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Questions tagged [weak-topology]

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2
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1answer
49 views

$A \in \mathcal{L}(X,Y) \implies A^* \in \mathcal{L}(Y^*_{w^*}, X^*_{w^*})$

Exercise : Let $X,Y$ be Banach spaces and $A \in \mathcal{L}(X,Y)$. Show that $ A^* \in \mathcal{L}(Y^*_{w^*}, X^*_{w^*})$. Attempt : The linearity is trivial. Τo show that $A^*$ is $w^*$ to $w^*$...
2
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1answer
44 views

$B \subseteq X$ is bounded $\Leftrightarrow$ $\forall U \in \mathcal{U}(0) ~ \exists \lambda_U: ~B \subseteq \lambda_u U $

I have some troubles with the following problem and hope some of you can help me. Let $X$ be a vector space, equipped with the $\sigma$-weak topology $\sigma(X,Y)$, where $Y$ is a subspace of the ...
0
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1answer
22 views

convergence for the weak-* topology

Let E be a Banach space. Let $(x^∗_n )$ be a sequence in $E^∗$ verifying $(<x^∗_n , x>)$ converges for any $x ∈ E$. Prove that $\exists x^∗ ∈ E^∗: (x^∗_n )$ converges vers $x^*$ for the weak-∗...
2
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0answers
32 views

reference request: $K$ is relatively weakly compact in $C(X)$ iff $K$ is relatively pointwise compact in $C(X)$

Let $X$ be a compact Hausdorff topological space and $K$ be a norm-bounded subset of $C(X)$(The space of all bounded continuous real valued function on $X$). Then $K$ is relatively compact in $C(X)$ ...
0
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0answers
39 views

Proof of the Alaoglu Theorem

I was reading through the proof of the Alaoglu theorem which states Let $X$ be a normed space Then the unit ball in $X^*=B^*$ is compact with respect to the $weak^*$ topology. The proof goes as ...
3
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0answers
32 views

Is the set $\overline{\text{conv}}^{w^*} C$ weakly* compact?

Exercise : Let $X$ be a Banach space and $C \subseteq X^*$ be $w^*-$compact. Is the set $\overline{\text{conv}}^{w^*} C$ $w^*-$compact ? Thoughts : I (think) that I know that $w^*-$compact sets ...
0
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1answer
24 views

$X$ Banach, $u_n \to u$ and $x^*_n \xrightarrow{w^*} x^*$ implie that $\langle x^*_n, u_n\rangle \to \langle x^*,u\rangle$.

Exercise : Let $X$ be a Banach space, $u_n \to u$ and $x^*_n \xrightarrow{w^*} x^*$. Show that $\langle x^*_n, u_n\rangle \to \langle x^*,u\rangle$. Attempt-Discussion : I know that a sequence $...
3
votes
2answers
54 views

$u_n$ converges weakly to $u$ $\Leftrightarrow$ $\{u_n\}$ is bounded and $\langle x^*, u_n \rangle \to \langle x^*, u \rangle$

Proof Request : I am seeking for a proof of the following Lemma defining Weak Convergence. I am aware of a similar statement regarding Hilbert spaces but it seems to differ. I know that for the $(\...
1
vote
1answer
69 views

On weak*-sequentially completeness

I want to prove that every dual space is weak*-sequentially complete. Let $X$ be a normed linear space and let $(f_n)$ be a weak* Cauchy sequence in $X^*$. Thus for all $x\in X$, $(f_n(x))$ is a ...
0
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1answer
30 views

Prove that the If $G(f)$ inherits the product topology from $X \times Y$, then $X$ is homeomorphic to $G(f)$ via the map $x \to (x,f(x))$.

Let $X$ and $Y$ be topological spaces and let $f\colon X \to Y$ be a continuous map. Define $G(f)={(x, f(x)| x \in X)} \subset X \times Y$. If $G(f)$ inherits the product topology from $X \times Y$, ...
0
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3answers
34 views

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 ...
1
vote
2answers
42 views

Weak continuity of the addition and scalar multiplication

Let $X$ be an infinite dimensional normed vector space. Show that vector addition and scalar multiplication are weakly continuous. $$+:X×X \rightarrow X; +(x,y)=x+y$$ $$•:\mathbb {R}×X \rightarrow X; •...
1
vote
1answer
48 views

Weak-* continuous functionals with bounded level sets

Let $X$ be an infinite-dimensional Banach space, $X^*$ its topological dual and $f:X^*\to\mathbb{R}$ some weak-* continuous functional (not necessarily linear). Is it possible for $f^{-1}(a)$ to ...
2
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1answer
23 views

Can you have a weakly convergent sequence of unbounded linear operators? (Example)

Is it at all feasible to have a weakly convergent sequence of unbounded linear operators? If so, what is a concrete example of a sequence of not necessarily bounded linear operators who converge in ...
2
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2answers
43 views

Show that $g_n(x)=ne^{-nx}$ is bounded, where $x\in\Omega=]0,1[$ for $n\in\Bbb{N}.$

Show that $g_n(x)=ne^{-nx}$ is bounded, where $x\in\Omega=]0,1[$ for $n\in\Bbb{N}.$ My trial I'm thought of it this way that if $g_n(x)=ne^{-nx}$ converges weakly in $L_1(\Omega)$ then it is bounded....
1
vote
2answers
70 views

$+:X\times X\to X,(x,y)\mapsto +(x,y)=x+y$ and $\cdot:\Bbb{R}\times X\to X,(\lambda,x)\mapsto \cdot(\lambda,y)=\lambda\cdot x$ are weakly continuous

$$+:X\times X\to X,\\(x,y)\mapsto +(x,y)=x+y$$ and $$\cdot:\Bbb{R}\times X\to X,\\(x,y)\mapsto \cdot(\lambda,y)=\lambda\cdot x$$ are weakly continuous, where $X$ is an infinite dimensional normed ...
3
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1answer
44 views

A question on Weak and Norm topology

How to prove that If $F$ is closed with respect to the weak topology then $F$ is closed with respect to the norm topology... Weak topology means let $V$ Banach space the weak topology on $V*$ is ...
2
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0answers
23 views

Weak-continuity of an operator on the vector space of measures

Hello and thanks in advance for your time. Let $\mathcal{S} \subset \mathbb{R}^d$ (we can also choose $\mathcal{S}$ compact if this helps here) for some integer $d$ and let $M(\mathcal{S})$ be the ...
1
vote
2answers
52 views

Unit ball of $X^{**}$ is weakly compact!

Is it true that the closed unit ball in $X^{**}$ is compact with respect to the weak topology on $X^{**}$, where $X$ is a Banach space? If so, how can we prove it?
6
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2answers
96 views

If $Y\subset X$ are Banach spaces such that $Y$ is dense in $X$, is it true that $X'$ is dense in $Y'$?

If $Y$ is a dense subspace of a Banach space $(X,\|\cdot\|_1)$ and $(Y,\|\cdot\|_2)$ is a Banach space such that the inclusion from $(Y,\|\cdot\|_2)$ into $(X,\|\cdot\|_1)$ is continuous, then it is ...
1
vote
1answer
31 views

Rudin's functional analysis theorem 3.21.

Small proof of Suppose $X$ is a topological vector space on which $X^*$ separates points. Suppose $A$ and $B$ are disjoint, nonempty, compact, convex sets in $X$. Then there exits $\Lambda \in X^*$ ...
1
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2answers
38 views

Literature on relationship between strong and weak* (weak star) convergence

I am trying to follow a proof in the paper Wasserstein Generative Adversarial Networks by Arjovsky et al. (proof A in supplementary material). They show that the convergenc of the total variation ...
0
votes
1answer
21 views

example of a Banach space X and a subspace Y such that Y is strongly closed but not wealky closed.

I am trying to solve this exercise: find a Banach space X and a subspace Y such that Y is strongly closed but is not wealky closed. I know Y can't be a convex subspace because strongly closed + ...
2
votes
2answers
52 views

A weakly convergent sequence in a compact set, is strongly convegnet

Let $E$ be a Banach space, and $K \subset E$, compact set for the strong topology. And let $(x_n)_n$ converges for the weak topology $\sigma(E,E^*)$ to $x$. Why $(x_n)_n$ converges for the strong ...
1
vote
1answer
43 views

Compactness in weak$^*$-topology on $l_1(\mathbb{N})$

Let $K$ denote the closed unit ball of $l_1(\mathbb{N})$ (considered as a vector space over $\mathbb{C}$). Is $\mathrm{co}(\mathrm{Ext}(K))$ (the convex hull of its extreme points) compact in the ...
1
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0answers
19 views

Relation between energy functionals of stationary and dynamic version of a PDE.

Actually, I am dealing with a very specific equation, but It is rather complex and goes along with many notations and long calculations so I would like to ask a question in a conceptual spirit. Given ...
1
vote
1answer
35 views

Proof theorem 3.17 Rudin's functional analysis

Consider the two theorems (3.15 and 3.16) of Rudin's functional analysis: Theorem 3.15: If $V$ is a neighborhood of $0$ in a topological vector space $X$ and if $$K = \left\{\Lambda \in X^* : |\...
0
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2answers
24 views

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 $B\subset E^*$,...
0
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0answers
92 views

Closed unit ball and convex hull of its extreme points

Let $K$ denote the closed unit ball of $l_1(\mathbb{N})$ (considered as a vector space over $\mathbb{C}$). (a) Show that the set of extreme points of $K$ is $\mathrm{Ext}(K)= \{ \lambda e_n :\lambda ...
1
vote
1answer
29 views

The image of the weak topology with the canonical injection $J$

Let $E$ be a Banach space. With the weak topology $\sigma(E,E^*)$. And let $J:E\rightarrow E^{**} $ be the canonical injection. Can we prove that the image of the weak topology with $J$ is exactly ...
0
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2answers
41 views

Rudin's functional analysis theorem 3.12

Suppose $E$ is a convex subset of a locally convex space $X$. Then the weak closure $\overline{E}_w$ of $E$ is equal to its original closure $\overline{E}$. The proof starts as follows $\overline{...
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0answers
48 views

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 $X'$. ...
6
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2answers
129 views

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 ...
0
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0answers
41 views

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 $C$ ...
0
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2answers
41 views

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?
2
votes
1answer
42 views

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 (...
3
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0answers
55 views

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-...
3
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3answers
85 views

$L^1([0,1])$ 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 ...
2
votes
1answer
41 views

$X^*$ 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 ...
0
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0answers
26 views

Neighborhood of zero on weak topology

I have a homework, I have tried so much on this question, but I couldn't solve it. How can I prove it? Can you help me? Thank you. $E$ is a normed space and $E'$ is the topological dual of $E$. If $...
0
votes
1answer
22 views

For an infinite dimensional Banach space, $X^*$ when given the weak* topology is of the first category in itself [duplicate]

Let $X$ be an infinite dimensional Banach space. Why is $X^*$ of the first category in itself when given the weak* topology. Very closely related to $X^*$ with its weak*-topology is of the first ...
3
votes
2answers
114 views

Question about a weak*-norm continuity of a linear operator

Let $X$ and $Y$ be infinite dimensional normed linear spaces and let $S:Y^*\to X^*$ be a one-one linear operator. I want to show that $S$ can not be weak*-norm continuous. My idea is to choose a ...
1
vote
1answer
59 views

Closed convex hull of pure states of non-unital $C^*$-algebras

It is known that, when $\mathcal{A}$ is a $C^{*}$-algebra with an identity element, the space $\mathcal{S}$ of states of $\mathcal{A}$ is a convex subset of the topological dual $\mathcal{A}^*$ of $\...
0
votes
2answers
72 views

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 ...
0
votes
1answer
24 views

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$^*$-...
1
vote
1answer
60 views

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-...
2
votes
1answer
142 views

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 ...
6
votes
2answers
211 views

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 ...
2
votes
1answer
38 views

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 ...
5
votes
1answer
57 views

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 ...