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 $\lim_{n\rightarrow \infty } \| x_n - x\| =0$

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strong and weak* limits

I have the sequence $\Phi_n(x)=\sin(nx) \ \forall x\in(0,\pi),n\in \mathbb{N}$ and I know that it weakly* converges to $0$ in $L^{\infty}((0,\pi))$. Now I have to show that it doesn't converge ...
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Strong convergence in $L^p(\mathbb{R}^N)$

I would like to ask your help about this: Let $q>1$ fixed. Consider $\{m_n\}$ a sequence of non-negative functions in $L^s(\mathbb{R}^N)$ for every $s\in[1,q)$ such that $\int_{\mathbb{R}^N}m_n=M$ ...
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Equivalent definitions for Strong Operator Topology in Banach Spaces

Strong Operator Topology (definition-$1$): We called a sequence $\{T_n\}$ in $\beta(X,Y),$ for $X,~Y$ are Banach spaces and $\beta(X,Y)$ denotes the family of bounded linear operators from $X$ to $Y$, ...
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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 ...
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Proof of a.s. convergence in Continuous Mapping Theorem

Suppose $g$ is a.e. continuous on the support of $X$. Then the continuous mapping theorem (CMT) asserts (among other things) that \begin{align*} X_n\overset{a.s.}{\rightarrow}X\implies g(X_n)\overset{...
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Brezis Exercise 3.13 item 3.

I work in the following question: Let $E$ be a space Banach, $\{x_n\}_{n=1}^{\infty} \subset E$, $x \in E$ and \begin{equation*} K_{n}= \overline{con(\bigcup_{i=n}^{\infty}\{xi\})}. \end{...
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Prove that if $T_n \rightarrow T$ and $x_n \rightarrow x$ then $T_n x_n \rightarrow T x$

$X$ - Banach space and $T_n$ bounded linear operators that is strongly convergent to $T$ as well as $x_n$ is strongly convergent to $x$. How to prove that $T_n x_n$ is strongly convergent to $Tx$? I ...
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Show strong operator convergence

I have a problem in which I need to prove that a sequence of operators is strongly convergent and that it isn't uniformly convergent. The operators are defined like so: $T_j: L^1 \to L^∞$ And the ...
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weak and strong $L^p$ convergence

for my class I need to find two sequences of functions such that each satisfies one of the conditions below: $f_n \in L^p(\Bbb{R})$ such that it converges weakly toward $0$ in $L^p(\Bbb{R})$ but ...
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Are strong convergence of measure and almost sure convergence of a random variable related?

I am studying measure theory and probability I am wondering if and how these two concepts are related. A sequence of random variables $X_n$ defined on some probability space $(\Omega,\mathscr{A},\mu)$ ...
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Does weak convergence in $L^\infty(\mathbb{R},L^p(\mathbb{R}^n))$ imply strong convergence in $L^p_{loc}(\mathbb{R}\times\mathbb{R}^n)$?

Does weak convergence in $L^\infty\left(\mathbb{R},L^p\left(\mathbb{R}^n\right)\right)$ imply strong convergence in $L^p_{loc}\left(\mathbb{R}\times\mathbb{R}^n\right)$ up to a subsequence ?
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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 ...
I have found an example that strong convergence does not imply convergence in norm. Let us take $T_{n} = P_{span\{e_{1},...,e_{n}\}}$ be a projection on $span\{e_{1},...,e_{n}\}$ in Hilbert's space ...
Let $|e|\leq1$ such that $e\neq0$, we define the lineal operator $F_e: C^1([-1,1],R)\rightarrow R$ as $$F_e x(t) =\frac{x(e) - x(-e)} {2 e} .$$ I need to know if the sequence of operators \${\{F_e\}...