# 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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### Space of test functions defined by norms

This is the problem assigned: So I know that a locally convex Hausdorff space is defined by a vector space and a family of seminorms. So is part $a$ just wanting me to show that $\|\phi\|_m$ is in ...
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### TVS on the reals which or which not induces convergence in norm

Recently I wondered, if convergence in some given metric $d$ on $\mathbb R^n$ induces convergence in norm. Of course, if $d(x,y) = \|f(x)-f(y)\|$, where $f$ is a bijection on $\mathbb R^n$, then this ...
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### Convex subsets of a topological vector space

I'm trying to prove: Let $X$ be a topological vector space and $A \subseteq X$. $A$ is convex if and only if $$\forall s,t \in \mathbb{R}_{+}, (s+t)A = sA + tA$$ Let $sx + ty \in sA + tA$. How ...
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### Topology :definition of neighbourhood of a point and basic questions

I am going through the basics of topology, in order to deal with topological vector space. I haven't taken any course of topology so I have some fundamental questions (I 've seen only some topological ...
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### Definition of Banach Limits on $\ell^\infty$. Proof of Linearity and Continuity

I want to show that the Banach Limit $\Lambda$ on the set $\ell^\infty$ is a continuous linear functional in the dual space $(\ell^\infty)^\star$. I know that the Banach Limit exists, is left-...
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### Topology on the space of universally integrable functions

Let $X$ be a compact space. Let us call a function $f:X\to {\mathbb C}$ universally integrable if it is integrable with respect to each regular Borel measure $\mu$ on $X$ (i.e. a positive functional ...
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### Nuclear Frechet space as inductive limit

Can a nuclear Frechet space also be defined as an countable inductive system of Banach spaces with nuclear maps?
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### Existence of Banach space in which nuclear space embeds densely

If $N$ is a nuclear space, does there exist a Banach space $X$, s.t. $N$ embeds densely in $X$?
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### Prove that $E ⊆ D_K$ is bounded if and only if $\{ \|\partial^\alpha\psi\|_{C(K)} : \psi\in E \}$ is bounded for every multiindex $\alpha$.

Let $K\in\mathbb{R^d}$ d be compact. Prove that $E ⊆ D_K$ is bounded if and only if $\{ \|\partial^\alpha\psi\|_{C(K)} : \psi\in E \}$ is bounded for every multiindex $\alpha$. $D_K$ is the space of ...
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### Continuity of Minkowski functional in locally convex topological vector space

Let $X$ be a locally convex topological vector space over $\mathbb{R}$ or $\mathbb{C}$ and let $p_C(x)=\inf (\lbrace t>0 \mid t^{-1}x \in C\rbrace)$ be the Minkowski functional for an arbitrary ...
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### Rudin's “Functional Analysis” theorem 6.5 [closed]

In the proof of part (a) of theorem 6.5 (pg. 139 of the first edition) he states that: Since $\mathcal{D}_k\cap W$ is open in $\mathcal{D}_k$, we have proved that $\mathcal{D}_k\cap V \in \tau_k$ ...
### The topology of $\mathbb{S}(\mathbb{R^d})$ induced by two different families of seminorms.
Let $\mathbb{S}(\mathbb{R^d})$ be the Schwartz class on $\mathbb{R^d}$ and define the following two families of seminorms.\rho_{\alpha\beta}(f):= \|x^\alpha\partial^\beta f\|_\infty \\\sigma_{\alpha\...