is Bounded",
"Definition:Series/Real",
"Definition:Convergent Series/Number Field",
"Comparison Test",
"Definition:Absolutely Convergent Series",
"Absolu... |
proofwiki-14073 | Convergent Trigonometric Series is Periodic | Let $\map S x$ be a trigonometric series:
:$\map S x = \dfrac {a_0} 2 + \ds \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$
Let $S$ be convergent.
Then $S$ is periodic:
:$\forall r \in \Z: \map S {x + 2 r \pi} = \map S x$ | Let $\map S x$ converge to some $L \in \R$.
Let $r \in \Z$ be arbitrary.
Then:
{{begin-eqn}}
{{eqn | l = \map S {x + 2 r \pi}
| r = \dfrac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos n \paren {x + 2 r \pi} + b_n \sin n \paren {x + 2 r \pi} }
| c = Definition of $\map S {x + 2 r \pi}$
}}
{{eqn | r... | Let $\map S x$ be a [[Definition:Trigonometric Series|trigonometric series]]:
:$\map S x = \dfrac {a_0} 2 + \ds \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$
Let $S$ be [[Definition:Convergent Series of Numbers|convergent]].
Then $S$ is [[Definition:Real Periodic Function|periodic]]:
:$\forall ... | Let $\map S x$ [[Definition:Convergent Series of Numbers|converge]] to some $L \in \R$.
Let $r \in \Z$ be arbitrary.
Then:
{{begin-eqn}}
{{eqn | l = \map S {x + 2 r \pi}
| r = \dfrac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos n \paren {x + 2 r \pi} + b_n \sin n \paren {x + 2 r \pi} }
| c = D... | Convergent Trigonometric Series is Periodic | https://proofwiki.org/wiki/Convergent_Trigonometric_Series_is_Periodic | https://proofwiki.org/wiki/Convergent_Trigonometric_Series_is_Periodic | [
"Trigonometric Series",
"Periodic Functions"
] | [
"Definition:Trigonometric Series",
"Definition:Convergent Series/Number Field",
"Definition:Periodic Function/Real"
] | [
"Definition:Convergent Series/Number Field",
"Sine and Cosine are Periodic on Reals"
] |
proofwiki-14074 | Sum of Infinite Series of Product of Power and Sine | Let $r \in \R$ such that $\size r < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^\infty r^k \sin k x
| r = r \sin x + r^2 \sin 2 x + r^3 \sin 3 x + \cdots
| c =
}}
{{eqn | r = \dfrac {r \sin x} {1 - 2 r \cos x + r^2}
| c =
}}
{{end-eqn}} | From Euler's Formula:
:$e^{i \theta} = \cos \theta + i \sin \theta$
Hence:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^\infty r^k \sin k x
| r = \map \Im {\sum_{k \mathop = 1}^\infty r^k e^{i k x} }
| c =
}}
{{eqn | r = \map \Im {\sum_{k \mathop = 0}^\infty \paren {r e^{i x} }^k}
| c = as $\map \I... | Let $r \in \R$ such that $\size r < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^\infty r^k \sin k x
| r = r \sin x + r^2 \sin 2 x + r^3 \sin 3 x + \cdots
| c =
}}
{{eqn | r = \dfrac {r \sin x} {1 - 2 r \cos x + r^2}
| c =
}}
{{end-eqn}} | From [[Euler's Formula]]:
:$e^{i \theta} = \cos \theta + i \sin \theta$
Hence:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^\infty r^k \sin k x
| r = \map \Im {\sum_{k \mathop = 1}^\infty r^k e^{i k x} }
| c =
}}
{{eqn | r = \map \Im {\sum_{k \mathop = 0}^\infty \paren {r e^{i x} }^k}
| c = as $\... | Sum of Infinite Series of Product of Power and Sine | https://proofwiki.org/wiki/Sum_of_Infinite_Series_of_Product_of_Power_and_Sine | https://proofwiki.org/wiki/Sum_of_Infinite_Series_of_Product_of_Power_and_Sine | [
"Sine Function",
"Trigonometric Series"
] | [] | [
"Euler's Formula",
"Sum of Infinite Geometric Sequence"
] |
proofwiki-14075 | Integral over 2 pi of Sine of m x by Sine of n x | Let $m, n \in \Z$ be integers.
Let $\alpha \in \R$ be a real number.
Then:
:$\ds \int_\alpha^{\alpha + 2 \pi} \sin m x \sin n x \rd x = \begin {cases} 0 & : m \ne n \\ \pi & : m = n \end {cases}$
That is:
:$\ds \int_\alpha^{\alpha + 2 \pi} \sin m x \sin n x \rd x = \pi \delta_{m n}$
where $\delta_{m n}$ is the Kronecke... | Let $m \ne n$.
{{begin-eqn}}
{{eqn | l = \int \sin m x \sin n x \rd x
| r = \frac {\sin \paren {m - n} x} {2 \paren {m - n} } - \frac {\sin \paren {m + n} x} {2 \paren {m + n} } + C
| c = Primitive of $\sin m x \sin n x$
}}
{{eqn | ll= \leadsto
| l = \int_\alpha^{\alpha + 2 \pi} \sin m x \sin n x \rd ... | Let $m, n \in \Z$ be [[Definition:Integer|integers]].
Let $\alpha \in \R$ be a [[Definition:Real Number|real number]].
Then:
:$\ds \int_\alpha^{\alpha + 2 \pi} \sin m x \sin n x \rd x = \begin {cases} 0 & : m \ne n \\ \pi & : m = n \end {cases}$
That is:
:$\ds \int_\alpha^{\alpha + 2 \pi} \sin m x \sin n x \rd x ... | Let $m \ne n$.
{{begin-eqn}}
{{eqn | l = \int \sin m x \sin n x \rd x
| r = \frac {\sin \paren {m - n} x} {2 \paren {m - n} } - \frac {\sin \paren {m + n} x} {2 \paren {m + n} } + C
| c = [[Primitive of Sine of a x by Sine of b x|Primitive of $\sin m x \sin n x$]]
}}
{{eqn | ll= \leadsto
| l = \int_\... | Integral over 2 pi of Sine of m x by Sine of n x | https://proofwiki.org/wiki/Integral_over_2_pi_of_Sine_of_m_x_by_Sine_of_n_x | https://proofwiki.org/wiki/Integral_over_2_pi_of_Sine_of_m_x_by_Sine_of_n_x | [
"Definite Integrals involving Sine Function"
] | [
"Definition:Integer",
"Definition:Real Number",
"Definition:Kronecker Delta"
] | [
"Primitive of Sine of a x by Sine of b x",
"Primitive of Square of Sine of a x"
] |
proofwiki-14076 | Integral over 2 pi of Cosine of m x by Cosine of n x | Let $m, n \in \Z$ be integers.
Let $\alpha \in \R$ be a real number.
Then:
:$\ds \int_\alpha^{\alpha + 2 \pi} \cos m x \cos n x \rd x = \begin {cases} 0 & : m \ne n \\ \pi & : m = n \end {cases}$
That is:
:$\ds \int_\alpha^{\alpha + 2 \pi} \cos m x \cos n x \rd x = \pi \delta_{m n}$
where $\delta_{m n}$ is the Kronecke... | Let $m \ne n$.
{{begin-eqn}}
{{eqn | l = \int \cos m x \cos n x \rd x
| r = \frac {\sin \paren {m - n} x} {2 \paren {m - n} } + \frac {\sin \paren {m + n} x} {2 \paren {m + n} } + C
| c = Primitive of $\cos m x \cos n x$
}}
{{eqn | ll= \leadsto
| l = \int_\alpha^{\alpha + 2 \pi} \cos m x \cos n x \rd ... | Let $m, n \in \Z$ be [[Definition:Integer|integers]].
Let $\alpha \in \R$ be a [[Definition:Real Number|real number]].
Then:
:$\ds \int_\alpha^{\alpha + 2 \pi} \cos m x \cos n x \rd x = \begin {cases} 0 & : m \ne n \\ \pi & : m = n \end {cases}$
That is:
:$\ds \int_\alpha^{\alpha + 2 \pi} \cos m x \cos n x \rd x =... | Let $m \ne n$.
{{begin-eqn}}
{{eqn | l = \int \cos m x \cos n x \rd x
| r = \frac {\sin \paren {m - n} x} {2 \paren {m - n} } + \frac {\sin \paren {m + n} x} {2 \paren {m + n} } + C
| c = [[Primitive of Cosine of a x by Cosine of b x|Primitive of $\cos m x \cos n x$]]
}}
{{eqn | ll= \leadsto
| l = \i... | Integral over 2 pi of Cosine of m x by Cosine of n x | https://proofwiki.org/wiki/Integral_over_2_pi_of_Cosine_of_m_x_by_Cosine_of_n_x | https://proofwiki.org/wiki/Integral_over_2_pi_of_Cosine_of_m_x_by_Cosine_of_n_x | [
"Definite Integrals involving Cosine Function"
] | [
"Definition:Integer",
"Definition:Real Number",
"Definition:Kronecker Delta"
] | [
"Primitive of Cosine of a x by Cosine of b x",
"Primitive of Square of Cosine of a x"
] |
proofwiki-14077 | Integral over 2 pi of Sine of m x by Cosine of n x | Let $m, n \in \Z$ be integers.
Let $\alpha \in \R$ be a real number.
Then:
:$\ds \int_\alpha^{\alpha + 2 \pi} \sin m x \cos n x \rd x = 0$ | Let $m \ne n$.
{{begin-eqn}}
{{eqn | l = \int \sin m x \cos n x \rd x
| r = \frac {-\map \cos {m - n} x} {2 \paren {m - n} } - \frac {\map \cos {m + n} x} {2 \paren {m + n} } + C
| c = Primitive of $\sin m x \cos n x$
}}
{{eqn | ll= \leadsto
| l = \int_\alpha^{\alpha + 2 \pi} \sin m x \cos n x \rd x
... | Let $m, n \in \Z$ be [[Definition:Integer|integers]].
Let $\alpha \in \R$ be a [[Definition:Real Number|real number]].
Then:
:$\ds \int_\alpha^{\alpha + 2 \pi} \sin m x \cos n x \rd x = 0$ | Let $m \ne n$.
{{begin-eqn}}
{{eqn | l = \int \sin m x \cos n x \rd x
| r = \frac {-\map \cos {m - n} x} {2 \paren {m - n} } - \frac {\map \cos {m + n} x} {2 \paren {m + n} } + C
| c = [[Primitive of Sine of a x by Cosine of b x|Primitive of $\sin m x \cos n x$]]
}}
{{eqn | ll= \leadsto
| l = \int_\a... | Integral over 2 pi of Sine of m x by Cosine of n x | https://proofwiki.org/wiki/Integral_over_2_pi_of_Sine_of_m_x_by_Cosine_of_n_x | https://proofwiki.org/wiki/Integral_over_2_pi_of_Sine_of_m_x_by_Cosine_of_n_x | [
"Definite Integrals involving Sine Function",
"Definite Integrals involving Cosine Function"
] | [
"Definition:Integer",
"Definition:Real Number"
] | [
"Primitive of Sine of a x by Cosine of b x",
"Primitive of Sine of a x by Cosine of a x"
] |
proofwiki-14078 | Integral over 2 pi of Sine of n x | Let $m \in \Z$ be an integer.
Then:
:$\ds \int_\alpha^{\alpha + 2 \pi} \sin m x \rd x = 0$ | Let $m \ne n$.
{{begin-eqn}}
{{eqn | l = \int \sin m x \rd x
| r = -\frac {\cos m x} m + C
| c = Primitive of $\sin m x$
}}
{{eqn | ll= \leadsto
| l = \int_\alpha^{\alpha + 2 \pi} \sin m x \rd x
| r = \intlimits {-\frac {\cos m x} m} \alpha {\alpha + 2 \pi}
| c =
}}
{{eqn | r = \paren {-\... | Let $m \in \Z$ be an [[Definition:Integer|integer]].
Then:
:$\ds \int_\alpha^{\alpha + 2 \pi} \sin m x \rd x = 0$ | Let $m \ne n$.
{{begin-eqn}}
{{eqn | l = \int \sin m x \rd x
| r = -\frac {\cos m x} m + C
| c = [[Primitive of Sine of a x|Primitive of $\sin m x$]]
}}
{{eqn | ll= \leadsto
| l = \int_\alpha^{\alpha + 2 \pi} \sin m x \rd x
| r = \intlimits {-\frac {\cos m x} m} \alpha {\alpha + 2 \pi}
| ... | Integral over 2 pi of Sine of n x | https://proofwiki.org/wiki/Integral_over_2_pi_of_Sine_of_n_x | https://proofwiki.org/wiki/Integral_over_2_pi_of_Sine_of_n_x | [
"Definite Integrals involving Sine Function"
] | [
"Definition:Integer"
] | [
"Primitive of Sine Function/Corollary",
"Sine of Zero is Zero"
] |
proofwiki-14079 | Integral over 2 pi of Cosine of n x | Let $m \in \Z$ be an integer.
Then:
:$\ds \int_\alpha^{\alpha + 2 \pi} \cos m x \rd x = \begin {cases} 0 & : m \ne 0 \\ 2 \pi & : m = 0 \end {cases}$ | Let $m \ne 0$.
{{begin-eqn}}
{{eqn | l = \int \cos m x \rd x
| r = \frac {\sin m x} m + C
| c = Primitive of $\cos m x$
}}
{{eqn | ll= \leadsto
| l = \int_\alpha^{\alpha + 2 \pi} \cos m x \rd x
| r = \intlimits {\frac {\sin m x} m} \alpha {\alpha + 2 \pi}
| c =
}}
{{eqn | r = \paren {\fra... | Let $m \in \Z$ be an [[Definition:Integer|integer]].
Then:
:$\ds \int_\alpha^{\alpha + 2 \pi} \cos m x \rd x = \begin {cases} 0 & : m \ne 0 \\ 2 \pi & : m = 0 \end {cases}$ | Let $m \ne 0$.
{{begin-eqn}}
{{eqn | l = \int \cos m x \rd x
| r = \frac {\sin m x} m + C
| c = [[Primitive of Cosine of a x|Primitive of $\cos m x$]]
}}
{{eqn | ll= \leadsto
| l = \int_\alpha^{\alpha + 2 \pi} \cos m x \rd x
| r = \intlimits {\frac {\sin m x} m} \alpha {\alpha + 2 \pi}
| ... | Integral over 2 pi of Cosine of n x | https://proofwiki.org/wiki/Integral_over_2_pi_of_Cosine_of_n_x | https://proofwiki.org/wiki/Integral_over_2_pi_of_Cosine_of_n_x | [
"Definite Integrals involving Cosine Function"
] | [
"Definition:Integer"
] | [
"Primitive of Cosine Function/Corollary",
"Cosine of Zero is One",
"Primitive of Constant"
] |
proofwiki-14080 | Coefficients of Cosine Terms in Convergent Trigonometric Series | Let $\map S x$ be a trigonometric series which converges to $\map f x$ on the interval $\openint \alpha {\alpha + 2 \pi}$:
:$\map f x = \dfrac {a_0} 2 + \ds \sum_{m \mathop = 1}^\infty \paren {a_m \cos m x + b_m \sin m x}$
Then:
:$\forall n \in \Z_{\ge 0}: a_n = \dfrac 1 \pi \ds \int_\alpha^{\alpha + 2 \pi} \map f x \c... | {{begin-eqn}}
{{eqn | l = \map f x
| r = \dfrac {a_0} 2 + \sum_{m \mathop = 1}^\infty \paren {a_m \cos m x + b_m \sin m x}
| c =
}}
{{eqn | ll= \leadsto
| l = \int_\alpha^{\alpha + 2 \pi} \map f x \cos n x \rd x
| r = \int_\alpha^{\alpha + 2 \pi} \paren {\dfrac {a_0} 2 + \sum_{m \mathop = 1}^\i... | Let $\map S x$ be a [[Definition:Trigonometric Series|trigonometric series]] which [[Definition:Convergent Series of Numbers|converges]] to $\map f x$ on the [[Definition:Closed Real Interval|interval]] $\openint \alpha {\alpha + 2 \pi}$:
:$\map f x = \dfrac {a_0} 2 + \ds \sum_{m \mathop = 1}^\infty \paren {a_m \cos m... | {{begin-eqn}}
{{eqn | l = \map f x
| r = \dfrac {a_0} 2 + \sum_{m \mathop = 1}^\infty \paren {a_m \cos m x + b_m \sin m x}
| c =
}}
{{eqn | ll= \leadsto
| l = \int_\alpha^{\alpha + 2 \pi} \map f x \cos n x \rd x
| r = \int_\alpha^{\alpha + 2 \pi} \paren {\dfrac {a_0} 2 + \sum_{m \mathop = 1}^\i... | Coefficients of Cosine Terms in Convergent Trigonometric Series | https://proofwiki.org/wiki/Coefficients_of_Cosine_Terms_in_Convergent_Trigonometric_Series | https://proofwiki.org/wiki/Coefficients_of_Cosine_Terms_in_Convergent_Trigonometric_Series | [
"Trigonometric Series",
"Fourier Series"
] | [
"Definition:Trigonometric Series",
"Definition:Convergent Series/Number Field",
"Definition:Real Interval/Closed"
] | [
"Integral over 2 pi of Cosine of n x",
"Integral over 2 pi of Sine of m x by Cosine of n x",
"Integral over 2 pi of Cosine of m x by Cosine of n x"
] |
proofwiki-14081 | Coefficients of Sine Terms in Convergent Trigonometric Series | Let $\map S x$ be a trigonometric series which converges to $\map f x$ on the interval $\openint \alpha {\alpha + 2 \pi}$:
:$\map f x = \dfrac {a_0} 2 + \ds \sum_{m \mathop = 1}^\infty \left({a_m \cos m x + b_m \sin m x}\right)$
Then:
:$\forall n \in \Z_{\ge 0}: b_n = \dfrac 1 \pi \ds \int_\alpha^{\alpha + 2 \pi} \map ... | {{begin-eqn}}
{{eqn | l = \map f x
| r = \dfrac {a_0} 2 + \sum_{m \mathop = 1}^\infty \paren {a_m \cos m x + b_m \sin m x}
| c =
}}
{{eqn | ll= \leadsto
| l = \int_\alpha^{\alpha + 2 \pi} \map f x \sin n x \rd x
| r = \int_\alpha^{\alpha + 2 \pi} \paren {\dfrac {a_0} 2 + \sum_{m \mathop = 1}^\i... | Let $\map S x$ be a [[Definition:Trigonometric Series|trigonometric series]] which [[Definition:Convergent Series of Numbers|converges]] to $\map f x$ on the [[Definition:Closed Real Interval|interval]] $\openint \alpha {\alpha + 2 \pi}$:
:$\map f x = \dfrac {a_0} 2 + \ds \sum_{m \mathop = 1}^\infty \left({a_m \cos m ... | {{begin-eqn}}
{{eqn | l = \map f x
| r = \dfrac {a_0} 2 + \sum_{m \mathop = 1}^\infty \paren {a_m \cos m x + b_m \sin m x}
| c =
}}
{{eqn | ll= \leadsto
| l = \int_\alpha^{\alpha + 2 \pi} \map f x \sin n x \rd x
| r = \int_\alpha^{\alpha + 2 \pi} \paren {\dfrac {a_0} 2 + \sum_{m \mathop = 1}^\i... | Coefficients of Sine Terms in Convergent Trigonometric Series | https://proofwiki.org/wiki/Coefficients_of_Sine_Terms_in_Convergent_Trigonometric_Series | https://proofwiki.org/wiki/Coefficients_of_Sine_Terms_in_Convergent_Trigonometric_Series | [
"Trigonometric Series",
"Fourier Series"
] | [
"Definition:Trigonometric Series",
"Definition:Convergent Series/Number Field",
"Definition:Real Interval/Closed"
] | [
"Integral over 2 pi of Sine of n x",
"Integral over 2 pi of Sine of m x by Cosine of n x",
"Integral over 2 pi of Sine of m x by Sine of n x"
] |
proofwiki-14082 | Sine of Angle plus Full Angle/Corollary | Let $n \in \Z$ be an integer.
Then:
:$\map \sin {x + 2 n \pi} = \sin x$ | From Sine of Angle plus Full Angle:
:$\map \sin {x + 2 \pi} = \sin x$
The result follows from the General Periodicity Property:
If:
:$\forall x \in X: \map f x = \map f {x + L}$
then:
:$\forall n \in \Z: \forall x \in X: \map f x = \map f {x + n L}$
{{qed}} | Let $n \in \Z$ be an [[Definition:Integer|integer]].
Then:
:$\map \sin {x + 2 n \pi} = \sin x$ | From [[Sine of Angle plus Full Angle]]:
:$\map \sin {x + 2 \pi} = \sin x$
The result follows from the [[General Periodicity Property]]:
If:
:$\forall x \in X: \map f x = \map f {x + L}$
then:
:$\forall n \in \Z: \forall x \in X: \map f x = \map f {x + n L}$
{{qed}} | Sine of Angle plus Full Angle/Corollary | https://proofwiki.org/wiki/Sine_of_Angle_plus_Full_Angle/Corollary | https://proofwiki.org/wiki/Sine_of_Angle_plus_Full_Angle/Corollary | [
"Sine Function"
] | [
"Definition:Integer"
] | [
"Sine of Angle plus Full Angle",
"General Periodicity Property"
] |
proofwiki-14083 | Fourier's Theorem | Let $\alpha \in \R$ be a real number.
Let $\map f x$ be a real function which is defined and bounded on the interval $\openint \alpha {\alpha + 2 \pi}$.
Let $f$ satisfy the Dirichlet conditions on $\openint \alpha {\alpha + 2 \pi}$:
{{:Definition:Dirichlet Conditions}}
Outside the interval $\openint \alpha {\alpha + 2 ... | === Lemma 1 ===
{{:Fourier's Theorem/Lemma 1}}{{qed|lemma}} | Let $\alpha \in \R$ be a [[Definition:Real Number|real number]].
Let $\map f x$ be a [[Definition:Real Function|real function]] which is defined and [[Definition:Bounded Real-Valued Function|bounded]] on the [[Definition:Real Interval|interval]] $\openint \alpha {\alpha + 2 \pi}$.
Let $f$ satisfy the [[Definition:Dir... | === [[Fourier's Theorem/Lemma 1|Lemma 1]] ===
{{:Fourier's Theorem/Lemma 1}}{{qed|lemma}} | Fourier's Theorem | https://proofwiki.org/wiki/Fourier's_Theorem | https://proofwiki.org/wiki/Fourier's_Theorem | [
"Fourier Series",
"Piecewise Continuous Functions",
"Fourier's Theorem"
] | [
"Definition:Real Number",
"Definition:Real Function",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Real Interval",
"Definition:Dirichlet Conditions",
"Definition:Real Interval",
"Definition:Periodic Function/Real",
"Definition:Fourier Series/Range 2 Pi",
"Definition:Convergent Series/Number... | [
"Fourier's Theorem/Lemma 1"
] |
proofwiki-14084 | Definite Integral of Step Function | Let $\alpha, \beta \in \R$ be a real numbers such that $\alpha < \beta$.
Let $\map f x$ be a step function defined on the interval $\closedint \alpha \beta$:
:$\map f x = \lambda_1 \chi_{\mathbb I_1} + \lambda_2 \chi_{\mathbb I_2} + \cdots + \lambda_n \chi_{\mathbb I_n}$
where:
:$\lambda_1, \lambda_2, \ldots, \lambda_n... | Each of the intervals $\mathbb I_k$ is such that $f \sqbrk {\mathbb I_k}$ is a constant function:
:$\forall x \in \mathbb I_k: \map f x = \lambda_k$
Thus:
{{begin-eqn}}
{{eqn | l = \int_{\mathbb I_k} \map f x \rd x
| r = \int_{\alpha_k}^{\beta_k} \lambda_k \rd x
| c =
}}
{{eqn | r = \lambda_k \paren {\beta... | Let $\alpha, \beta \in \R$ be a [[Definition:Real Number|real numbers]] such that $\alpha < \beta$.
Let $\map f x$ be a [[Definition:Step Function|step function]] defined on the [[Definition:Real Interval|interval]] $\closedint \alpha \beta$:
:$\map f x = \lambda_1 \chi_{\mathbb I_1} + \lambda_2 \chi_{\mathbb I_2} + ... | Each of the [[Definition:Real Interval|intervals]] $\mathbb I_k$ is such that $f \sqbrk {\mathbb I_k}$ is a [[Definition:Constant Function|constant function]]:
:$\forall x \in \mathbb I_k: \map f x = \lambda_k$
Thus:
{{begin-eqn}}
{{eqn | l = \int_{\mathbb I_k} \map f x \rd x
| r = \int_{\alpha_k}^{\beta_k} \la... | Definite Integral of Step Function | https://proofwiki.org/wiki/Definite_Integral_of_Step_Function | https://proofwiki.org/wiki/Definite_Integral_of_Step_Function | [
"Step Functions",
"Definite Integrals"
] | [
"Definition:Real Number",
"Definition:Step Function",
"Definition:Real Interval",
"Definition:Real Number",
"Definition:Constant",
"Definition:Real Interval",
"Definition:Set Partition",
"Definition:Characteristic Function (Set Theory)/Set",
"Definition:Definite Integral",
"Definition:Interval/Ord... | [
"Definition:Real Interval",
"Definition:Constant Mapping",
"Integral of Constant/Definite",
"Category:Step Functions",
"Category:Definite Integrals"
] |
proofwiki-14085 | Step Function satisfies Dirichlet Conditions | Let $\alpha, \beta \in \R$ be a real numbers such that $\alpha < \beta$.
Let $\map f x$ be a step function defined on the interval $\openint \alpha \beta$.
Then $f$ satisfies the Dirichlet conditions. | Recall the definition of step function:
:{{Definition:Step Function}}
Recall the Dirichlet conditions:
{{:Definition:Dirichlet Conditions}}
We inspect the Dirichlet conditions in turn. | Let $\alpha, \beta \in \R$ be a [[Definition:Real Number|real numbers]] such that $\alpha < \beta$.
Let $\map f x$ be a [[Definition:Step Function|step function]] defined on the [[Definition:Real Interval|interval]] $\openint \alpha \beta$.
Then $f$ satisfies the [[Definition:Dirichlet Conditions|Dirichlet condition... | Recall the definition of [[Definition:Step Function|step function]]:
:{{Definition:Step Function}}
Recall the [[Definition:Dirichlet Conditions|Dirichlet conditions]]:
{{:Definition:Dirichlet Conditions}}
We inspect the [[Definition:Dirichlet Conditions|Dirichlet conditions]] in turn. | Step Function satisfies Dirichlet Conditions | https://proofwiki.org/wiki/Step_Function_satisfies_Dirichlet_Conditions | https://proofwiki.org/wiki/Step_Function_satisfies_Dirichlet_Conditions | [
"Step Functions",
"Dirichlet Conditions"
] | [
"Definition:Real Number",
"Definition:Step Function",
"Definition:Real Interval",
"Definition:Dirichlet Conditions"
] | [
"Definition:Step Function",
"Definition:Dirichlet Conditions",
"Definition:Dirichlet Conditions",
"Definition:Step Function",
"Definition:Step Function",
"Definition:Dirichlet Conditions"
] |
proofwiki-14086 | Parseval's Theorem/Formulation 2 | Let $f$ be a real function which is square-integrable over the interval $\openint {-\pi} \pi$.
Let $f$ be expressed by the Fourier series:
:$\map f x = \ds \sum_{n \mathop = -\infty}^\infty c_n e^{i n x}$
where:
:$c_n = \ds \frac 1 {2 \pi} \int_{-\pi}^\pi \map f t e^{-i n t} \rd t$
Then:
:$\ds \frac 1 {2 \pi} \int_{-\p... | {{begin-eqn}}
{{eqn | l = \frac 1 {2 \pi} \int_{-\pi}^\pi \size {\map f x}^2 \rd x
| r = \frac 1 {2 \pi} \int_{-\pi}^\pi \map f x \overline {\map f x} \rd x
| c = Modulus in Terms of Conjugate
}}
{{eqn | r = \frac 1 {2 \pi} \int_{-\pi}^\pi \sum_{n \mathop = -\infty}^\infty c_n e^{i n x} \overline {\sum_{m \... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Square-Integrable Function|square-integrable]] over the [[Definition:Open Real Interval|interval]] $\openint {-\pi} \pi$.
Let $f$ be expressed by the [[Definition:Fourier Series over Range 2 Pi|Fourier series]]:
:$\map f x = \ds \sum_{n \ma... | {{begin-eqn}}
{{eqn | l = \frac 1 {2 \pi} \int_{-\pi}^\pi \size {\map f x}^2 \rd x
| r = \frac 1 {2 \pi} \int_{-\pi}^\pi \map f x \overline {\map f x} \rd x
| c = [[Modulus in Terms of Conjugate]]
}}
{{eqn | r = \frac 1 {2 \pi} \int_{-\pi}^\pi \sum_{n \mathop = -\infty}^\infty c_n e^{i n x} \overline {\sum_... | Parseval's Theorem/Formulation 2 | https://proofwiki.org/wiki/Parseval's_Theorem/Formulation_2 | https://proofwiki.org/wiki/Parseval's_Theorem/Formulation_2 | [
"Parseval's Theorem"
] | [
"Definition:Real Function",
"Definition:Square-Integrable Function",
"Definition:Real Interval/Open",
"Definition:Fourier Series/Range 2 Pi"
] | [
"Modulus in Terms of Conjugate",
"Sum of Complex Conjugates",
"Fubini's Theorem",
"Integral over 2 pi of Exponential of i by n x",
"Category:Parseval's Theorem"
] |
proofwiki-14087 | Parseval's Theorem/Formulation 1 | Let $f$ be a real function which is square-integrable over the interval $\openint {-\pi} \pi$.
Let $f$ be expressed by the Fourier series:
:$\map f x \sim \dfrac {a_0} 2 + \ds \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$
{{explain|What does $\sim$ mean?}}
Then:
:$\ds \frac 1 \pi \int_{-\pi}^\pi \s... | {{ProofWanted}}
{{Namedfor|Marc-Antoine Parseval}}
Category:Parseval's Theorem
9h5wuhgjgv3kv4f7zvpg9echh2xe9sf | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Square-Integrable Function|square-integrable]] over the [[Definition:Open Real Interval|interval]] $\openint {-\pi} \pi$.
Let $f$ be expressed by the [[Definition:Fourier Series over Range 2 Pi|Fourier series]]:
:$\map f x \sim \dfrac {a_0... | {{ProofWanted}}
{{Namedfor|Marc-Antoine Parseval}}
[[Category:Parseval's Theorem]]
9h5wuhgjgv3kv4f7zvpg9echh2xe9sf | Parseval's Theorem/Formulation 1 | https://proofwiki.org/wiki/Parseval's_Theorem/Formulation_1 | https://proofwiki.org/wiki/Parseval's_Theorem/Formulation_1 | [
"Parseval's Theorem"
] | [
"Definition:Real Function",
"Definition:Square-Integrable Function",
"Definition:Real Interval/Open",
"Definition:Fourier Series/Range 2 Pi"
] | [
"Category:Parseval's Theorem"
] |
proofwiki-14088 | Fourier Series/Square of x minus pi, Square of pi | :400pxrightthumb$\map f x$ and $5$th order expansion
Let $\map f x$ be the real function defined on $\openint 0 {2 \pi}$ as:
:$\map f x = \begin{cases} \paren {x - \pi}^2 & : 0 < x \le \pi \\ \pi^2 & : \pi < x < 2 \pi \end{cases}$
Then its Fourier series can be expressed as:
{{begin-eqn}}
{{eqn | l = \map f x
| o... | By definition of Fourier series:
:$\displaystyle \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$
where:
{{begin-eqn}}
{{eqn | l = a_n
| r = \dfrac 1 \pi \int_0^{2 \pi} \map f x \cos n x \rd x
}}
{{eqn | l = b_n
| r = \dfrac 1 \pi \int_0^{2 \pi} \map f x \sin n... | :[[File:Sneddon-1-2-Example1.png|400px|right|thumb|$\map f x$ and $5$th order expansion]]
Let $\map f x$ be the [[Definition:Real Function|real function]] defined on $\openint 0 {2 \pi}$ as:
:$\map f x = \begin{cases} \paren {x - \pi}^2 & : 0 < x \le \pi \\ \pi^2 & : \pi < x < 2 \pi \end{cases}$
Then its [[Definiti... | By definition of [[Definition:Fourier Series over Range 2 Pi|Fourier series]]:
:$\displaystyle \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$
where:
{{begin-eqn}}
{{eqn | l = a_n
| r = \dfrac 1 \pi \int_0^{2 \pi} \map f x \cos n x \rd x
}}
{{eqn | l = b_n
|... | Fourier Series/Square of x minus pi, Square of pi | https://proofwiki.org/wiki/Fourier_Series/Square_of_x_minus_pi,_Square_of_pi | https://proofwiki.org/wiki/Fourier_Series/Square_of_x_minus_pi,_Square_of_pi | [
"Examples of Fourier Series"
] | [
"File:Sneddon-1-2-Example1.png",
"Definition:Real Function",
"Definition:Fourier Series/Range 2 Pi"
] | [
"Definition:Fourier Series/Range 2 Pi",
"Cosine of Zero is One",
"Primitive of Constant",
"Primitive of Power",
"Linear Combination of Integrals/Definite",
"Sum of Integrals on Adjacent Intervals for Continuous Functions",
"Integral over 2 pi of Cosine of n x",
"Primitive of x by Cosine of a x",
"Si... |
proofwiki-14089 | Derivation of Fourier Series over General Range | Let $\alpha \in \R$ be a real number.
Let $\lambda \in \R_{>0}$ be a strictly positive real number.
Let $f: \R \to \R$ be a function such that $\ds \int_{\mathop \to \alpha}^{\mathop \to \alpha + 2 \lambda} \map f x \rd x$ converges absolutely.
Let:
:$\ds f \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \... | By definition of Fourier series over the range of integration $\openint \alpha {\alpha + 2 \pi}$:
:$(1): \quad \dfrac {a_0} 2 + \ds \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$
where:
{{begin-eqn}}
{{eqn | l = a_n
| r = \dfrac 1 \pi \int_\alpha^{\alpha + 2 \pi} \map f x \cos n x \rd x
}}
{{eq... | Let $\alpha \in \R$ be a [[Definition:Real Number|real number]].
Let $\lambda \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]].
Let $f: \R \to \R$ be a [[Definition:Function|function]] such that $\ds \int_{\mathop \to \alpha}^{\mathop \to \alpha + 2 \lambda} \map f x \rd x$... | By definition of [[Definition:Fourier Series over Range 2 Pi|Fourier series]] over the [[Definition:Range of Integration|range of integration]] $\openint \alpha {\alpha + 2 \pi}$:
:$(1): \quad \dfrac {a_0} 2 + \ds \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$
where:
{{begin-eqn}}
{{eqn | l = a_n
... | Derivation of Fourier Series over General Range | https://proofwiki.org/wiki/Derivation_of_Fourier_Series_over_General_Range | https://proofwiki.org/wiki/Derivation_of_Fourier_Series_over_General_Range | [
"Fourier Series"
] | [
"Definition:Real Number",
"Definition:Strictly Positive/Real Number",
"Definition:Function",
"Definition:Absolutely Convergent Integral",
"Definition:Fourier Series/Fourier Coefficient"
] | [
"Definition:Fourier Series/Range 2 Pi",
"Definition:Definite Integral/Limits of Integration",
"Definition:Absolutely Convergent Integral",
"Definition:Limit of Real Function/Right",
"Definition:Limit of Real Function/Left",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Bounded Mapping/Real-Value... |
proofwiki-14090 | Fourier Series/4 minus x squared over Range of 2 | Let $\map f x$ be the real function defined on $\openint 0 2$ as:
:600pxthumbright$\map f x$ and its $7$th approximation
:$\map f x = 4 - x^2$
Then its Fourier series can be expressed as:
:$\map f x \sim \ds \frac 8 3 - \frac 4 {\pi^2} \sum_{n \mathop = 1}^\infty \frac {\cos n \pi x} {n^2} + \frac 4 \pi \sum_{n \mathop... | By definition of Fourier series:
:$\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos n \pi x + b_n \sin n \pi x}$
where:
{{begin-eqn}}
{{eqn | l = a_n
| r = \int_0^2 \map f x \cos n \pi x \rd x
}}
{{eqn | l = b_n
| r = \int_0^2 \map f x \sin n \pi x \rd x
}}
{{end-eqn}}
for all ... | Let $\map f x$ be the [[Definition:Real Function|real function]] defined on $\openint 0 2$ as:
:[[File:Sneddon-1-3-Example2.png|600px|thumb|right|$\map f x$ and its $7$th approximation]]
:$\map f x = 4 - x^2$
Then its [[Definition:Fourier Series|Fourier series]] can be expressed as:
:$\map f x \sim \ds \frac 8 3 -... | By definition of [[Definition:Fourier Series|Fourier series]]:
:$\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos n \pi x + b_n \sin n \pi x}$
where:
{{begin-eqn}}
{{eqn | l = a_n
| r = \int_0^2 \map f x \cos n \pi x \rd x
}}
{{eqn | l = b_n
| r = \int_0^2 \map f x \sin n \pi... | Fourier Series/4 minus x squared over Range of 2 | https://proofwiki.org/wiki/Fourier_Series/4_minus_x_squared_over_Range_of_2 | https://proofwiki.org/wiki/Fourier_Series/4_minus_x_squared_over_Range_of_2 | [
"Examples of Fourier Series"
] | [
"Definition:Real Function",
"File:Sneddon-1-3-Example2.png",
"Definition:Fourier Series"
] | [
"Definition:Fourier Series",
"Cosine of Zero is One",
"Primitive of Power",
"Linear Combination of Integrals/Definite",
"Primitive of Cosine Function/Corollary",
"Sine of Integer Multiple of Pi",
"Primitive of x squared by Cosine of a x",
"Sine of Integer Multiple of Pi",
"Cosine of Integer Multiple... |
proofwiki-14091 | Fourier Series/1 over -1 to 0, Cosine of Pi x over 0 to 1 | Let $\map f x$ be the real function defined on $\openint {-1} 1$ as:
:800pxthumbright$\map f x$ and its $7$th approximation
:$\map f x = \begin{cases}
1 & : -1 < x < 0 \\
\map \cos {\pi x} & : 0 < x < 1
\end{cases}$
Then its Fourier series can be expressed as:
:$\map f x \sim \displaystyle \dfrac 1 2 + \frac {\cos \pi ... | By definition of Fourier series:
:$\displaystyle \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos n \pi x + b_n \sin n \pi x}$
where:
{{begin-eqn}}
{{eqn | l = a_n
| r = \int_{-1}^1 \map f x \cos n \pi x \rd x
}}
{{eqn | l = b_n
| r = \int_{-1}^1 \map f x \sin n \pi x \rd x
}}
{{en... | Let $\map f x$ be the [[Definition:Real Function|real function]] defined on $\openint {-1} 1$ as:
:[[File:Sneddon-1-3-Example3.png|800px|thumb|right|$\map f x$ and its $7$th approximation]]
:$\map f x = \begin{cases}
1 & : -1 < x < 0 \\
\map \cos {\pi x} & : 0 < x < 1
\end{cases}$
Then its [[Definition:Fourier Seri... | By definition of [[Definition:Fourier Series|Fourier series]]:
:$\displaystyle \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos n \pi x + b_n \sin n \pi x}$
where:
{{begin-eqn}}
{{eqn | l = a_n
| r = \int_{-1}^1 \map f x \cos n \pi x \rd x
}}
{{eqn | l = b_n
| r = \int_{-1}^1 \ma... | Fourier Series/1 over -1 to 0, Cosine of Pi x over 0 to 1 | https://proofwiki.org/wiki/Fourier_Series/1_over_-1_to_0,_Cosine_of_Pi_x_over_0_to_1 | https://proofwiki.org/wiki/Fourier_Series/1_over_-1_to_0,_Cosine_of_Pi_x_over_0_to_1 | [
"Examples of Fourier Series"
] | [
"Definition:Real Function",
"File:Sneddon-1-3-Example3.png",
"Definition:Fourier Series"
] | [
"Definition:Fourier Series",
"Cosine of Zero is One",
"Primitive of Power",
"Primitive of Cosine Function/Corollary",
"Sine of Integer Multiple of Pi",
"Primitive of Cosine Function/Corollary",
"Sine of Integer Multiple of Pi",
"Primitive of Cosine of a x by Cosine of b x",
"Sine of Integer Multiple... |
proofwiki-14092 | Dilogarithm of Square | :$\map {\Li_2} z + \map {\Li_2} {-z} = \dfrac 1 2 \map {\Li_2} {z^2}$ | {{begin-eqn}}
{{eqn | l = \map {\Li_2} z + \map {\Li_2} {-z}
| r = -\paren {\int_0^z \frac {\map \ln {1 - t} } t \rd t + \int_0^z \frac {\map \ln {1 + t} } t \rd t}
| c = {{Defof|Dilogarithm Function}}
}}
{{eqn | r = -\int_0^z \frac {\map \ln {\paren {1 - t} \paren {1 + t} } } t \rd t
| c = Linear Com... | :$\map {\Li_2} z + \map {\Li_2} {-z} = \dfrac 1 2 \map {\Li_2} {z^2}$ | {{begin-eqn}}
{{eqn | l = \map {\Li_2} z + \map {\Li_2} {-z}
| r = -\paren {\int_0^z \frac {\map \ln {1 - t} } t \rd t + \int_0^z \frac {\map \ln {1 + t} } t \rd t}
| c = {{Defof|Dilogarithm Function}}
}}
{{eqn | r = -\int_0^z \frac {\map \ln {\paren {1 - t} \paren {1 + t} } } t \rd t
| c = [[Linear C... | Dilogarithm of Square/Proof 1 | https://proofwiki.org/wiki/Dilogarithm_of_Square | https://proofwiki.org/wiki/Dilogarithm_of_Square/Proof_1 | [
"Dilogarithm of Square",
"Spence's Function"
] | [] | [
"Linear Combination of Integrals/Definite",
"Sum of Logarithms",
"Difference of Two Squares"
] |
proofwiki-14093 | Dilogarithm of Square | :$\map {\Li_2} z + \map {\Li_2} {-z} = \dfrac 1 2 \map {\Li_2} {z^2}$ | {{begin-eqn}}
{{eqn | l = \map {\Li_2} z + \map {\Li_2} {-z}
| r = \sum_{n \mathop = 1}^\infty \frac {z^n} {n^2} + \sum_{n \mathop = 1}^\infty \frac {\paren {-z}^n} {n^2}
| c = Power Series Expansion for Spence's Function
}}
{{eqn | r = \paren {z + \frac {z^2} {2^2} + \frac {z^3} {3^2} + \frac {z^4} {4^2} +... | :$\map {\Li_2} z + \map {\Li_2} {-z} = \dfrac 1 2 \map {\Li_2} {z^2}$ | {{begin-eqn}}
{{eqn | l = \map {\Li_2} z + \map {\Li_2} {-z}
| r = \sum_{n \mathop = 1}^\infty \frac {z^n} {n^2} + \sum_{n \mathop = 1}^\infty \frac {\paren {-z}^n} {n^2}
| c = [[Power Series Expansion for Spence's Function]]
}}
{{eqn | r = \paren {z + \frac {z^2} {2^2} + \frac {z^3} {3^2} + \frac {z^4} {4^... | Dilogarithm of Square/Proof 2 | https://proofwiki.org/wiki/Dilogarithm_of_Square | https://proofwiki.org/wiki/Dilogarithm_of_Square/Proof_2 | [
"Dilogarithm of Square",
"Spence's Function"
] | [] | [
"Power Series Expansion for Spence's Function",
"Definition:Odd Integer",
"Definition:Even Integer"
] |
proofwiki-14094 | Power Series Expansion for Spence's Function | Spence's function has a power series expansion:
:$\ds \map {\Li_2} z = \sum_{n \mathop = 1}^\infty \frac {z^n} {n^2}$
for $z \in \C$.
This converges for $\size z \le 1$. | {{begin-eqn}}
{{eqn | l = \map {\Li_2} z
| r = -\int_0^z \frac {\map \ln {1 - t} } t \rd t
| c = {{Defof|Spence's Function}}
}}
{{eqn | r = -\int_0^z \frac 1 t \sum_{n \mathop = 1}^\infty \paren {-\frac {t^n} n} \rd t
| c = Power Series Expansion for $\map \ln {1 - x}$
}}
{{eqn | r = \sum_{n \mathop ... | [[Definition:Spence's Function|Spence's function]] has a [[Definition:Power Series|power series expansion]]:
:$\ds \map {\Li_2} z = \sum_{n \mathop = 1}^\infty \frac {z^n} {n^2}$
for $z \in \C$.
This [[Definition:Convergent Series|converges]] for $\size z \le 1$. | {{begin-eqn}}
{{eqn | l = \map {\Li_2} z
| r = -\int_0^z \frac {\map \ln {1 - t} } t \rd t
| c = {{Defof|Spence's Function}}
}}
{{eqn | r = -\int_0^z \frac 1 t \sum_{n \mathop = 1}^\infty \paren {-\frac {t^n} n} \rd t
| c = [[Power Series Expansion for Logarithm of 1 - x|Power Series Expansion for $\... | Power Series Expansion for Spence's Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Spence's_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Spence's_Function | [
"Spence's Function",
"Examples of Power Series"
] | [
"Definition:Spence's Function",
"Definition:Power Series",
"Definition:Convergent Series"
] | [
"Power Series Expansion for Logarithm of 1 - x",
"Fubini's Theorem",
"Primitive of Power",
"Category:Spence's Function",
"Category:Examples of Power Series"
] |
proofwiki-14095 | Odd Function of Zero is Zero | Let $f: \R \to \R$ be an odd function.
Let $f$ be defined at the point $x = 0$.
Then:
:$\map f 0 = 0$ | By definition of odd function:
:$\map f {-x} = -\map f x$
and so:
{{begin-eqn}}
{{eqn | l = \map f {-0}
| r = \map f 0
| c =
}}
{{eqn | r = -\map f 0
| c =
}}
{{end-eqn}}
The only real number $a$ for which $a = -a$ is $0$.
Hence the result.
{{qed}} | Let $f: \R \to \R$ be an [[Definition:Odd Function|odd function]].
Let $f$ be defined at the point $x = 0$.
Then:
:$\map f 0 = 0$ | By definition of [[Definition:Odd Function|odd function]]:
:$\map f {-x} = -\map f x$
and so:
{{begin-eqn}}
{{eqn | l = \map f {-0}
| r = \map f 0
| c =
}}
{{eqn | r = -\map f 0
| c =
}}
{{end-eqn}}
The only [[Definition:Real Number|real number]] $a$ for which $a = -a$ is $0$.
Hence the result.
{... | Odd Function of Zero is Zero | https://proofwiki.org/wiki/Odd_Function_of_Zero_is_Zero | https://proofwiki.org/wiki/Odd_Function_of_Zero_is_Zero | [
"Odd Functions"
] | [
"Definition:Odd Function"
] | [
"Definition:Odd Function",
"Definition:Real Number"
] |
proofwiki-14096 | Fourier Cosine Coefficients for Even Function over Symmetric Range | Let $\map f x$ be an even real function defined on the interval $\openint {-\lambda} \lambda$.
Let the Fourier series of $\map f x$ be expressed as:
:$\map f x \sim \dfrac {a_0} 2 + \ds \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$
Then for all $n \in \Z_{\ge ... | As suggested, let the Fourier series of $\map f x$ be expressed as:
:$\map f x \sim \dfrac {a_0} 2 + \ds \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$
By definition of Fourier series:
:$a_n = \dfrac 1 \lambda \ds \int_{-\lambda}^{-\lambda + 2 \lambda} \map f x... | Let $\map f x$ be an [[Definition:Even Function|even]] [[Definition:Real Function|real function]] defined on the [[Definition:Real Interval|interval]] $\openint {-\lambda} \lambda$.
Let the [[Definition:Fourier Series|Fourier series]] of $\map f x$ be expressed as:
:$\map f x \sim \dfrac {a_0} 2 + \ds \sum_{n \mathop... | As suggested, let the [[Definition:Fourier Series|Fourier series]] of $\map f x$ be expressed as:
:$\map f x \sim \dfrac {a_0} 2 + \ds \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$
By definition of [[Definition:Fourier Series|Fourier series]]:
:$a_n = \dfra... | Fourier Cosine Coefficients for Even Function over Symmetric Range | https://proofwiki.org/wiki/Fourier_Cosine_Coefficients_for_Even_Function_over_Symmetric_Range | https://proofwiki.org/wiki/Fourier_Cosine_Coefficients_for_Even_Function_over_Symmetric_Range | [
"Even Functions",
"Fourier Series"
] | [
"Definition:Even Function",
"Definition:Real Function",
"Definition:Real Interval",
"Definition:Fourier Series"
] | [
"Definition:Fourier Series",
"Definition:Fourier Series",
"Cosine Function is Even",
"Even Function Times Even Function is Even",
"Definition:Even Function"
] |
proofwiki-14097 | Fourier Sine Coefficients for Even Function over Symmetric Range | Let $\map f x$ be an even real function defined on the interval $\openint {-\lambda} \lambda$.
Let the Fourier series of $\map f x$ be expressed as:
:$\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$
Then for all $n \in \Z_{> 0}$... | As suggested, let the Fourier series of $\map f x$ be expressed as:
:$\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$
By definition of Fourier series:
{{begin-eqn}}
{{eqn | l = b_n
| r = \frac 1 \lambda \int_{-\lambda}^{-\... | Let $\map f x$ be an [[Definition:Even Function|even]] [[Definition:Real Function|real function]] defined on the [[Definition:Real Interval|interval]] $\openint {-\lambda} \lambda$.
Let the [[Definition:Fourier Series|Fourier series]] of $\map f x$ be expressed as:
:$\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop ... | As suggested, let the [[Definition:Fourier Series|Fourier series]] of $\map f x$ be expressed as:
:$\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$
By definition of [[Definition:Fourier Series|Fourier series]]:
{{begin-eqn}}
... | Fourier Sine Coefficients for Even Function over Symmetric Range | https://proofwiki.org/wiki/Fourier_Sine_Coefficients_for_Even_Function_over_Symmetric_Range | https://proofwiki.org/wiki/Fourier_Sine_Coefficients_for_Even_Function_over_Symmetric_Range | [
"Even Functions",
"Fourier Series"
] | [
"Definition:Even Function",
"Definition:Real Function",
"Definition:Real Interval",
"Definition:Fourier Series"
] | [
"Definition:Fourier Series",
"Definition:Fourier Series",
"Sine Function is Odd",
"Odd Function Times Even Function is Odd",
"Definition:Odd Function"
] |
proofwiki-14098 | Fourier Series for Even Function over Symmetric Range | Let $\map f x$ be an even real function defined on the interval $\openint {-\lambda} \lambda$.
Then the Fourier series of $\map f x$ can be expressed as:
:$\map f x \sim \dfrac {a_0} 2 + \ds \sum_{n \mathop = 1}^\infty a_n \cos \frac {n \pi x} \lambda$
where for all $n \in \Z_{\ge 0}$:
:$a_n = \dfrac 2 \lambda \ds \int... | By definition of the Fourier series for $f$:
:$\map f x \sim \dfrac {a_0} 2 + \ds \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$
From Fourier Cosine Coefficients for Even Function over Symmetric Range:
:$a_n = \ds \dfrac 2 \lambda \int_0^\lambda \map f x \cos \... | Let $\map f x$ be an [[Definition:Even Function|even]] [[Definition:Real Function|real function]] defined on the [[Definition:Real Interval|interval]] $\openint {-\lambda} \lambda$.
Then the [[Definition:Fourier Series|Fourier series]] of $\map f x$ can be expressed as:
:$\map f x \sim \dfrac {a_0} 2 + \ds \sum_{n \... | By definition of the [[Definition:Fourier Series|Fourier series]] for $f$:
:$\map f x \sim \dfrac {a_0} 2 + \ds \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$
From [[Fourier Cosine Coefficients for Even Function over Symmetric Range]]:
:$a_n = \ds \dfrac 2 ... | Fourier Series for Even Function over Symmetric Range | https://proofwiki.org/wiki/Fourier_Series_for_Even_Function_over_Symmetric_Range | https://proofwiki.org/wiki/Fourier_Series_for_Even_Function_over_Symmetric_Range | [
"Even Functions",
"Fourier Series"
] | [
"Definition:Even Function",
"Definition:Real Function",
"Definition:Real Interval",
"Definition:Fourier Series"
] | [
"Definition:Fourier Series",
"Fourier Cosine Coefficients for Even Function over Symmetric Range",
"Fourier Sine Coefficients for Even Function over Symmetric Range"
] |
proofwiki-14099 | Fourier Cosine Coefficients for Odd Function over Symmetric Range | Let $\map f x$ be an odd real function defined on the interval $\openint {-\lambda} \lambda$.
Let the Fourier series of $\map f x$ be expressed as:
:$\ds \map f x \sim \dfrac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$
Then for all $n \in \Z_{\ge 0... | As suggested, let the Fourier series of $\map f x$ be expressed as:
:$\ds \map f x \sim \dfrac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$
By definition of Fourier series:
{{begin-eqn}}
{{eqn | l = a_n
| r = \frac 1 \lambda \int_{-\lambda}^{-... | Let $\map f x$ be an [[Definition:Odd Function|odd]] [[Definition:Real Function|real function]] defined on the [[Definition:Real Interval|interval]] $\openint {-\lambda} \lambda$.
Let the [[Definition:Fourier Series|Fourier series]] of $\map f x$ be expressed as:
:$\ds \map f x \sim \dfrac {a_0} 2 + \sum_{n \mathop =... | As suggested, let the [[Definition:Fourier Series|Fourier series]] of $\map f x$ be expressed as:
:$\ds \map f x \sim \dfrac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$
By definition of [[Definition:Fourier Series|Fourier series]]:
{{begin-eqn}}... | Fourier Cosine Coefficients for Odd Function over Symmetric Range | https://proofwiki.org/wiki/Fourier_Cosine_Coefficients_for_Odd_Function_over_Symmetric_Range | https://proofwiki.org/wiki/Fourier_Cosine_Coefficients_for_Odd_Function_over_Symmetric_Range | [
"Odd Functions",
"Fourier Series"
] | [
"Definition:Odd Function",
"Definition:Real Function",
"Definition:Real Interval",
"Definition:Fourier Series"
] | [
"Definition:Fourier Series",
"Definition:Fourier Series",
"Cosine Function is Even",
"Odd Function Times Even Function is Odd",
"Definition:Odd Function"
] |
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