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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Reflexitivity of sobolev spaces
Let $\Omega \subset \mathbb{R}^n$ open
I'm interested to understand the reflexivity property of Sobolev spaces $W^{1, p}(\Omega)$ for $p \in [1, +\infty]$, using the fact that $L^p(\Omega)$ are ...
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Weak-star closed condition
In Dunford-Schwartz,Linear Operator, I, General Theory, to prove the Eberlein-Smulian theorem we use the following fact:
Let $X$ be Banach space,
Let $B$ be the norm closed unite ball of $X^*$ and $x^...
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Operator continuous with range in weak topology iff continuous with range in the norm topology
I'm self studying MacCluer's book Elementary Functional Analysis, and I came across the following problem.
Problem 3.18.
Let $X$ and $Y$ be normed linear spaces and suppose $T \colon X \to Y$ is ...
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An interesting family of seminorms $\mathcal F$ and comparison between the topology generated by this seminorms and the Weak Operator Topology.
I am learning functional analysis and I am stuck with the following questions from Strong Operator Topology and Weak Operator topology on $\mathcal B(H)=\{T:H\to H:T$ is Op-Norm continuous,linear $\}$....
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Completeness of Y (normed tvs) in the proof of th. 2 in ch. V section 3 of Yosida's Functional Analysis ed 6
I'm stuck on the proof of theorem 2 in chapter V, section 3 of Yosida's Functional Analysis edition 6 (pages 140,141).
Theorem 2 says : A locally convex linear topological space X is reflexive iff it ...
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Exercise about convex hull and weak convegence
Suposse $X$ is a normed space, $(x_k)_{k \in \mathbb{N}}$ a sequence in $X$, $x \in X$ such that $x_k \underset{weakly}{\rightarrow} x$.
Let $co$ denote the convex hull.
Show that, there exists $y_k \...
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The base for the weak topology is actually a base
I am reading Real and Functional Analysis by Serge Lang and I am having problems with this:
Let $Y$ be a topological space and let $\mathcal{F}$ be a family of mappings $f:X\rightarrow Y$ of $X$ into $...
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Weak closure of a subset of the unit sphere of $\ell_1$
It is a well-known and standard fact that for every infinite-dimensional Banach space $E$ the weak closure $\overline{S_E}^w$ of the unit sphere $S_E$ of $E$ is equal to the closed unit ball $B_E$ of $...
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Understanding the definiton of weakly open sest and weak convergence
I am learning about weak and weak* topology. In the book I am reading the following is mentioned
Definiton (weak topology) If X is a LCS, the weak topology on X, is
the topoloty defined by the family ...
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Difference between the weak and weak* topology (using seminorms to define the topologies)
A few days ago, I was interested in the weak topology and the fact that the weak topology is the coarsest topology such that $f:X \rightarrow \mathbb{K}$ is continuous.
(How to show that, the weak ...
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How to show that, the weak topology is the coarsest topology such that all $f:E \rightarrow \mathbb{K}$ are continuous?
Let E be a normed space, $E':=\{f:E \rightarrow \mathbb{K}| f \text{ is continuous and linear}\}$. Define $p_f(x):=|f(x)| where f \in E'$ and $x \in E$. Consider the family of seminorms $\mathcal{P}=\{...
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Is the strict topology stronger than the weak* topology on the Fourier-Stieltjes algebra?
Let $G$ be a locally compact group and $B(G)$ its Fourier-Stieltjes algebra.
It can be defined as either the dual of the group C$^*$-algebra $C^*(G)$ or the linear span of continuous positive definite ...
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$B(X^{**})$ is the $w^*$-closure of $B(X)$ in $X^{**}$
I am reading Bollobás' Linear Analysis. Chapter 8. Theorem 6., as the title says:
$B(X^{**})$ is the $w^*$-closure of $B(X)$ in $X^{**}$
The proof starts by saying that i) $B(X^{**}$) is $w^*$-...
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Simple way to show that standard basis on $\ell^p$ is weakly pre-compact?
Consider the sequence $e_n = (0,0,\ldots,0,1,0,\ldots)$ in $\ell^1$ which is weakly convergent to zero in $\ell^p$. for all $1\leq p \leq \infty.$ It is then obvious, from the theorem that sequences ...
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Concave conjuguate and weak star topology
I consider a TVS locally convex and separated. I define on it the concave conjuguate of a concave and upper semi continuous function as
$$
f^{*}(x^{*}) =\inf_{y\in X}\left\{x^{*}(y) - f(y)\right\},\...
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