Questions tagged [weak-topology]

Let $X=(X,\tau)$ be a topological vector space whose continuous dual $X^*$ separates points (i.e., is T2). The weak topology $\tau_w$ on $X$ is defined to be the coarsest/weakest topology (that is, the topology with the fewest open sets) under which each element of $X^*$ remains continuous on $X$.

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Setwise convergence of measures implies weak convergence under special hypothesis

I'm struggling with producing a proof of the following result: Let $X = \overline{\mathbb{C}}$ be the Riemann sphere, and consider $M(X)$ the space of finite Borel measures on $X$ with norm given by ...
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Compare weak and finest locally convex topology on $\mathbb{R} [x]$

Let $V = \mathbb{R} [x] \cong \bigoplus_{\mathbb{N}} \mathbb{R}$ be the vector space of univariate polynomials, or the space of real sequences that have all but finitely many elements equal to zero. ...
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5 votes
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Weak* separability of dual unit ball of D[0,1]

Let $D[0,1]$ be the space of all right-continuous left-limited functions $f\colon [0,1]\to \mathbb{R}$ equipped with the supremum norm $f\mapsto \|f\|_\infty = \sup_{t\in[0,1]} |f(t)|$. This is a non-...
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Existence of measure within KL and Levy-Prokhorov distance

Let $p_1,q_1,p_2$ be probability measures. Does there exist a measure $q_2$ such that $$ KL(q_2||p_2) \leq KL(q_1||p_1) \quad \text{and} \quad \pi_{LP}(q_1,q_2) \leq \pi_{LP}(p_1,p_2) $$ where $KL(\...
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Is the set of rank 1 matrices positive matrices that sum to $1$ closed?

This question arises from my exploratory research on causality. Let $M=\{ A\in\mathbb R^{n\times n} : \forall i,j~A_{i,j}\geq 0,\sum_{i,j} A_{i,j} = 1 \}$ be the simplex of matrices of size $n\times n$...
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An explicit description for a certain type of infinite-dimensional homogeneous polynomials

Denote by $X_p$ ($1 \le p \le \infty$) the Banach spaces of complex sequences with finite $p$-norm and limit $0$. Suppose $(x_i):=(x_0, x_1, \dotsc)$, then the "degree-$d$ Veronese map" can ...
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1 vote
1 answer
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continuous functions in strong(norm) topolgy and weak topology

While reading Wasserstein GAN paper and in Appendix A, it says that The norm topology is very strong. Therefore, we can expect that not many functions $\theta \mapsto \mathbb{P}_\theta$ will be ...
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Does dense inclusion of dual space implies reflexive

If $X$ and $Y$ are Banach spaces that $Y \subsetneq X$, suppose $i : Y \hookrightarrow X$ and $i^{\star} : X^{\star} \hookrightarrow Y^{\star}$ are both continuous and norm dense. Must $Y$ be ...
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The weak topology on a dual pair $(X,Y)$ is metrizable iff the dimension of $Y$ is at most countable.

Here $X,Y$ are assumed to be vector spaces, and $Y$ a subset of the algebraic dual of $X$. The weak topology if of course generated by the family of seminorms $\{ |y(x)| < \epsilon, y \in Y, \...
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Pontryagin Dual and Continuity involving w* topology

For $G$, a locally compact group, we've defined $\hat{G}$ to be the group of all continuous homomorphisms from $G$ to the torus $\mathbb{T}$. With this definition, $\hat{G}\subseteq L^\infty(G)$ as ...
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1 answer
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Showing $L^{\infty}(\mathbb R)^*\neq L^1(\mathbb R)$ using the weak star compactness of unit ball in $L^{\infty}(\mathbb R)^*$.

$\lambda_n(f)=\frac{1}{2n}\int_{-n}^{n}f$ defines a dual element of $ L^{\infty}(\mathbb R)$. It is easy to see that $\lambda_n\in L^{\infty}(\mathbb R)^*_1.$ By using the weak-star compactness of ...
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weakly convergence and Hahn-Banach theorem in Reflexive Banach space

Here, it is the question. Let $E$ be a reflexive Banach space with dual topological space $E^{*}$. Assume that $\{x_{n}\}$ be a sequence weakly convergent to $x_{0}$ in ${E}$ and $\Vert x_{n}\Vert_{...
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1 vote
1 answer
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Apparent contradiction between weak metrizability results

I will fix $E \doteq \ell^2$, an infinite dimensional space, with separable dual. In here, there was a discussion that the weak topology of $E$ is not metrizable. By a result given in here, the disk $...
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Continuity of identiy mapping in the weak topology

I am trying to show that the identity mapping $$id :(M,\parallel \parallel) \longrightarrow (E,\sigma(E,E^*)) $$ is continuous. such that $E$ is a normed space and $M$ a vector subspace of $E$. Using ...
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1 answer
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Weak* convergence on $L^\infty(\Omega)$ and almost everywhere convergence [duplicate]

Let $\Omega$ be finite measure space. Suppose that $f_n \to f$ in $L^\infty(\Omega)$ for the weak* topology. Does there exists a subsequence (or a subnet) $(f_{n_k})$ such that $f_{n_k} \to f$ almost ...
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1 answer
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Brezis's Ex 3.22: If $E$ is reflexive, there is a sequence of norm $1$ that weakly converges to $0$

I'm doing Ex 3.22 in Brezis's book of Functional Analysis. Let $E$ be an infinite-dimensional Banach space satisfying one of the following assumptions: (a) $E'$ is separable (b) $E$ is reflexive. ...
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  • 1,153
1 vote
1 answer
59 views

Closed subspace of dual space is the whole space if the intersection of kernels is 0

Let $X$ be a Banach space, and $E \subset X^*$ a subspace of the dual $X^*$ that is closed in the weak-* topology. Show that if $\cap_{\lambda \in E} \ker(\lambda) = 0$, then $E = X^*$. The analogous ...
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2 answers
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Is the hypothesis $\mu \ge 0$ redundant in Brezis's Ex 3.15?

I'm doing Ex 3.15 in Brezis's book of Functional Analysis. Let $(E, |\cdot|)$ be a reflexive Banach space. In the following, the convex subset $K$ of $E$ is equipped with the weak topology $\sigma (E,...
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0 votes
1 answer
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The closure of linear subspace E is intersection of kernels (weak topology)

Let $X$ be real vector space and let $F$ be a linear subspace of the space of all linear functional $X \to \mathbb{R}$. And let $U_F$ be the weakest topology on $X$ such that all elements of $F$ are ...
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1 answer
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Can Brezis's Ex 3.9 be generalized to arbitrary subset $M$ of $E$?

I'm doing Ex 3.9 in Brezis's book of Functional Analysis. Let $E$ be a Banach space; let $M \subset E$ be a linear subspace, and let $f \in E'$. Prove that there exists some $g \in M^{\perp}$ such ...
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0 votes
1 answer
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Weakly convergent sequence on a strongly compact set is strongly convergent

I'm doing Ex 3.7 in Brezis's book of Functional Analysis. Could you have a check on my attempt? Let $E$ be a Banach space and let $K \subset E$ be a subset of $E$ that is compact in the strong ...
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1 vote
2 answers
60 views

Weakly compact set is bounded in norm

I'm doing Ex 3.1 in Brezis's book of Functional Analysis. Let $(E, | \cdot |)$ be a normed space and let $A \subset E$ be a subset that is compact in the weak topology $\sigma\left(E, E'\right)$. ...
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  • 1,153
0 votes
1 answer
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A weak$^\star$ open neighborhood contains a line

I'm reading @Nate Eldredge's answer: Here's a counterexample. Let $X$ be any infinite-dimensional Banach space. Let $\mathcal{U}$ be the collection of all weak-* open neighborhoods of $0 \in X^*$. ...
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  • 1,153
1 vote
0 answers
50 views

Corollary of Eberlein Smulian theorem

I've read Conway's chapter 4-13. An Eberlein-Smulian Theorem says that if $X$ is a Banach space and $A \subseteq X$, then TFAE. Each sequence of elements of $A$ has a weakly convergent subsequence ...
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1 answer
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A simpler proof of Milman–Pettis theorem by diameter argument

Recently, I have come across an elegant proof of Milman–Pettis theorem. Surprisingly, I'm able to make this proof even simpler. I'm very happy to share it with you and receive your suggestion. Let $E$...
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  • 1,153
0 votes
1 answer
20 views

If a weakly convergent net is bounded by $r$, is its limit bounded by $r$?

Let $(E, | \cdot |)$ be a normed space and $E'$ its topological dual. Let $\sigma(E,E')$ be the weak topology of $E$. Let $(x_d)_{d\in D}$ be a net in $E$ that converges in $\sigma(E,E')$ to $x\in E$. ...
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  • 1,153
1 vote
1 answer
47 views

A more direct proof of Goldstine theorem

I'm trying to give a more direct proof of Goldstine theorem. Could you have a check on my attempt? Let $E$ be a normed space, $E'$ its dual, and $E''$ its bidual. Let $B_E$ and $B_{E''}$ be the ...
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  • 1,153
4 votes
1 answer
118 views

Where does my proof of Milman-Pettis theorem break down?

I'm trying to prove Milman-Pettis's theorem. Let $E$ be a uniformly convex Banach space. Then $E$ is reflexive. Clearly, my attempt is not correct because I have not used the uniform convexity of $E$...
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  • 1,153
0 votes
1 answer
18 views

The weak$^\star$ topology $\sigma(E',E)$ coincides with the subspace topology that $\tau$ induces $E'$

I'm reading an elegant proof Banach–Alaoglu theorem from here. The proof depends on the following observation. Let $E$ be a normed vector space, $E'$ its topological dual, and $\sigma(E', E)$ the ...
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  • 1,153
0 votes
1 answer
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$\sigma(M, M')$ coincides with the subspace topology that $\sigma(E, E')$ induces on $M$

I'm trying to prove below result stated without proof in my textbook. Could you have a check on my attempt? Let $E$ be a locally convex t.v.s. and $M$ its linear subspace. Let $\sigma(E, E')$ and $\...
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  • 1,153
7 votes
2 answers
159 views

Weak convergence in the Hilbert cube

I'm very curious about the following problem: How can I show that in the Hilbert cube defined as $$C=\{x=(x_1,x_2,\dots) \in l^p: |x_n|\leq \frac{1}{n}\,\,\, \forall n \in \mathbb{N}\}, 1\leq p < \...
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0 votes
1 answer
39 views

A simpler proof that bounded sequence has a convergent subsequence in weak topology

I'm proving Theorem 3.18 in Brezis's book of Functional Analysis. Assume that $E$ is a reflexive normed linear space and $E^\star$ its topological dual. Let $\left(x_{n}\right)$ be a bounded sequence ...
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  • 1,153
0 votes
0 answers
33 views

Affine function on the space of Borel probability measures under weak$^*$ topology

I am curious about any property of affine continuous functions from $\mathcal{P}(X) \to \mathbb{R}$ under weak$^*$ topology. Here $X$ is a locally compact Polish space and $\mathcal{P}(X)$ is the set ...
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1 vote
1 answer
76 views

Weak topology and weak convergenge in probability spaces

Let $X$ be a Polish space (metrizable, complete, separable) with $\mathcal{B}(X)$ its borel sigma algebra. Let us consider $\mathcal{P}(X)$ the space of probability measures on $\mathcal{B}(X)$. We ...
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1 answer
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The weak$^\star$ topology is compatible with the vector space structure

After proving that the weak topology is compatible with the vector space structure, I try to do the same for weak$^\star$ topology, i.e., Let $E$ be a topological vector space and $E^\star$ its ...
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  • 1,153
2 votes
1 answer
38 views

The weak topology is compatible with the vector space structure

I'm trying to prove this result Let $E$ be a topological vector space and $E^\star$ its topological dual. Let $\sigma(E, E^\star)$ be the weak topology on $E$. We denote by $E_w$ the vector space $E$ ...
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  • 1,153
0 votes
1 answer
36 views

Let $\varphi:E^\star \to \mathbb R$ be linear and continuous in $\sigma(E^\star, E)$. Then there is $e \in E$ such that $\varphi = Je$

I'm trying to prove Prop 3.14 in Brezis' book of Functional Analysis. I posted my proof as an answer below. Could you have a check on my attempt? Let $(E, | \cdot|)$ be a normed linear space and $E^\...
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5 votes
0 answers
61 views

If $f_n \overset{\star}{\rightharpoonup} f$ in $\sigma(E^\star, E)$, then $\|f\| \le \liminf \|f_n\|$

I'm trying to prove this result. Could you have a check on my proof? Let $(E, | \cdot|)$ be a normed linear space and $E^\star$ its topological dual. Let $\sigma(E^\star, E)$ be the weak$^\star$ ...
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  • 1,153
2 votes
1 answer
92 views

Is there some relationship between weak topology on $E$ and weak$^\star$ topology on $E^\star$?

Let $(E, | \cdot|)$ be a normed linear space and $E^\star$ its topological dual. Let $\sigma(E, E^\star)$ be the weak topology on $E$, and $\sigma(E^\star, E)$ the weak$^\star$ topology on $E^\star$. ...
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1 vote
0 answers
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A question on weak topology induced by a family of functionals

Let $X$ be a normed linear space and $F\subset X^*$. Let $\overline{F}$ denotes the closure of $F$ in $X^*$ in the norm topology. Give $X$ the weak topology induced by the family $F$ denoted by $\...
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  • 1,542
0 votes
1 answer
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Show that $g \in L^1(0,1)$, if $S_g$ is weakly compact

I'm preparing for my Functional Analysis final by solving old exam exercises. I'm working on a two-part exercise, where I think I have a solution for the first part, but am unable to do the second. ...
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2 votes
0 answers
42 views

Non-empty weakly open set is unbounded in the norm topology

In proving that weak topology is not metrizable, I come across below lemma. Could you have a check on my attempt? Let $(E, | \cdot|)$ be an infinite-dimensional normed linear space and $E^\star$ its ...
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  • 10.2k
1 vote
1 answer
36 views

What is the definition of a weakly compact operator from a Banach space to another?

What is the definition of a weakly compact operator from a Banach space $X$ to another $Y$? I think the definition needs to know the definition of weak topology but I don't know what is the the weak ...
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  • 567
1 vote
1 answer
49 views

How wild are weakly continuous curves in a Banach space?

For one reason or the other I work in a reflexive infinite dimensional Banach space $X$ and I am interested in curves $$ \gamma:[0,1]\to X $$ that are only weakly continuous. I have no idea how to ...
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  • 1,420
0 votes
0 answers
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If the weak topology is metrizable, then the dual space has a countable algebraic basis

I'm solving Ex 3.8.1. in Brezis' book of Functional Analysis. Could you have a check on my attempt? Let $(E, | \cdot |)$ be a Banach space and $d$ a metric on $E$ that induces the weak topology $\...
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  • 10.2k
1 vote
1 answer
57 views

A simpler proof of the continuity equivalence between norm and weak topologies

I'm trying to simplify the proof of Theorem 3.10 in Brezis' book of Functional Analysis. My proof is much simpler than the original. I'm afraid that I made subtle mistakes. Could you have a check on ...
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  • 10.2k
0 votes
0 answers
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A simpler proof of $\sigma (E, E^\star)\boxtimes\sigma (F, F^\star)=\sigma\big(E \times F,(E \times F)^\star \big)$ for the product of weak topologies

I'm trying to prove this identity. Previously, I gave it a try here but the proof there is not correct. I found another proof here which is quite complicated. Could you have a check on my below ...
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  • 10.2k
2 votes
0 answers
50 views

Prove that $\sigma (E, E^\star) \boxtimes \sigma (F, F^\star) = \sigma \big (E \times F, (E \times F)^\star \big)$ for the product of weak topologies

I'm proving this identity about the product of weak topologies. Could you have a check on my attempt? Let $E,F$ be T.V.S. and $E^\star, F^\star$ their continuous dual respectively. Let $\sigma (E, E^\...
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0 votes
0 answers
29 views

Sequence in weak and weak* topology

I try to find the example to show this If $f_{n}\overset{w*}{\rightarrow} f$ and $x_{n}\overset{w}{\rightarrow}x$ not necessarily $f_{n}(x_{n}) \rightarrow f(x)$. prove the $f_{n}(x_{n})\rightarrow ...
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  • 35
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Does continuous imply pseudo-monotone?

Let $E$ and $F$ be real Banach spaces, brought into duality by $\langle f, e \rangle$ for $f\in F$ and $e\in E$. Let $E$ be equipped with a topology $\tau_E$ finer than the weak topology $\sigma(E, F)$...
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