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Combined display of all available logs of ProofWiki. You can narrow down the view by selecting a log type, the username (case-sensitive), or the affected page (also case-sensitive).
- 01:17, 10 February 2022 Mag487 talk contribs created page 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...")
- 01:03, 10 February 2022 Mag487 talk contribs created page 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 $\...")
- 05:41, 3 February 2022 Mag487 talk contribs created page Resolvent Mapping is Analytic (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)...")
- 01:48, 3 February 2022 Mag487 talk contribs created page Resolvent Mapping is Continuous (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
- 00:56, 3 February 2022 Mag487 talk contribs created page 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}...")
- 00:31, 3 February 2022 Mag487 talk contribs created page 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 =...")
- 18:49, 28 January 2022 Mag487 talk contribs created page 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_...")
- 07:28, 20 January 2022 Mag487 talk contribs created page 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
- 00:04, 25 May 2020 Mag487 talk contribs created page 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 $\lambda > 0$...")
- 22:18, 22 August 2009 User account Mag487 talk contribs was created