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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1answer
45 views

Sequence of functions that converges strongly in $L^2(\mathbb R)$ but not pointwise

I am trying to find a sequence of functions $\{f_j\}$ and a function $f$ such that $f_j\to f$ strongly in $L^2(\mathbb R)$ but $f_j$ does not converge to $f$ pointwise. The definition of strong ...
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75 views

Some doubts about proof of Strong Law of Large Numbers

I quote Jacod-Protter. Theorem: Let $\left(X_n\right)_{n\geq1}$ be independent and identically distributed and defined on the same space. Let$$\mu=\mathbb{E}\{X_j\}$$ $$\sigma^2=\sigma_{X_j}^2<\...
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14 views

Strong continuity of unitary representation of a Locally-compact Hausdorff Topological Group $G$.

Let $H_\pi$ be a hilbert space. $(\pi,H_{\pi})$ is an unitary representation of a Locally-compact Hausdorff Topological Group $G$ on $H_\pi$. We call a representation is strongly continuous if for ...
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2answers
53 views

Weak and strong convergence of a sequence of operators in $L^2(\mathbb{R})$

Let $M_\alpha = sin(\frac{x}{\alpha})f(x)$ a sequence of operators in $L^2(\mathbb{R})$. Prove that $M_\alpha$ does not converge strongly to $0$, but converges weakly to $0$, for $\alpha \to 0$. ...
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1answer
15 views

Tail of increasing convergent net of self-adjoint operators is bounded

Let $H$ be a Hilbert space and $(T_\alpha)$ an increasing net of self-adjoint operators that converges (in some topology) to an operator $T$. Then $(T_\alpha)$ is not necessarily norm-bounded I think (...
2
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1answer
61 views

When does strong convergence imply convergence in operator norm?

I have a $C_0$-semigroup $(T_t)_{t\ge0}$ and I want to show $\lim_{t \to \infty} T_t =0 $ with respect to the operator norm. After some effort, I was able to prove $\lim_{t \to \infty} T_t =0 $ ...
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1answer
24 views

Let $B_nx=x(t-\frac{1}{n})\in L^2(\mathbb{R})$. Show that $B_n\overset{s}{\to}Id$ but $B_n\not\rightrightarrows Id$

Let $B_nx=x(t-\frac{1}{n})$ be a sequence of operators in $L^2(\mathbb{R})$. Show that $B_n\overset{s}{\to}Id$ but $B_n\not\rightrightarrows Id$ where $Id$ is the identity operator. So for the strong ...
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0answers
43 views

Weak and Pointwise Convergence question

I found an interesting property in some lecture notes on weak convergence: lemma 3.2 (2) on page 10: https://www.uio.no/studier/emner/matnat/math/MAT4380/v06/Weakconvergence.pdf . The idea is as ...
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1answer
29 views

Range of strong limit of a semigroup belongs to the fixed space?

Let $\left(T(t)\right)_{t\geq0}$ be a $C_0$-semigroup on a Banach lattice $E$ such that $T(t)$ converges strongly to a positive operator $S$ as $t \to \infty.$ Then $$T(t)S=S \text{ for all }t\geq0.$$ ...
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1answer
28 views

If $H$ is Hilbert and $T\in B(H)$ is left shift w.r.t. the OB $(e_{k})_{k\in\mathbb{N}}$, then $T^{n}x\to0$ for all $x\in H$

Let $H$ be Hilbert and $T\in B(H)$ the left shift operator w.r.t. the orthonormal basis $(e_{k})_{k\in\mathbb{N}}$. That is, $Te_{1}=0$ and $Te_{k}=e_{k-1}$ for $k>1$. Then how do I show that $T^{n}...
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22 views

PDEs: Dirichlet problem with cubic term and first derivative

I have following problem: $$ -\Delta u + u^3 + \sum b_i \frac{\partial u}{\partial x_i} = f $$ with $u = 0$ on boundary. With $b_i \in L^{\infty}$, $d \in \{2, 3\}$ and $f \in L^2$, $\Omega$, as ...
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1answer
41 views

Does weak convergence imply strong convergence for parameterized measure?

Define for some density $q_{\kappa_n}$ (with respect to the Lebesgue measure) on $\Theta$, parameterized by $\kappa_n \in \mathbb{R}^d$. Next, define the sequence of measures $\mu_n$ as $$\mu_n(A) = \...
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1answer
40 views

Laws of large numbers problem: Convergence of a symmetric sequence [closed]

I am looking for good solutions to this problem. Could you please help me with this? Any solution would be appreciated. Let $\mathit{X_n}$ be independent and identically distributed. Assume $\...
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1answer
39 views

Weak, Strong, a.e. convergence parameters in $L^2(R^3)$ excercise

I'm doing this exercise, and I wish for corrections on my solution: a) Find the index set $I:=\{ \alpha \in R\} : \{f_n\} \in L^2(R^3), > f_n(x) = n^\alpha X_{B_{1/n}}$ converge weakly in $L^2(R^...
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1answer
32 views

Operator satisfying certain properties

Give an example of a Hilbert space $H$ and a sequence of compact operators $(S_n)_{n=1}^{\infty}$ on H such that: (i) $||S_n||\leq 1$ (ii) The operators $V_N=\sum_{n=1}^N\dfrac{1}{n}S_n$ converge ...
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0answers
75 views

Poke holes in this proof of the SLLN

I have a proof (sketch) of the Strong Law of Large Numbers, at least the "sufficiency" half of it, that seems a little too easy. This is the version where you only assume i.i.d. random variables, and ...
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75 views

About weak convergence and convergence of norms

Today in an exam on functional analysis the following question was posed: Let $H$ be a Hilbert space and $(x_n)_{n\in \Bbb{N}} \subseteq H$ be a sequence that converges weakly to $x\in H$ satisfying $...
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1answer
71 views

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 ...
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1answer
86 views

Example that strong convergence does not imply convergence in norm

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 ...
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0answers
43 views

Weak and strong convergence of operator sequence

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\}...
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1answer
58 views

Almost sure convergence of quadratic form x'Ax

Let $x_n$ be an $n\times 1$ vector of random variables, and $A_n=(a_{ij,n})$ be an $n\times n$ constant matrix. Suppose that $n^{-1}x_n'x_n$ converges almost surely to some limit as $n\rightarrow \...
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1answer
24 views

Does this conclude that $ \int_E f_jg_j dx \to \int_E fgdx$?

Suppose $f_j \to f$ strongly in $L^2(E)$ and $g_j \to g \ $ weakly in $L^2(E)$, where $E \subset \mathbb{R}^d$ is measurable. Show that $ \ \int_Ef_jg_jdx \to \int_E fgdx$. Answer: I am quoting the ...
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1answer
35 views

Boundedness and Strong convergence

$f_n\rightarrow f$ in $L^2(0,1)$, $\{ f,f_1,f_2,\ldots \}\subset H^1(0,1)$, $||f_n||_{H^1(0,1)}\leq M,\ \forall n\geq 1 $, Is is true that $f_n\rightharpoonup f$ in $H^1(0,1)$? If not, then what is a ...
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2answers
209 views

Bounded sequence in $L^\infty$ which converges in $L^1$

I have sequence in $L^\infty(\mathbb{R}^n)\cap L^1(\mathbb{R}^n)$ such that 1. $(u_n)_n$ is bounded in $L^\infty$ : there exists $a>0$ such that $\|u_n\|_{L^\infty}\leq a$. 2.$(u_n)_n$ converges (...
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1answer
93 views

Problem about weak convergence

I am reading a lecture note about Radon Riesz theorem, the resources is here: https://faculty.etsu.edu/gardnerr/5210/notes/Radon-Riesz.pdf At page 4, fourth line from the bottom, it says convergence ...
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1answer
22 views

Under what condition on the space X, any Continuous operator will be Completely continuous.

Categorise the spaces $X$ for which $B_{00}(X,X)=B(X,X)$, where $B(X,X)$ is the set of bounded linear operators and $ B_{00}(X,X)$ the set of completely continuous operators, i.e. operators which take ...
2
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1answer
45 views

Spectrum and Convergence of an operator

can you help me to solve this exercice? The first point is ok but I have problems with the others. Let $S_\varepsilon$ an continuous operator on $L^2(\mathbb{R})$ define as $$ S_\varepsilon[f](x)= \...
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2answers
69 views

Strongly convergent subsequence $+$ point-wise convergence $\Rightarrow$ strong convergence?

Let $X$ be a Hilbert space containing functions defined over a bounded region $\Omega\subset \mathbb{R}^N$. Assume $f_n\in X$ has a strongly convergent subsequence, say $f_{n_k}$. Also, assume $f_n\to ...
3
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2answers
426 views

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 ...
2
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1answer
100 views

Strongly convergence and Uniformly convergence in Banach space

Let $X,Y$ be Banach spaces, $T_n:X\to Y$ are bounded linear operators and $S:X\to Y$ is compact operator. Suppose for all $x\in X$, $||T_nx||\leq ||Sx||$ and $T_n$ strong convergent to $T$(bounded ...
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0answers
75 views

Double Limit of operators converges weakly, does single limit converge?

I am working on a problem from Reed and Simon, which states: Suppose $\{A_\alpha\}$ and $\{B_\alpha\}$, $\alpha \in I$, are nets. Let $A_\alpha^* \to A^*$ and $B_\alpha \to B$ in the Strong Operator ...
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2answers
62 views

Example Not convergent uniformly

Are there Hilbert space $H$ and adjoint operator $T$ s.t., $||T||=1$, $(Tx,x) (\forall x \in H)$ and $T^n$ not convergent uniformly?   I proved $T^n$ convergent strongly by using spectrum theory. ...
2
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1answer
115 views

Convergence in trace-class norm

Let $A_n\to 0$ strongly and each $A_n$ be trace-class. It would have been nice if $A_n\to0$ in trace-class norm, but this is false. Here is a possible counter-example: Let the Hilbert space be $\...
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1answer
155 views

Book of Functional analysis

I'm looking for a book/note or something else of functional analysis which has a lot of exercises and examples. In particular I need convergence (strong, weakly) in C^m spaces, L^p and also in sobolev ...
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1answer
85 views

Exercise on Sobolev Spaces and strong/weak convergence

I have to solve this exercise: "Fix $v \in \mathcal{C}^\infty_c(\mathbb{R})$. Discuss the strong and the weak convergence of the sequence $u_n(x) = \frac{v(nx-n^2)}{n}$ in the spaces $W^{k,p}(\mathbb{...
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1answer
168 views

Mazur Lemma for sequence of Lipschitz functions

This question regards to the Proposition 5.1 in Godefroy and Kalton (2003). Let $E$ be a finite-dimension Banach space and let $(L_n)_{n\in \mathbb{N}}$ be a sequence of functions $L_n: E\to \mathcal{...
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1answer
91 views

Divergence free property preserved under weak convergence in $L^2$

Given $\Psi =\{ u \in H^1_0(\Omega) : \mathrm{div}(u)=0 \in \Omega\}$ equipped with $ H^1(\Omega)$ norm , and $\Phi$ is the closure of $\Psi$ for the $L^2(\Omega)$ norm ,it is equipped with $L^2(\...
2
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2answers
131 views

Prove strong convergence in Hilbert space

Let $H$ be Hilbert space, $T:H \to H$ be bounded self adjoint operator. Suppose $\|T\|=1$ and for all $x \in H, \langle Tx,x \rangle \geq 0$. Then $\{T^n\}$ strongly convergent, but there exists $T$ ...
3
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1answer
81 views

Understanding a problem and the construction of a proof

Hello Math Stack Exchange. I'm currectly studying the functional analysis, and I am kind of rusty to construct a proof. I have been thinking a lot about the problem, that I have been unsure how to ...
2
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0answers
364 views

Weakly convergent sequence with a strongly convergence subsequence

Suppose we have a sequence in a Hilbert Space, which converges weakly to some limit, and it has a subsequence which converges strongly to that same limit. Does this imply the sequence is strongly ...
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2answers
577 views

Why do this sequence converges weakly but not strongly?

Let $f_n(x)=\sin(nx)g(x)$ a sequence in $L^2(\mathbf{R})$, where $g(x)\in L^2(\mathbf{R})$ is such that $g(x)>0$. I have to prove that $$f_n(x)\rightharpoonup 0 \ \ \ \mbox{weakly}$$ but $$f_n(x)\...
5
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3answers
966 views

Counter example: strong convergence does not imply convergence in operator norm [closed]

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\...
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2answers
170 views

$A$ is strongly dense in its double commutant $A^{''}$

The picture below comes from Murphy's book. My Questions: (1) In the sixth line of the picture, why $u(x)\in K$? Note that from $pu=up$, we only know that $K$ reduces $u$. (2)I think the proof can ...
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1answer
476 views

Uniform, strong and weak convergence of the operators

First of I just want to provide the definitions I have for the three concepts. Let $A_n: X \to Y$. $A_n$is said to converge to $ A:(n\to \infty)$ $$\text{Uniformly : }\|A_n-A\|=0 \\ \text{Strongly : ...
4
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2answers
3k views

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_{...
2
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2answers
109 views

Let $T_{n}:X \rightarrow Y$ bounded s.t $\left\|T_{n}\right\| \rightarrow \infty$, Then $\exists x_{0}\in X$ s.t $|T_{n}(x_{0})|\rightarrow \infty$

Let $X$ be Banach space an $Y$ normed vector space. Let $T_{n}:X \rightarrow Y$ lineal and bounded operators such that $\left\|T_{n}\right\| \rightarrow \infty$ as $n\rightarrow \infty$. Then there ...
2
votes
3answers
427 views

A subsequence of a weakly convergent sequence $x_n$ is strongly convergent to $x$, then the sequence $x_n$ is strongly convergent to $x$?

If $x_n$ in a normed space $X$ is weakly convergent to $x$, and if there is a subsequence $y_k$ of it such that this subsequence is strongly convergent to $x$, then can we say that the sequence $x_n$ ...
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0answers
528 views

“Proof” that weak convergence implies strong convergence, even for infinite dimensional spaces

We know a result that if for each bounded linear functional $f$ from a normed space $X$ to $K$(its field), $f(x)=0$ then $x=0$. Now if $x_n$ in $X$ is weakly convergent to $x$ then for each bounded ...
4
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0answers
58 views

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 $y$, $$ f(X_n,y)\overset{\text{a.s.}}{\longrightarrow}g(y). $$...
0
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1answer
653 views

Weak convergence and $\|f_{n}\|\rightarrow\|f\|$ implies strong convergence [duplicate]

Let $\left\{ f_{n}\right\} _{n=1}^{\infty}$ be a sequence in $L^{p}\left([0,1]\right)$ for $p\geq1$. Suppose that there exists $f\in L^{p}\left([0,1]\right)$ satisfying $\lim_{n\rightarrow\infty}...