# 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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### Norm continuity of weakly continuous bounded holomorphic function

Let $X$ be a Banach space and consider a function $f\colon\overline{S}\to X$ where $S$ is the strip $$S=\{z\in\mathbb{C}:0<\operatorname{Re}z<1\}.$$ Suppose that $f$ is continuous with respect ...
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### Unable to find weak topology T in $\mathbb{R}$

I am self studying topology and I was unable to solve this question. Wayne Patty's Question 2.4.5. Let $U$ be usual topology on $\mathbb{R}$ and let $T$ be the weak Topology on $\mathbb{R}$ induced ...
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### How to describe weak topologies T on the $\mathbb{R}$

I was unable to solve this question of Exercise 2.4 (Weak Topologies) Question : Let U be the usual topology on $\mathbb{R}$ . Describe the weak topology T on $\mathbb{R}$ induced by each of the ...
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### Compactness of a subset of $\ell^2$

Let $K \subset \ell^2(\mathbb{N})$ be a set defined as follows: $$K := \left\{x = (x_1, x_2, \dots) \in \ell^2(\mathbb{N}) \,|\, |x_n| \le \frac{1}{n}\right\}.$$ Since $\ell^2(\mathbb{N})$ is a ...
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### How can I conclude that the weak topology on $~\mathcal l^2~$ is a proper subset of the norm topology from what I've done?
I'm doing a problem in topology. In a) I proved that the weak topology is coarser than the norm topology, and in b) I proved that the standard one sequence $~(e_n)~$ in $~\mathcal l^2~$ approaches \$~(...