# Questions tagged [weak-convergence]

For questions about weak convergence, which can concern sequences in normed/ topological vectors spaces, or sequences of measures. Please use other tags like (tag: functional-analysis) or (tag: probability-theory).

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### Strong and weak convergence in $\ell^1$

Let $\ell^1$ be the space of absolutely summable real or complex sequences. Let us say that a sequence $(x_1, x_2, \ldots)$ of vectors in $\ell^1$ converges weakly to $x \in \ell^1$ if for every ...
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### If $X_n \stackrel{d}{\to} X$ and $c_n \to c$, then $c_n \cdot X_n \stackrel{d}{\to} c \cdot X$

Let $X_n$, $X$ random variables on a probability space $(\Omega,\mathcal{A},\mathbb{P})$ and $(c_n)_n \subseteq \mathbb{R}$, $c \in \mathbb{R}$ such that $c_n \to c$ and $X_n \stackrel{d}{\to} X$. ...
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### How to prove that $\mu_n \rightharpoonup \mu$ IFF $\mu_n \overset{*}{\rightharpoonup} \mu$ and $\mu_n (X) \to \mu (X)$?

Let $X$ be a metric space, $\mathcal M(X)$ the space of all finite signed Borel measures on $X$, $\mathcal C_b(X)$ be the space of real-valued bounded continuous functions, $\mathcal C_0(X)$ be the ...
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### Is the weak topology sequential on some infinite-dimensional Banach space?

Recall that a topological space is sequential, iff every sequentially closed set is already closed. Is there an infinite-dimensional Banach space on which the weak topology is sequential? I ...
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### Every bounded sequence has a weakly convergent subsequence in a Hilbert space

I tried to prove the following theorem and was wondering if someone could please tell me if my proof can be fixed somehow... Theorem: Let $H$ be a Hilbert space and $x_n\in H$ a bounded sequence. ...
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### Schur's Theorem: In $\ell^1$ weak convergence of $x_n$ is the same as convergence in the norm

I'm having a really hard time with nearly every part of this proof, any help would be appreciated. Schur's Theorem: In $\ell^1$ weak convergence of $x_n$ is the same as convergence in the norm. ...
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### Proving the closed unit ball of a Hilbert space is weakly sequentially compact

I bumped into this statement in Hofer-Zehnder in the middle of proving a Hamiltonian field always has a periodic orbit over a level set of the hamiltonian if that set is a regular compact and strictly ...
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### continuity in the strong topology implies continuity in the weak one

I have to prove that if $T:(E,\|\cdot\|_E)\rightarrow (F,\|\cdot\|_F)$ is a continuous and linear operator, and $x_h\rightharpoonup x$ in $E$, than $Tx_h\rightharpoonup Tx$ in $F$. So we know that $T$ ...
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### Lévy's metric on $\mathbb{R}^d$

I know that a sequence of measures on $\mathbb{R}$ converges in distribution if and only if the corresponding Lévy's metric converges (Relationship to weak toplogy (Lévy metric)). According to ...
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### Functional weakly lower-semicontinuous [duplicate]

If $X$ is a topological space, then a functional $\varphi:X\to\mathbb{R}$ is lower-semicontinuous (l.s.c) if $\varphi^{-1}(a,\infty)$ is open in $X$ for any $a\in\mathbb{R}$. If $X$ is a Hilbert space,...
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### How to show that the space of probability measures on $\mathbb{R}$ is separable under Lévy metric

The Lévy metric between distribution functions $F$ and $G$ is given by: $$\rho(F,G) = \inf\left\{\epsilon : F(x-\epsilon)-\epsilon\leq G(x)\leq F(x+\epsilon)+\epsilon\right\}.$$ Another way to ...
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### Closed convex hull of weak convergent sequence: How to show that $x$ is the only element in $\bigcap K_n$?

Le $H$ be a Hilbert space and Suppose $x_n$ converges weakly to $x$ in $H$. Let $K_n$ be the closed convex hull $\bar{co}\{x_k:k\geq n\}$. I would like to show $\bigcap K_n=\{x\}$. What I know so far ...
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### Does the sequence $(\sqrt{n} \cdot 1_{[0, 1/n]})_n$ converge weakly in $L^2$?

Let $f_n (x) = \sqrt n 1_{[0,1/n ] } (x) \in L^2 (\mathbb R)$. Does $f_n$ converges weakly to $0$ in $L^2$? I tried to prove it by using Hölder's inequality or the Lebesgue differentiation theorem, ...
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### Convexity and strong lower semicontinuity imply weak lower semicontinuity

I have seen that if a set $K$ on an Hilbert space $H$ is convex and strongly sequentially-closed, it is weakly closed. The teacher said that if you take a convex and weakly lower semicontinuous ...
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### Weak convergence in probability implies uniform convergence in distribution functions

Exercise 1: Let $\mu_n$, $\mu$ be probability measures on $\left(\mathbb{R}, \mathcal{B}\left(\mathbb{R}\right)\right)$ with distribution functions $F_n$, $F$. Show: If $\left(\mu_n\right)$ converges ...
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### Confusion with the narrow and weak* convergence of measures

Think of a LCH space $X.$ Consider the spaces $C_{0}(X)$ of continuous functions "vanishing at infinity" and the space $BC(X)$ of bounded continuous functions. Consider as well the space of Radon (...
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In Hilbert space why does convergence in weak topology $x_n$ to $x$ imply that $x_n$ converges to $x$ in norm? Thank you very much for your answers. What if I put a condition on weak convergence i.e....