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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The algebraic dual space of a TVS is complete

(Treves Exercise 5.4) Let $E$ be a TVS and $E^*$ its algebraic dual. Provide $E^*$ with the topology of pointwise convergence in $E$. A basis of neighborhoods of zero in this topology is provided by ...
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Is a strongly holomorphic function automatically continuous?

I haven't found any sources that state this explicitly for arbitrary topological vector spaces (most sources are concerned with Frechet spaces, where even weakly holomorphic implies continuous). I'm ...
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Topological vector spaces book recommendation

I'm currently taking a class covering the theory of topological vector spaces using the book Topological Vector Spaces, Distributions, and Kernels by Francois Treves. I find the subject to be very ...
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Infinite dimensional topological vectorspaces with dense finite dimensional subspaces

Consider $\mathbb R$ as a $\mathbb Q$ vector space. Using the usual metric on $\mathbb R$, we find: $\mathbb Q \subset \mathbb R$ is dense and one dimensional (indeed every non-zero subspace appears ...
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Closed subsets of $\mathbb{C}^*$ proper for multiplication

Let $S_1$ and $S_2$ be two proper closed subsets of $\mathbb{C}^*$. Let's denote by $\overline{S_1}$ and $\overline{S_2}$ their closure in $\mathbb{C}_{\infty}.$ (Alexandrov compactification) ...
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Dense convex set in $*$-weak topology

Let $X$ be a Hausdorff topological vector space over $\mathbb{K}$. Suppose $W$ is a convex subset of its topological dual $X'$. How to prove that if for any $x\in X\setminus\{0\}$ set $\{f(x):f\in W\}$...
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Bounded neighbourhood of zero in TVS

Is is true that in any topological vector space, which is $T_1$ there exists bounded neighbourhood of zero ? Is is still true if we omit $T_1$ axiom ?
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Space of polynomials as a continuous image of F-space

Let $X=\mathbb{R}[a,b]$. Is there any norm $\|\cdot\|$ on $X$ s.th. $X$ is a continuous image of some $F$-space. ($F$-space means that there exists complete metric s.th. $d(x+z,y+z)=d(x,y)$) ? My ...
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When is a diffeomorphism analytic?

I've read somewhere "a class $C^\infty$ diffeomorphism is said to be analytic" but I forgot to write down where I read this and now I can not find it, which makes me wonder if it's true? I'm working ...
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Locally compact Hausdorff topological vector spaces are finite-dimensional

It's an important fact that locally compact Hausdorff topological vector spaces are finite-dimensional, a proof can be found here. I'm somewhat stuck with the proof. If ${U}$ is a neighbourhood of ...
I know, that if $X$ and $Y$ is second countable, then $X_{/Y}$ is second countable. Is it true, that if $X$ is normed vector space, $Y$ is closed subspace of $X$, and $Y$ is second countable, and ...