Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [strong-convergence]

A sequence $(x_n)$ in a normed space $X$ is said to be strongly convergent if there is an $x\in X$ such that $\lim_{n\rightarrow \infty } | x_n - x| =0 $

Compare weak convergence : A sequence $(x_n)$ in a normed space $X$ is said to be weakly convergent if there is an $x\in X$ such that for every $f\in X'$, $\lim_{n\rightarrow \infty } f( x_n ) = f(x) $ where $X'$ is a dual space. In fact if ${\rm dim}\ X<\infty$ they are same. In general weak convergene does not imply strong convergence

115 questions
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Does left-multiplication by compact operators turn strong-convergence into norm-convergence?

If $\{T_i\}_{i\in I}$ is a bounded net of operators on a Hilbert space $\mathscr H$, converging strongly to some operator $T$, and if $K$ is a compact operator on $\mathscr H$, then the net $\{T_iK\}_i$ is known to converge in norm to…
Ruy
  • 20,073
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What distinguishes weak and strong convergence of bounded linear operator in Banach spaces?

I'm self-studying using Applied Analysis by John Hunter and Bruno Nachtergaele. In chapter 5 on Banach space, the authors defined strong convergence and weak convergence as followed: A sequence ($T_{n}$) in $\mathcal{B}(X,Y)$ converges strongly…
userrandom
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Counter example: strong convergence does not imply convergence in operator norm

Consider a family of operators $T_n\in\mathcal{L}(X)$ where $X$ is a separable Hilbert space. Find examples in which $T_n$ converges strongly to $T\in \mathcal{L}(X)$, i.e. $\|Tx-T_nx\|\to 0$ as $n\to \infty$ for all $x\in X$, but not in operator…
Saj_Eda
  • 803
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Weak convergence to strong convergence

Let $u_n:[0,1]\to[0,\infty)$ are functions such that $u_n\rightharpoonup u$ in $L^2[0,1]$ and $u_n \log u_n\rightharpoonup u\log u$ in $L^1[0,1]$. Does this imply $u_n$ strongly converges to $u$ in $L^1[0,1]$? I remember doing this problem a few…
mudok
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Confusion about weak and strong convergence

This might be rather elementary but I can't seem to find what's wrong with the following argument. I suppose it's something rather trivial. For operators $T\in B(\mathcal{H})$ We say that $T_i\to T$ weakly if $\forall x,y\in \mathcal{H}$ we…
El Ruño
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On a sequence $f_k$ in $L^{2+\frac{1}{k}}$

Suppose that $f_k\in L^{2+\frac{1}{k}}(\Omega)$ with the property that $\|f_k\|_{L^{2+\frac{1}{k}}(\Omega)} = 1$ for all $k\ge 1$. $\Omega$ is a bounded domain in $\mathbb{R}^n$. Can such a sequence of functions strongly converge to ZERO in…
Carl Lincoln
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Weak convergence implies strong convergence in $L^1$ for Fourier series?

We say $\{f_n\}$ weakly converge to $f$ in $L^1[-π,π]$ if for each $g \in L^\infty[-π,π]$, $$\lim_{n\to\infty}\int_{-π}^{π}f_n(x)g(x)dx=\int_{-π}^{π}f(x)g(x)dx.$$ There is a question in my homework which I can't prove it: For each $f \in…
Luke Chen
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Transposition operator is continuous? (topology of bounded convergence)

Given a continuous map of TVS $u:E\to F$, we can associate to it the transpose map $u^t:F'\to E'$. Here, $E',F'$ are the continuous duals of $E,F$, respectively. The map $u^t$ is continuous, so long as $E', F'$ carry a reasonable topology (the weak…
Or Kedar
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How does the Glivenko-Cantelli theorem improve the stochastic convergence of the empirical distribution $F_n(x)$?

Let $X_i$ be iid random variables with empirical cumulative distribution function $F_n(x)$ and CDF $F(x)$. From the central limit theorem and the strong law of large numbers, we know that $F_n\stackrel{d/a.s.}{\to}F$. The Glivenko-Cantelli theorem…
Disjoint Union
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If $f(X_n,y)\to g(y)$ almost surely for each fixed $y$, $\Pr[f(X_n,Y)\leq\alpha]\to\Pr[g(Y)\leq\alpha]$?

Suppose that $Y$ is a random variable, $\{X_n\}$ a sequence of random variables, and $f,g$ functions. It's given that for each nonrandom $y$, $$ f(X_n,y)\overset{\text{a.s.}}{\longrightarrow}g(y). $$ For a nonrandom $\alpha\in(0,1)$, can we say…
yurnero
  • 10,675
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probability convergence as compactness for a.s. convergence

I stumbled upon the sequential characterization of probability convergence. $ X_n \xrightarrow{\mathbb{P}} X$ if and only if for each subsequence $X_{n_k}$, there exists a subsequence $X_{n_{k_j}}$ such that $X_{n_{k_j}} \xrightarrow{a.s.}…
Simone Licciardi
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Relationship between strong and weak operator topologies

I am looking for citations in a book or article for the following fact: Let $T_n$ be a sequence of operators in Hilbert space and $T$ also be an operator in Hilbert space. If $T_n$ is convergent in a weak operator topology to $T$ and $T_n^*T_n$ is…
james stan
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Norm of a Toeplitz operator

Hello and thank you for visiting my Stack Exchange post. I am going through the book called Introduction to large truncated Toeplitz matrices by Albrecht Böttcher & Bernd Silbermann and I am on the first chapter. I am trying to fill in the details…
snafu
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$\sigma$-Weakly Continuous Bounded Linear Functional on a von Neumann Algebra is Normal

I have been working on Exercise 4 in Chapter 4 of Murphy's "$C^*$-Algebras and Operator Theory", which is as follows: Let $A$ be a von Neumann algebra on $H$, and suppose that $\tau$ is a bounded linear functional on $A$. We say $\tau$ is normal if,…
LSK21
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Will $L^1\log L^1$ bound gives strong $L^1$ convergence?

I am trying to prove this statement: given a sequence $\{f_n | f_n > 0, c_1 \leq \int_{\Omega} f_n \log{f_n} \leq c_2 \}$, here $\Omega$ is a bounded domain; can we prove the $L^1$ strong convergence of $f_n$? With the given information, now I can…
Ziyao Yu
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