# User contributions for Caliburn

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- 16:38, 1 June 2023 diff hist +23 Talk:Main Page
- 16:37, 1 June 2023 diff hist +487 Talk:Main Page
- 15:50, 1 June 2023 diff hist +290 Talk:Main Page
- 11:19, 23 May 2023 diff hist −29 m Evaluation Linear Transformation on Normed Vector Space is Weak to Weak-* Continuous Embedding into Second Normed Dual removed Category:Weak-* Topologies on Topological Vector Spaces; added Category:Weak-* Topologies using HotCat current
- 11:19, 23 May 2023
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**+1,869** N Evaluation Linear Transformation on Normed Vector Space is Weak to Weak-* Continuous Embedding into Second Normed Dual Created page with "== Theorem == <onlyinclude> Let $\Bbb F \in \set {\R, \C}$. Let $X$ be a normed vector space over $\Bbb F$. Let $X^\ast$ be the normed dual of $X$. Let $X^{\ast \ast}$ be the second normed dual of $X$. Let $w$ be the weak topology on $X$. Let $w^\ast$ be the weak-$\ast$ topology on $X^{\ast \ast}$. Let..." - 11:05, 23 May 2023 diff hist −33 m Evaluation Linear Transformation on Normed Vector Space is Linear Transformation from Space to Second Normed Dual removed Category:Second Normed Duals using HotCat current
- 11:04, 23 May 2023 diff hist −1 m Category:Evaluation Linear Transformations (Normed Vector Spaces) removed Category:Normed Vector Spaces; added Category:Second Normed Duals using HotCat current
- 11:04, 23 May 2023 diff hist −63 Fundamental Property of Norm on Bounded Linear Functional −Category:Second Normed Duals; ±Category:Evaluation Linear Transformations (Normed Vector Spaces)→Category:Bounded Linear Functionals using HotCat current
- 18:51, 9 May 2023 diff hist 0 m Image of Weakly Convergent Sequence under Compact Linear Transformation is Convergent Caliburn moved page User:Caliburn/s/fa/Image of Weakly Convergent Sequence under Compact Linear Transformation is Convergent to Image of Weakly Convergent Sequence under Compact Linear Transformation is Convergent without leaving a redirect
- 18:51, 9 May 2023
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**+1,641** Image of Weakly Convergent Sequence under Compact Linear Transformation is Convergent - 18:35, 9 May 2023
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**+605** N User:Caliburn/s/fa/Initial Topology Generated by Countable Family of Functions Separating Points is Metrizable Created page with "== Theorem == <onlyinclude> Let $X$ be a set. For each $n \in \N$, let $\struct {Y_n, d_n}$ be a metric space. Let $\family {f_n}_{n \in \N}$ be a indexed family of functions such that: :for each $x, y \in X$ with $x \ne y$ there exists $n \in \N$ such that $\map {f_n} x = \map {f_n} y$. Let $\tau$ be the initial topology on $X$ gen..." current - 14:09, 9 May 2023 diff hist +241 Talk:Invertibility of Identity Minus Operator
- 09:57, 9 May 2023 diff hist +412 Talk:Invertibility of Identity Minus Operator
- 19:44, 8 May 2023 diff hist +171 N Category:Element of Unital Banach Algebra on Boundary of Group of Units of Subalgebra is Not Invertible in Algebra Created page with "{{SubjectCategory|result = Element of Unital Banach Algebra on Boundary of Group of Units of Subalgebra is Not Invertible in Algebra}} Category:Unital Banach Algebras" current
- 19:44, 8 May 2023 diff hist +2 Element of Unital Banach Algebra on Boundary of Group of Units of Subalgebra is Not Invertible in Algebra
- 19:44, 8 May 2023
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**+1,851** N Element of Unital Banach Algebra on Boundary of Group of Units of Subalgebra is Not Invertible in Algebra Created page with "== Lemma == <onlyinclude> Let $\struct {A, \norm {\, \cdot \,} }$ be a unital Banach algebra. Let $\map G A$ be the group of units of $A$. Let $B$ be a closed subalgebra of $A$. Let $\map G B$ be the group of units of $A$. Let $x \in \partial \map G B$, where $\partial \map G B$ is the Definition:Boundary (T..." - 19:38, 8 May 2023
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**+2,413** N Element of Unital Banach Algebra on Boundary of Group of Units of Subalgebra is Not Invertible in Algebra/Lemma Created page with "== Lemma == <onlyinclude> Let $\struct {A, \norm {\, \cdot \,} }$ be a unital Banach algebra. Let $\map G A$ be the group of units of $A$. Let $x \in \partial \map G A$, where $\partial \map G A$ is the topological boundary of $\map G A$. Then there exists a sequence $\sequence {z_n}_{n \in \N}$ in $A$ such that $\norm {z_n} = 1$ for each $n..." - 13:17, 8 May 2023
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**+973** N Norm of Inverse of Sequence of Invertible Elements Converging to Non-Invertible Element in Unital Banach Algebra Created page with "== Theorem == <onlyinclude> Let $\struct {A, \norm {\, \cdot \,} }$ be a unital Banach algebra. Let $\map G A$ be the group of units of $A$. Let $x \in A \setminus \map G A$. Let $\sequence {x_n}_{n \in \N}$ be a sequence in $\map G A$ such that $x_n \to x$. Then $\norm {x_n^{-1} } \to \infty$ as $n \to \infty$. </onlyinclude> == Proof == Note that if there existed $n \in..." current - 11:02, 8 May 2023
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**+2,060** N Inverse Mapping on Group of Units in Unital Banach Algebra is Continuous Created page with "== Theorem == <onlyinclude> Let $\struct {A, \norm {\, \cdot \,} }$ be a unital Banach algebra. Let $\map G A$ be the group of units of $A$. Define $\phi : \map G A \to \map G A$ by: :$\map \phi x = x^{-1}$ for each $x \in \map G A$. Then $\phi$ is continuous. </onlyinclude> == Proof == Let $x \in \map G A$ and $y \in A$ be such that: :$\ds \norm {x - y} < \..." current - 22:38, 1 April 2023 diff hist +11 Talk:Nonexistence of Complex Matrices whose Commutator equals Identity current
- 22:38, 1 April 2023 diff hist +237 N Talk:Nonexistence of Complex Matrices whose Commutator equals Identity Created page with "The result is true in a general Banach algebra FWIW. In fact there are no complex matrices whose commutator is equal to a scalar multiple of the identity. ~~~~"
- 09:48, 19 March 2023
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**+1,102** N Definition:Measurable Function/Banach Space Valued Function Created page with "== Definition == <onlyinclude> Let $\GF \in \set {\R, \C}$. Let $I$ be a real interval. Let $X$ be a Banach space over $\GF$. Let $f : I \to X$ be a function. We say that $f$ is '''measurable''' if there exists a sequence of simple functions $\sequence {f_n}_{n \mathop \in \N}$ such that: :$\ds \map f t = \lim_{n \m..." current - 09:37, 19 March 2023 diff hist +165 Definition:Measurable Function current
- 21:26, 18 March 2023 diff hist +72 Uniformly Continuous Semigroup Bounded on Compact Intervals current
- 21:25, 18 March 2023 diff hist −24 Uniformly Continuous Semigroup Bounded on Compact Intervals
- 21:24, 18 March 2023 diff hist +373 Identity Mapping on Normed Vector Space is Bounded Linear Operator current
- 21:18, 18 March 2023 diff hist +36 Norm of Continuous Function is Continuous I'm not sure how necessary this is but sure
- 19:03, 18 March 2023 diff hist −23 Bound on C0 Semigroup typo, I guess I'm off the ball today current
- 19:02, 18 March 2023 diff hist −56 Semigroup of Bounded Linear Operators is C0 iff Point Evaluations Continuous yep accident current
- 15:30, 18 March 2023
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**+630** N Norm of Continuous Function is Continuous Created page with "== Theorem == <onlyinclude> Let $\GF \in \set {\R, \C}$. Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be normed vector spaces over $\GF$. Let $f : X \to Y$ be a continuous. Then $\norm f_Y : X \to \hointr 0 \infty$ is continuous. </onlyinclude> == Proof == Follows immediately from c..." - 15:11, 18 March 2023
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**+3,951** N Semigroup of Bounded Linear Operators is C0 iff Point Evaluations Continuous Created page with "== Theorem == <onlyinclude> Let $\GF \in \set {\R, \C}$. Let $X$ be a Banach space over $\GF$. Let $\family {\map T t}_{t \ge 0}$ be a semigroup of bounded linear operators. For each $x \in X$, define $x^\wedge : \hointr 0 \infty \to X$ by: :$\map {x^\wedge} t = \map T t x$ for each $t \in \hointr 0 \infty$. Then $\family {\map T t}_{t \ge 0}$ is a Definition:C0 Semigroup|$C_0$..." - 14:54, 18 March 2023 diff hist +85 N Category:C0 Semigroups Created page with "{{SubjectCategory|C0 Semigroup}} Category:Semigroups of Bounded Linear Operators" current
- 14:07, 18 March 2023
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**+3,572** N Bound on C0 Semigroup Created page with "== Theorem == <onlyinclude> Let $\GF \in \set {\R, \C}$. Let $X$ be a Banach space over $\GF$. Let $\family {\map T t}_{t \ge 0}$ be a $C_0$ semigroup. Let $\struct {\map B X, \norm {\, \cdot \,}_{\map B X} }$ be the space of bounded linear transformations equipped with the canonical norm. The..." - 13:56, 18 March 2023 diff hist +8 Uniformly Continuous Semigroup Bounded on Compact Intervals
- 13:40, 18 March 2023 diff hist +3 Definition:C0 Semigroup current
- 13:40, 18 March 2023
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**+1,136** N Definition:C0 Semigroup Created page with "== Theorem == <onlyinclude> Let $\GF \in \set {\R, \C}$. Let $X$ be a Banach space over $\GF$. Let $\family {\map T t}_{t \ge 0}$ be a semigroup of bounded linear operators. We say that $\family {\map T t}_{t \ge 0}$ is a '''$C_0$ semigroup''' {{iff}}: :$\ds \lim_{t \mathop \to 0^+} \map T t x = x$ for each $x \in X$. </onlyinclude> == Also known as == A $C_0$ semigroup may also be..." - 13:38, 18 March 2023 diff hist +84 Definition:Uniformly Continuous Semigroup current
- 13:18, 18 March 2023 diff hist +69 Semigroup of Bounded Linear Operators Uniformly Continuous iff Continuous as Map from Non-Negative Reals to Bounded Linear Operators
- 14:35, 17 March 2023 diff hist +14 Semigroup of Bounded Linear Operators Uniformly Continuous iff Continuous as Map from Non-Negative Reals to Bounded Linear Operators
- 14:35, 17 March 2023 diff hist −12 Semigroup of Bounded Linear Operators Uniformly Continuous iff Continuous as Map from Non-Negative Reals to Bounded Linear Operators
- 13:55, 17 March 2023 diff hist +103 N Category:Uniformly Continuous Semigroups Created page with "{{SubjectCategory|Uniformly Continuous Semigroup}} Category:Semigroups of Bounded Linear Operators" current
- 13:55, 17 March 2023 diff hist +52 m Semigroup of Bounded Linear Operators Uniformly Continuous iff Continuous as Map from Non-Negative Reals to Bounded Linear Operators added Category:Semigroups of Bounded Linear Operators using HotCat
- 13:54, 17 March 2023 diff hist +85 N Category:Semigroups of Bounded Linear Operators ] current
- 11:16, 17 March 2023
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**+3,715** N Semigroup of Bounded Linear Operators Uniformly Continuous iff Continuous as Map from Non-Negative Reals to Bounded Linear Operators Created page with "== Theorem == <onlyinclude> Let $\GF \in \set {\R, \C}$. Let $X$ be a Banach space over $\GF$. Let $\family {\map T t}_{t \ge 0}$ be a semigroup of bounded linear operators. Let $\struct {\map B X, \norm {\, \cdot \,}_{\map B X} }$ be the space of bounded linear transformations equipped with the Definition:Norm on Space of Bounded..." - 10:50, 17 March 2023
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**+2,517** N Uniformly Continuous Semigroup Bounded on Compact Intervals Created page with "== Theorem == <onlyinclude> Let $\GF \in \set {\R, \C}$. Let $X$ be a Banach space over $\GF$. Let $\family {\map T t}_{t \ge 0}$ be a uniformly continuous semigroup. Let $\struct {\map B X, \norm {\, \cdot \,}_{\map B X} }$ be the space of bounded linear transformations equipped with the Definition:Norm on Space of Bounded Linear Transf..." - 10:24, 17 March 2023 diff hist +225 User talk:Caliburn
- 20:52, 16 March 2023 diff hist −33 Definition:Semigroup of Bounded Linear Operators current
- 20:52, 16 March 2023 diff hist −33 Definition:Uniformly Continuous Semigroup
- 20:52, 16 March 2023 diff hist −33 Definition:Infinitesimal Generator of Semigroup first edition, copied this by mistake current
- 20:22, 16 March 2023 diff hist 0 Definition:Uniformly Continuous Semigroup