# Tagged Questions

The study of vector space with a topology which makes the maps which sums two vectors and which multiply a vector by a scalar continuous. It's a natural generalization of normed spaces.

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### Schwartz space of functions versus Schwartz space in a more general sense?

Part of me is afraid that this isn't a well-formed question, but try as I might, I can't seem to figure out anything reasonable on this topic. I'm hoping someone here can help. In functional ...
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### Norm of a space - How to show its one

Let $C^1([0,1])$ be the space of all functions having continuous derivative. For each $f\in C^1([0,1])$, set $$\|f\|=\left(\int_0^1 (|f|^2+|f'|^2)dx\right)^{(1/2)}$$ Show that $\|\cdot\|$ is a norm ...
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### Compactness and (global) convergence in measure

Let $B$ denote the unit ball of $L^\infty$. Question: is $B$ sequentially compact for the topology of convergence in measure ? I am not necessarily assuming that the measure is finite (but $\sigma$ ...
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### Elements near the identity of a linear subspace

I am currently trying to understand a proof and ran into the following problem. The proof states (everything takes place in a commutative, unital Banach-Algebra): A linear subspace $X$ with ...
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### Show that continuous linear maps on the space of test functions take $C_K^\infty(\Omega)$ into some $C_{K_N}^\infty(\Omega)$

Let $\Omega$ be a nonempty open subset of $\mathbb{R}^n$, and let $\cup_{n=1}^\infty K_n = \Omega$ be an exhaustion of $\Omega$ by compact sets. Let $\mathcal{D}(\Omega) = \mathcal{D}$ be the standard ...
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### How to prove this: $\overline{A\cap B}=\overline{A\cap \overline{B}}$?

Let $A$ be an open set of $E$ an normed linear space, and $B\subset E$, then I have to prove that $$\overline{A\cap B}=\overline{A\cap \overline{B}}$$ (I'm stuck in the two $\subset$'s) Any help ...
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### familyof seminorms on normed spaces

Let $(X,\|\cdot\|)$ be a normed space. It is known that the norm $\|\cdot\|$ induces a topology, known as the norm topology $\tau$ on $X$. Then the pair $(X,\tau)$ is a locally convex topological ...
I'm reading a book in which it's claimed that for a strongly continuous representation, $U: G\rightarrow Aut(E)$ of a Lie group, G, on a locally convex, complete, Hausdorff topological vector space,E, ...