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 $X$.
562 questions
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Reference request: Kadec-Klee property for $\ell^1$
I am looking for references (textbooks or articles) where it is shown that the Banach space $\ell^1$ has the Kadec–Klee property, i.e. that the weak topology and the norm topology coincide on the unit ...
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Topology literature focusing on continuous functions and measures
I am looking for literature dealing with topologies on spaces of continuous functions ($C_0$, $C_c$, $C_b$, $\ldots$), particularly with regard to their application when dealing with topologies on ...
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What’s the geometric intuition behind the Milman-Pettis theorem (uniform convexity ⇒ reflexivity)?
The Milman-Pettis theorem says that every uniformly convex Banach space is reflexive.
I’ve read and understood the proof, but I’m struggling with the motivation. The statement feels mysterious: ...
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upper bound on the 1-Wasserstein by maximum mean discrepancy (MMD) distance
I want to make sure that my understanding from these two papers, and whether we can provide an upper bound on the 1-Wasserstein by MMD is correct. Below $\gamma_k$ refers to maximum mean discrepancy (...
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Extension of continuous functions from dense subsets of $\mathbb{R}^T$
Let $T$ be an uncountable index set and consider the product space $\mathbb{R}^T$ with the product topology. Suppose $Y \subset \mathbb{R}^T$ is a dense linear subspace.
Is it true that every ...
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A sub-collection of a sub-basis to a topology generates the same topology - proof assistance
Background and context
This is once again another follow-up question to this question I asked yesterday. Attempting to understand the concept of sub-basis to topologies and the weak topology induced ...
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How can $\mathcal{H}$ be a sub-basis if no topology is given?
This is a second follow up question to this question I asked today (this is the first followup question).
In the original question I asked to prove the following:
Let $\left\{ \left(X_{\alpha} , \...
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Generating collection of subsets to the weak topology on a set $Y$ induced by functions from $Y$ to a collection of topological spaces
Definitions
Let $X$ be a set. A collection of subsets $\mathcal{B} \subseteq \mathcal{P} \left( X \right)$ is said to be a basis for a topology on $X$ if:
For every $x\in X$ there exists $B \in \...
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Exercise 2.9 (b) - Topics in Banach Space Theory (Albiac, Kalton)
I'm trying to solve Exercise 2.9 from Albiac and Kalton's book Topics in Banach Space Theory. The exercise in question is the following:
Let $X$ be a Banach space.
(a) Show that for every $x^{**} \in ...
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A question about Fabian et al's proof of the Banach-Dieudonné Theorem
The Banach-Dieudonné Theorem states:
Let $X$ be a real Banach space and let $A$ be a convex subset of $X^\ast$. Suppose that $A \cap n B_{X^\ast}$ is $w^\ast$-closed for each $n \in \mathbb N$. Then $...
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How is the limit inferior defined for nets and subnets?
For sequences in $\mathbb{R}$ we usually define
$$
\liminf_{n \to \infty} x_n = \sup_{n} \inf_{k \geq n} x_k.
$$
For a subsequence $(x_{n_j})_j$ one can define analogously
$$
\liminf_{j \to \infty} x_{...
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Is every bounded subset of $X^*$ relatively weak$^*$-compact?
Let $X$ be a Banach space and let $Y \subseteq X^*$.
I came across the following proposition:
Proposition. If $Y$ is bounded in $X^*$, then $Y$ is relatively compact in $X^*$ with respect to the weak*...
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If $P:X^* \to X^*$ is a projection with weak$^*$-closed range, is $P$ weak$^*$-continuous?
Let $X$ be a Banach space and $P:X^* \to X^*$ a (bounded linear) projection, i.e., $P^2 = P$. I would like to know whether the following statement is true:
Claim. If $\operatorname{Im}(P)$ is $\sigma(...
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If the kernel of an operator is $\omega^*$-closed, is the operator $\omega^*$-continuous?
Let $X$ be an infinite-dimensional Banach space and $Y$ a finite-dimensional Banach space.
It is a standard fact that if $T: X \to Y$ is a linear operator and $\ker(T)$ is norm-closed, then $T$ is ...
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Exercise 2.6 - Topics in Banach Space Theory (Albiac, Kalton)
We are still in the saga of solving Kalton's problems and another one has stumped me:
2.6 Suppose $X$ is a Banach space whose dual is separable. Suppose that $\sum x_n^*$ is a series in $X^*$ which ...
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