# 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 $X$.

472 questions
Filter by
Sorted by
Tagged with
38 views

### Prove $X$ has the weak topology where $p:X\to Y$ is a covering map.

This question has been answered here, but the method used is different than the one I learned in my algebraic topology course, which I believe I should use. I showed that given that $Y$ is a CW ...
58 views

### Extension of a continuous function $f:X\rightarrow X$ to a function $g: E\rightarrow E$ such that $X$ is embedded in $E$, a Banach space.

We consider a compact metric space $X$ and a continuous function $f:X\rightarrow X$. I know it's possible to embed $X$ into a Banach or separable Hilbert space $E$ using the weak star topology. ...
35 views

### Exercise 3.1.23 in Salamon's Functional Analysis

I'm reading the book Functional Analysis written by Dietmar A. Salamon.The following problem is the exercise 3.1.23 in the book. Let X be a Banach space and suppose the dual space of X is separable....
46 views

### Proof verification regarding weak-continuity

I want to prove that For $V$ a normed space, $f: V \rightarrow \mathbb{K}$ a linear form that is continuous with respect to the weak topology, f is strongly continuous. By definition, the weak ...
1 vote
26 views

### The map $T: (H^1 (I), \|\cdot\|_{L^2}) \to \mathbb R, u \mapsto \|u\|_{H^1}$ is lower semi-continuous

Let $I$ be the open interval $(0, 1)$. Consider the map $$T: (H^1 (I), \|\cdot\|_{L^2}) \to \mathbb R, \quad u \mapsto \|u\|_{H^1}.$$ I would like to verify that $T$ is lower semi-continuous (l.s.c.)...
1 vote
50 views

33 views

### Brezis' exercise 8.4: weak convergence of a sequence in $W^{1, p} (\mathbb R)$

I'm trying to solve below exercise in Brezis' Functional Analysis, i.e., Exercise 8.4 Fix a function $\varphi \in C^\infty_c (\mathbb R)$ such that $\varphi \neq 0$. Let $u_n (x) := \varphi (x+n)$ ...
126 views

1 vote
65 views

### Weak and strong relative topologies coincide on compact sets

Let $\left(\mathcal{X}, \tau\right)$ be a locally convex topological space and $K$ a compact subset. Is there a reference for the proof that on $K$ the relative topology coincides with the relative ...
1 vote
69 views

### Weak* closure of sets

I struggle with the following exercise about the weak* closure of a set. Weak and Weak* topology is a bit a weak (pun intended) spot of mine so i would like to ask if somebody could take a look, ...
1 vote