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• 11:19, 23 May 2023 Caliburn talk contribs created page 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...")
• 18:52, 9 May 2023 Caliburn talk contribs deleted page User:Caliburn/s/fa/Locally Convex Space Induces Topology (content was: "== Theorem == Let $\struct {X, \mathcal P}$ be a locally convex space. <onlyinclude> Define the collection of sets $\tau \subseteq \map {\mathcal P} X$ by: :$U \in \tau$ {{iff}} $U \subseteq X$ and for each $x \in U$, there exists finitely many $p_1, p_2, \ldots, p_n \in \mathcal P$ and $\epsilon > 0$ such that: ::$\set {y \in X : \ma...", and the only contributor was "Caliburn" (talk))
• 18:52, 9 May 2023 Caliburn talk contribs deleted page User:Caliburn/s/fa/Resolvent Set of Linear Operator is Open (content was: "== Theorem == <onlyinclude> Let $\struct {X, \norm \cdot_X}$ be a Banach space over $\C$. Let $A : X \to X$ be a linear operator. Let $\map \rho A$ be the resolvent set of $A$. Then $\map \rho A$ is open. </onlyinclu...", and the only contributor was "Caliburn" (talk))
• 18:51, 9 May 2023 Caliburn talk contribs 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
• 18:37, 9 May 2023 Caliburn talk contribs deleted page User:Caliburn/s/fa/Definition:Resolvent of Linear Operator (content was: "== Definition == <onlyinclude> Let $\struct {X, \norm \cdot_X}$ be a Banach space over $\C$. Let $A : X \to X$ be a linear operator. Let $\map \rho A$ be the resolvent set of $A$. Let $\lambda \in \map \rho A$. Let: :$R_\lambda = A - \lambda...", and the only contributor was "Caliburn" (talk))
• 18:35, 9 May 2023 Caliburn talk contribs created page 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...")
• 19:44, 8 May 2023 Caliburn talk contribs created page 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")
• 19:44, 8 May 2023 Caliburn talk contribs created page 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 Caliburn talk contribs created page 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 Caliburn talk contribs created page 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...")
• 11:02, 8 May 2023 Caliburn talk contribs created page 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} < \...")
• 22:38, 1 April 2023 Caliburn talk contribs created page 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 Caliburn talk contribs created page 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...")
• 15:30, 18 March 2023 Caliburn talk contribs created page 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 Caliburn talk contribs created page 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 Caliburn talk contribs created page Category:C0 Semigroups (Created page with "{{SubjectCategory|C0 Semigroup}} Category:Semigroups of Bounded Linear Operators")
• 14:07, 18 March 2023 Caliburn talk contribs created page 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...")
• 13:40, 18 March 2023 Caliburn talk contribs created page 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...")
• 13:55, 17 March 2023 Caliburn talk contribs created page Category:Uniformly Continuous Semigroups (Created page with "{{SubjectCategory|Uniformly Continuous Semigroup}} Category:Semigroups of Bounded Linear Operators")
• 13:54, 17 March 2023 Caliburn talk contribs created page Category:Semigroups of Bounded Linear Operators (])
• 11:16, 17 March 2023 Caliburn talk contribs created page 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 Caliburn talk contribs created page 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...")
• 20:21, 16 March 2023 Caliburn talk contribs created page Definition:Infinitesimal Generator of Semigroup (Created page with "== Definition == <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 $\hointr 0 \infty$-indexed family of bounded linear transformations $\map T t : X \to X$. Define: :$\ds \map D A = \set {x \in X : \lim_{t \mathop \to 0^+} \frac {\map T t x - x} t \text { exists} }$ Define $A : \map D A \to X$...")
• 20:14, 16 March 2023 Caliburn talk contribs created page Category:Definitions/Uniformly Continuous Semigroups (Created page with "{{DefinitionCategory|def = Uniformly Continuous Semigroup|Semigroups of Bounded Linear Operators}}")
• 20:13, 16 March 2023 Caliburn talk contribs created page Category:Definitions/Semigroups of Bounded Linear Operators (Created page with "{{DefinitionCategory|def = Semigroup of Bounded Linear Operators|Banach Spaces}}")
• 20:09, 16 March 2023 Caliburn talk contribs created page Definition:Uniformly Continuous Semigroup (Created page with "== Definition == <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 $\hointr 0 \infty$-indexed family of bounded linear transformations $\map T t : X \to X$. Let $\struct {\map B X, \norm {\, \cdot \,}_{\map B X} }$ be the Definition:Space of Bounded Linear Transformations|space of bounded linea...")
• 20:04, 16 March 2023 Caliburn talk contribs created page Definition:Semigroup of Bounded Linear Operators (gonna get as far as I can with semigroup theory, which will be 3 pages without being able to do integration)
• 16:57, 16 March 2023 Caliburn talk contribs created page Definition:Simple Function/Banach Space (working towards Banach space integration)
• 12:49, 15 March 2023 Caliburn talk contribs created page Liouville's Theorem (Complex Analysis)/Banach Space (Created page with "== Theorem == <onlyinclude> Let $\struct {X, \norm {\, \cdot \,} }$ be a Banach space over $\C$. Let $f : \C \to X$ be an analytic function that is bounded. Then $f$ is constant. </onlyinclude> == Proof == Take $M \ge 0$ such that: :$\norm {\map f x} \le M$ for each $x \in X$. Let $\struct...")
• 12:25, 15 March 2023 Caliburn talk contribs created page Banach Space Valued Function is Analytic iff Weakly Analytic (Created page with "== Theorem == <onlyinclude> Let $U$ be an open subset of $\C$. Let $\struct {X, \norm {\, \cdot \,} }$ be a Banach space over $\C$. Let $f : U \to X$ be a function. Then $f$ is analytic {{iff}} it is weakly analytic. </onlyinclude> == Proof == === Necessary Condition ===...")
• 12:12, 15 March 2023 Caliburn talk contribs created page Definition:Weakly Analytic Function (Created page with "== Definition == <onlyinclude> Let $U$ be an open subset of $\C$. Let $\struct {X, \norm {\, \cdot \,} }$ be a Banach space over $\C$. Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,} }$. Let $f : U \to X$ be a function. We say that $f$ is '''weakly analytic''' {{iff}}: :for...")
• 22:53, 14 March 2023 Caliburn talk contribs created page Category:Definitions/Analytic Functions (Created page with "{{DefinitionCategory|def = Analytic Function|Analysis}}")
• 22:48, 14 March 2023 Caliburn talk contribs created page Definition:Analytic Function/Banach Space Valued Function (Created page with "== Definition == <onlyinclude> Let $U$ be an open subset of $\C$. Let $\struct {X, \norm {\, \cdot \,} }$ be a Banach space over $\C$. Let $f : U \to E$ be a function. We say that $f$ is '''analytic''' if the limit: :$\ds \lim_{w \to z} \frac {\map f w - \map f z} {w - z}$ exists for each $z \in U$. </onlyinclude> ==...")
• 22:14, 14 March 2023 Caliburn talk contribs created page Group of Units in Unital Banach Algebra is Open (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 $A$. Then $\map G A$ is open in $A$. </onlyinclude> == Proof == Let $x \in \map G A$. We find an open neighborhood of $x$ contained in $\map G A$. Clearly $x^{-1} \ne \mathbf 0_A$, so...")
• 21:59, 14 March 2023 Caliburn talk contribs moved page Disjoint Open Sets remain Disjoint with one Closure to Open Set Disjoint from Set is Disjoint from Closure
• 21:59, 14 March 2023 Caliburn talk contribs moved page Talk:Disjoint Open Sets remain Disjoint with one Closure to Talk:Open Set Disjoint from Set is Disjoint from Closure
• 17:41, 14 March 2023 Caliburn talk contribs created page Talk:Disjoint Open Sets remain Disjoint with one Closure (Created page with "To check - $A$ doesn't actually have to be open here right? The chain of reasoning only needs $S \setminus B$ to be closed. ~~~~")
• 17:17, 14 March 2023 Caliburn talk contribs created page Category:Normed Algebras (Created page with "{{SubjectCategory|Normed Algebra}} Category:Normed Vector Spaces Category:Algebras")
• 17:16, 14 March 2023 Caliburn talk contribs created page Category:Definitions/Normed Algebras (Created page with "{{DefinitionCategory|def = Normed Algebra|Algebras|Normed Vector Spaces}}")
• 17:15, 14 March 2023 Caliburn talk contribs created page Definition:Unital Normed Algebra (Created page with "== Definition == <onlyinclude> Let $\Bbb F \in \set {\R, \C}$. Let $\struct {A, \norm \cdot}$ be a normed algebra over $\Bbb F$ that is unital as an algebra. Let $\mathbf 1_A$ be the identity element of $A$. We say that $A$ is a '''unital normed algebra''' {{iff}}: :$\norm {\mathbf 1_A} = 1$ </onlyinclude> == Sources == * {{BookReference|Introduction to Banach Spaces and...")
• 17:12, 14 March 2023 Caliburn talk contribs created page Definition:Unitization of Normed Algebra (Created page with "== Definition == <onlyinclude> Let $\GF \in \set {\R, \C}$. Let $\struct {A, \norm {\, \cdot \,} }$ be a normed algebra that is not unital as an algebra. Let $A_+$ be the unitization of $\struct {A, \norm {\, \cdot \,} }$. Define $\norm {\, \cdot \,}_{A_+} : A_+ \to \hointr 0 \infty$ by: :$\norm {\tuple {x, \lambda} }_{A_+} = \norm x + \cmod \lambda$ for e...")
• 16:59, 14 March 2023 Caliburn talk contribs created page Definition:Unitization of Algebra over Field (Created page with "== Definition == <onlyinclude> Let $K$ be a field. Let $A$ be an algebra over $K$. Let $A_+ = A \times K$ be the direct product of $A$ and $K$ as vector spaces over $K$ with vector addition $+_{A \times K}$ and scalar multiplication $\cdot_{A \times K}$. Define multiplica...")
• 13:31, 13 March 2023 Caliburn talk contribs created page Riesz's Lemma/Proof 1 (Created page with "== Theorem == {{:Riesz's Lemma}} == Proof == <onlyinclude> Since $Y < X$: :$X \setminus Y$ is non-empty. Since $Y$ is closed: :$X \setminus Y$ is open. Let $x \in X \setminus Y$. Then there exists $\epsilon > 0$ such that: :$\map {B_\epsilon} x \subset X \setminus Y$ So, for all $y \in Y$, we must have: :$\norm {x - y} \ge \epsilon$ That is: :$\inf \set {\norm {x - y} \colon y...")
• 13:31, 13 March 2023 Caliburn talk contribs created page Riesz's Lemma/Proof 2 (Created page with "== Theorem == {{:Riesz's Lemma}} == Proof == <onlyinclude> Consider the normed quotient vector space $X/Y$ with quotient mapping $\pi$. From Operator Norm of Quotient Mapping in Quotient Normed Vector Space is 1, we have: :$\norm \pi_{\map B {X, X/Y} } = 1$ Since $\alpha \in \openint 0 1$, there exists $x_\alpha \in X$ with $\norm {x_\alpha} = 1$ and: :$\norm {\map \pi x_\alpha}_{X/Y}...")
• 13:27, 13 March 2023 Caliburn talk contribs created page Operator Norm of Quotient Mapping in Quotient Normed Vector Space is 1 (Created page with "== Theorem == <onlyinclude> Let $\Bbb F \in \set {\R, \C}$. Let $X$ be a normed vector space over $\Bbb F$. Let $N$ be a closed linear subspace of $X$. Let $\struct {X/N, \norm {\, \cdot \,}_{X/N} }$ be the normed quotient vector space associated with quotient vector space $X/N$. Let $\pi : X \to X/N$ be the Defi...")
• 21:53, 12 March 2023 Caliburn talk contribs created page Category:Unital Banach Algebras (Created page with "{{SubjectCategoryNodef|Unital Banach Algebra}} Category:Banach Algebras")
• 20:51, 12 March 2023 Caliburn talk contribs created page Sum Rule for Sequence in Normed Vector Space (Created page with "== Theorem == <onlyinclude> Let $\GF \in \set {\R, \C}$. Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space over $\GF$. Let $\sequence {x_n}_{n \in \N}$ and $\sequence {y_n}_{n \in \N}$ be convergent sequences such that: :$x_n \to x$ and: :$y_n \to y$ Then: :$x_n + y_n \to x + y$ </onlyinclude> == Proof == For each $n \in \N$, we have: {{begin-eqn}}...")
• 20:47, 12 March 2023 Caliburn talk contribs created page Convergent Sequence in Normed Vector Space is Bounded (Created page with "== Theorem == <onlyinclude> Let $\GF \in \set {\R, \C}$. Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space. Let $\sequence {x_n}_{n \in \N}$ be a convergent sequence. Then there exists $M > 0$ such that: :$\norm {x_n} \le M$ for each $n \in \N$. </onlyinclude> == Proof == Suppose that $x_n \to x$. From Convergent Sequence is Cauchy Sequence, $\se...")
• 20:43, 12 March 2023 Caliburn talk contribs moved page Product Rule for Limits in Normed Algebra to Product Rule for Sequence in Normed Algebra without leaving a redirect (more consistent)
• 20:42, 12 March 2023 Caliburn talk contribs created page Product Rule for Limits in Normed Algebra (Created page with "== Theorem == <onlyinclude> Let $\GF \in \set {\R, \C}$. Let $\struct {A, \norm {\, \cdot \,} }$ be a normed algebra over $\GF$. Let $\sequence {a_n}_{n \in \N}$ and $\sequence {b_n}_{n \in \N}$ be sequences in $A$ converging to $a$ and $b$ respectively. Then: :$a_n b_n \to a b$ </onlyinclude> == Proof == From Convergent Sequence in Normed Vector...")

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