# 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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### 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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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(\... 0 votes 1 answer 40 views ### 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$n\times n$... 1 vote 0 answers 86 views ### 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 ... 1 vote 1 answer 51 views ### 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 ... 0 votes 0 answers 31 views ### 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 ... 1 vote 1 answer 31 views ### 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, \... 11 views

### 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 ...
1 vote
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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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### 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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### 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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### 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. ...
1 vote
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### 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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