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18 October 2022
- 21:3921:39, 18 October 2022 diff hist −14 m Scheffé's Lemma Fix Latex Tag: Visual edit
18 February 2022
- 05:3805:38, 18 February 2022 diff hist +8 m Scheffé's Lemma No edit summary Tag: Visual edit: Switched
10 February 2022
- 01:2101:21, 10 February 2022 diff hist +17 m Spectrum of Bounded Linear Operator is Non-Empty No edit summary
- 01:1901:19, 10 February 2022 diff hist −3 m Spectrum of Bounded Linear Operator is Non-Empty No edit summary Tag: Visual edit: Switched
- 01:1701:17, 10 February 2022 diff hist +3,372 N Spectrum of Bounded Linear Operator is Non-Empty Created page with "== Theorem == Suppose $B$ is a Banach space, $\mathfrak{L}(B, B)$ is the set of bounded linear operators from $B$ to itself, and $T \in \mathfrak{L}(B, B)$. Then the spectrum of $T$ is non-empty. == Proof == Let $f : \Bbb C \to \mathfrak{L}(B,B)$ be the resolvent mapping defined as $f(z) = (T - zI)^{-1}$. Suppose the spectrum of $T$ is empty, so that $f(z)$ is well-defined for all $z\in\Bbb C$. We first show that $\|f(z)\|_*$ is unifo..."
- 01:0301:03, 10 February 2022 diff hist +1,629 N Resolvent Mapping Converges to 0 at Infinity Created page with "== Theorem == Let $B$ be a Banach space. Let $\map \LL {B, B}$ be the set of bounded linear operators from $B$ to itself. Let $T \in \map \LL {B, B}$. Let $\map \rho T$ be the resolvent set of $T$ in the complex plane. Then the resolvent mapping $f : \map \rho T \to \map \LL {B, B}$ given by $\map f z = \paren {T - z I}^{-1}$ is such that $\lim_{z\to\infty} \|f(z)\|_* = 0$. == Proof == Pick $z \in \Bbb C$ with $|z| > 2\|T\|_*$. Then $\..."
7 February 2022
- 06:3806:38, 7 February 2022 diff hist +17 m Resolvent Mapping is Analytic/Bounded Linear Operator No edit summary
6 February 2022
- 04:0304:03, 6 February 2022 diff hist +17 m Resolvent Mapping is Continuous/Bounded Linear Operator No edit summary
- 03:5803:58, 6 February 2022 diff hist +9 m Resolvent Mapping is Continuous/Bounded Linear Operator No edit summary
- 03:5403:54, 6 February 2022 diff hist +17 m Resolvent Mapping is Continuous/Bounded Linear Operator No edit summary
3 February 2022
- 06:1706:17, 3 February 2022 diff hist +1 m Resolvent Mapping is Analytic/Bounded Linear Operator No edit summary
- 06:1606:16, 3 February 2022 diff hist +12 m Resolvent Mapping is Analytic/Bounded Linear Operator No edit summary
- 05:4805:48, 3 February 2022 diff hist −51 m Resolvent Mapping is Analytic/Bounded Linear Operator No edit summary
- 05:4605:46, 3 February 2022 diff hist +8 m Resolvent Mapping is Analytic/Bounded Linear Operator No edit summary
- 05:4205:42, 3 February 2022 diff hist −42 Resolvent Mapping is Analytic/Bounded Linear Operator No edit summary
- 05:4105:41, 3 February 2022 diff hist +1,244 N Resolvent Mapping is Analytic/Bounded Linear Operator Created page with "== Theorem == Suppose $B$ is a Banach space, $\mathfrak{L}(B, B)$ is the set of bounded linear operators from $B$ to itself, and $T \in O$. Let $\rho(T)$ be the resolvent set of $T$ in the complex plane. Then the resolvent mapping $f : \rho(T) \to \mathfrak{L}(B,B)$ given by $f(z) = (T - zI)^{-1}$ is analytic and $$ \lim_{h\to 0} \frac{\norm{f(z+h) - f(z)}_*}{|h|} = (T-zI)^{-2}. $$ == Proof == For any $a\in \rho(T)$, define $R_a = (T - aI)..."
- 02:3002:30, 3 February 2022 diff hist +45 Resolvent Mapping is Continuous/Bounded Linear Operator No edit summary
- 02:2902:29, 3 February 2022 diff hist +106 Resolvent Mapping is Continuous/Bounded Linear Operator No edit summary
- 02:2802:28, 3 February 2022 diff hist +50 m Resolvent Mapping is Continuous/Bounded Linear Operator No edit summary
- 02:0102:01, 3 February 2022 diff hist +35 Resolvent Mapping is Continuous/Bounded Linear Operator No edit summary
- 01:5901:59, 3 February 2022 diff hist −80 Resolvent Mapping is Continuous/Bounded Linear Operator No edit summary
- 01:5101:51, 3 February 2022 diff hist +2 m Resolvent Mapping is Continuous/Bounded Linear Operator No edit summary
- 01:4801:48, 3 February 2022 diff hist +2,437 N Resolvent Mapping is Continuous/Bounded Linear Operator Created page with "== Theorem == Suppose $B$ is a Banach space, $\mathfrak{L}(B, B)$ is the set of bounded linear operators from $B$ to itself, and $T \in O$. Let $\rho(T)$ be the resolvent set of $T$ in the complex plane. Then the resolvent mapping $f : \rho(T) \to \mathfrak{L}(B,B)$ given by $f(z) = (T - zI)^{-1}$ is continuous in the operator norm $\|\cdot\|_*$. == Proof == Pick $z\in\rho(T)$. Since $z\in\rho(T)$, the operator $R_z = (T - zI)^{-1}$ exists..." Tag: Visual edit: Switched
- 00:5800:58, 3 February 2022 diff hist +13 m Invertibility of Identity Minus Operator No edit summary Tag: Visual edit: Switched
- 00:5800:58, 3 February 2022 diff hist +4 m Invertibility of Identity Minus Operator No edit summary Tag: Visual edit: Switched
- 00:5600:56, 3 February 2022 diff hist +2,099 N Invertibility of Identity Minus Operator Created page with "== Theorem == Suppose $B$ is a Banach space and $T \in \mathfrak{L}(B, B)$, the space of bounded linear operators on $B$. If $\| T \|_* < 1$ in the operator norm $\|\cdot\|_*$:, then $I - T$ is invertible and has inverse $$(I - T)^{-1} = \sum_{n\in \Bbb N} T^n.$$ == Proof == Define $S_n = I + T + T^2 + \ldots + T^n$. We first argue that $S_n$ converges to a bounded linear operator $S\in \mathfrak{L}(B,B)$. For any $n > m$, {{begin-eqn}} {{eqn | l = \norm{S_n - S_m}..."
- 00:3100:31, 3 February 2022 diff hist +778 N Operator Norm on Banach Space is Submultiplicative Created page with "== Theorem == Let $B$ be a Banach space, and let $S, T \in \mathfrak{L}(B, B)$ be bounded linear operators on $B$. Then $\|ST\|_* \leq \|S\|_* \|T\|_*$, where $\|\cdot\|_*$ is the operator norm. == Proof == Let $x\in B$ have norm $1$. Then {{begin-eqn}} {{eqn | l = \norm{(ST)x}_B | r = \norm{S(Tx)}_B }} {{eqn | o = \leq | r = \norm{S}_* \norm{Tx}_B | c = by definition of $\norm{S}_*$ }} {{eqn | o =..."
28 January 2022
- 19:1119:11, 28 January 2022 diff hist +2 m Convergence in Measure Implies Convergence a.e. of Subsequence No edit summary Tag: Visual edit: Switched
- 19:1019:10, 28 January 2022 diff hist 0 m Convergence in Measure Implies Convergence a.e. of Subsequence No edit summary Tag: Visual edit
- 18:4918:49, 28 January 2022 diff hist +1,291 N Convergence in Measure Implies Convergence a.e. of Subsequence Created page with "== Theorem == Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $f_n: D \to \R$ be a sequence of $\Sigma$-measurable functions for $D \in \Sigma$. Let $f_n$ converge in measure to a function $f$ on $D$. Then there is a subsequence $f_n$ of $f$ that converges a.e. to $f$. == Proof == For each $n, k\geq 1$, define $B_..."
- 17:5117:51, 28 January 2022 diff hist +61 m Scheffé's Lemma No edit summary Tag: Visual edit: Switched
- 17:4917:49, 28 January 2022 diff hist +1,087 Scheffé's Lemma No edit summary Tag: Visual edit: Switched
20 January 2022
- 07:3607:36, 20 January 2022 diff hist +14 m Scheffé's Lemma No edit summary Tag: Visual edit: Switched
- 07:3507:35, 20 January 2022 diff hist +16 m Scheffé's Lemma No edit summary Tag: Visual edit: Switched
- 07:3007:30, 20 January 2022 diff hist +83 Scheffé's Lemma No edit summary Tag: Visual edit: Switched
- 07:2807:28, 20 January 2022 diff hist +3 m Scheffé's Lemma fix error Tag: Visual edit
- 07:2807:28, 20 January 2022 diff hist +2,420 N Scheffé's Lemma Created page with "==Theorem== Let $\struct {X, \Sigma, \mu}$ be a measure space and $f_n$ be a sequence of $\mu$-integrable functions that converge almost everywhere to another $\mu$-integrable function $f$. Then $f_n$ converges to $f$ in $L^1$ if and only if $\int_X f_n d\mu$ converges to $\int_X f d\mu$. ==Proof of First Direction== Suppose $f_n \to f$ in $L^1$. Then {{begin-eqn}} {{eqn | l = \size{ \int_X \si..." Tag: Visual edit: Switched
25 May 2020
- 00:4000:40, 25 May 2020 diff hist −32 m Integral of Distribution Function No edit summary
- 00:0400:04, 25 May 2020 diff hist +1,669 N Integral of Distribution Function Created page with "== Theorem == Let $\struct {X, \Sigma, \mu}$ be a measure space and $f$ be a $\mu$-measurable function. Let $p > 0, r \geq 0$. For $\lambda > 0$..."
21 August 2010
- 06:2906:29, 21 August 2010 diff hist −2,215 Measure of Empty Set is Zero No edit summary
- 06:2606:26, 21 August 2010 diff hist −152 Definition:Measure (Measure Theory) No edit summary
28 January 2010
- 21:1221:12, 28 January 2010 diff hist +220 m Definition:Lipschitz Condition No edit summary
- 03:2703:27, 28 January 2010 diff hist +150 Definite Integral of Partial Derivative No edit summary
- 03:2003:20, 28 January 2010 diff hist −1 m Definite Integral of Partial Derivative No edit summary
- 03:1603:16, 28 January 2010 diff hist +45 m Definite Integral of Partial Derivative No edit summary
- 03:1203:12, 28 January 2010 diff hist −1 m Definite Integral of Partial Derivative No edit summary
27 January 2010
- 21:5721:57, 27 January 2010 diff hist −3 m Definite Integral of Partial Derivative No edit summary
- 21:5521:55, 27 January 2010 diff hist +32 m Definite Integral of Partial Derivative No edit summary
- 21:5521:55, 27 January 2010 diff hist +2 m Definite Integral of Partial Derivative No edit summary
- 21:4921:49, 27 January 2010 diff hist +3 m Definite Integral of Partial Derivative No edit summary