# 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 $X$.

353 questions
Filter by
Sorted by
Tagged with
39 views

### 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 ...
• 1,514
1 vote
55 views

### 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. ...
97 views

### 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-...
8 views

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
58 views

### 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 ...
14 views

31 views

### 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 ...
56 views

### 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 ...
• 263
60 views

### 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. ...
• 1,153
1 vote
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 ...
• 386
165 views

• 1,153
159 views

• 1,420
21 views

• 10.2k