# Questions tagged [weak-topology]

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103 questions
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### How to prove or disprove the following statement on Hausdorff topology

Let $X$ be a set and let $Y$ be a Hausdorff space. Let $f\colon X \to Y$ be a given mapping. Define $U \subset X$ to be open in $X$ if, and only if $U = f^{-1}(V)$ for some set $V$ open in $Y$. This ...
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### Is a metrizable and compact subset of $E^*$ is a metrizable complete subset ? for the weak* topology

If we have a Banach space $E$. and we consider it's dual $E^*$, it is a Banach space. so we consider the weak* topology ($\sigma(E^*,E)$) on $E^*$. So My question is : If we have a set $B\subset E^*$,...
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### Rudin's functional analysis theorem 3.10, proof that multiplication is continuous

Suppose $X$ is a vector space and $X'$ is a separating vector space of linear functionals on $X$. Then the $X'$-topology $\tau'$ makes $X$ into a locally convex space whose dual space is $X'$. ...
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### Why are weak topologies useful in functional analysis?

I've been reading through chapter 3, Rudin's Functional Analysis, and an important point is the one of weak topology. From the theorems it seems to me weak topologies are somehow the result of ...
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### A set $C$ that is not closed in the weak* topology

I have a problem, but first I would like to understand your statement. The statement is about a subset $C\subset l^{\infty}$ that is convex and closed in $C(l^{\infty}, |.|_{sup})$. I know that if $C$ ...
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### Prove the mapping $(x,y)\mapsto x+y$ from $X \times X \to X$ is continuous when $X$ is given a weak topology.

Let X be a Banach space. Prove the mapping $(x,y)\mapsto x+y$ from $X \times X \to X$ is continuous when $X$ is given a weak topology. Could anybody give me some hints to start?
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### Rudin functional analysis, section 3.8 (Topological preliminaries)

A quote from section 3.8. of Rudin's functional analysis: Suppose next that $X$ is a set and $\mathcal{F}$ is a nonempty family of mappings $f: X \to Y_f$, where each $Y_f$ is a topological space (...
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### Weak closure of subsets of the unitary sphere of a Banach space.

Assume that $(X,\|\cdot\|)$ is a Banach space with $\|\cdot\|$ strictly convex. Define $S=\{x\in X:\|x\|=1\}$. Suppose that $\varepsilon>0$ and $x_0\in S$ and define  B_\varepsilon=\{x\in X:\|x-...
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### $L^1([0,1])$ closed unit ball is not weakly compact

So I would like to prove this result by constructing a sequence of functions $u_n$ in $L^1([0,1])$, such that $\|u_n\|_{L^1}\leq 1$ for all $n$, but this subsequence does not have a convergent ...
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### $X^*$ with $weak^*$-topology is first category in itself when $X$ is infinite dimensional Fréchet space

I am reading Rudin's Functional Analysis and stuck at the problem in exercise 11 on page 87 : Let $X$ be infinite dimensional Fréchet space , then $X^*$ with it's $weak^*$-topology is of first ...
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### Can we prove that in a convex space, Weakly closed=> weak* closed ??

We know that weak* Topology is smaller than weak topology. So weak* closed sets are weakly closed. Banach Mazur theorem says "Strongly closed implies weakly closed if space is convex." can we expect ...
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### Is the space of finite Radon measures, equipped with weak topology, locally compact and second countable?

I saw someone in the following link claims the space of finite Radon measures is locally compact with weak topology. Is the space of finite Radon measures, equipped with weak topology, locally ...
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### If for every $e\in E\setminus\{0\},$ there exists $e^*\in A$ such that $e^*(e)\neq 0,$ then $A$ is weak-star dense in $E^*?$

Suppose that $E$ is a Banach space and $E^*$ is a dual space of $E,$ that is, $E^*$ contains all bounded linear functionals on $E.$ Let $A$ be a closed subspace of $E^*.$ Equip $E^*$ with weak$^*$-...
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### Space of measures has the Dunford-Pettis property?

I'm trying to prove Corollary 5.4.6. from Albiac, Kalton - Topics in Banach Space Theory: If $K$ is a compact Hausdorff space then $\mathcal{M}(K)$ has (DPP). I want to use The Dunford-...
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### Compactness / sequentially compact / reflexive space / separable spaces / weak topology

I'm a bit confuse with all theses notions. Let $E$ a normed vector space of infinite dimension (also Banach, but it's probably not important). The theorem of Eberlin Smulian theorem says that : all ...
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### In “Analyse fonctionnelle” of Brezis, in chapter III why do we need Banach spaces ? (especially for Kakutani's theorem)

In the book of Brezis : "Analyse fonctionnelle : Théorie et application", chapter III (i.e. construction of weak topology, weak-* topology reflexives spaces...), why do we need "Banach spaces" ? Isn't ...
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### When $E$ has finite dimension, then the weak topology and the strong topologie are the same.

Let $E$ a normed vector space of finite dimension. Then the strong topology and the weak topology are the same. To prove it, we take $x_0\in E$ and $U$ an open set (for the strong topology) that ...
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### When does the continuous dual of the weak operator topology consist only of finite linear combinations of evaluations?

Let $V$ and $W$ be topological vector spaces, say, over $\mathbb C$. Let $L(V, W)$ be the vector space of continuous linear maps equipped with the weak operator topology, which is the initial topology ...
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### Do operators from $L(X', Y')$ preserve weak*-convergence?

I am wondering whether the following is true: Let $X, Y$ be normed spaces and $T \in L(X', Y')$. If $x_n' \overset{*}{\rightharpoonup} x'$ in $X'$, then $Tx_n' \overset{*}{\rightharpoonup} Tx'$ ...
Let $\mu_k$ be a sequence of probability (Borel) measures on $\mathbb{R}^n$. We say, $\mu_k$ converges to a probability measure $\mu$ weakly if $\int gd\mu_k \to \int gd\mu$ for every continuous ...
### How to show a normed space $X$ is reflexive if the weak topology and weak* topology on $X^*$ are equal?
Let $X$ be a normed space such that the weak topology and weak* topology on $X^*$ are the same. I want to show $X$ is reflexive. My attempt is: Since we can use $\sigma(X^*,\widehat{X})$ to denote ...