# Questions tagged [topological-vector-spaces]

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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### Examples of topological spaces that closed+boundness implies compactness

One familiar example of such one is the Euclideans. Recently I learnt that the space of holomorphic functions over an open set $U\subset\mathbb{C}^n$ also has this property. The formal statement is as ...
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### empty interior and openness [closed]

In my teacher's lecture notes have these two notes • Every proper subspace of a topological linear space has empty interior. • Every proper subspace of a topological linear space is not open. My ...
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### Is the family of pseudometrics defining the topology of a TVS always equivalent to a translation-invariant family?

Rudin's Functional analysis Theorem 1.24 states, for a metrizable TVS, there always exists a translation-invariant metric matching the topology. But since every TVS, not necessarily Hausdorff, is ...
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### Compatibility - Topological modules contra vector spaces

So Tréves in his book on topological vector spaces shows that a filter $\mathcal{F}$ on a $\textit{vector space}$ $E$ is the filter of neighbourhoods of zero compatible with the linear structure if ...
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### Do "halves" of open sets exist in locally convex vector spaces?

Let $V$ be a locally convex Hausdorff topological vector space (over $\mathbb{R}$) and let $U\subseteq V$ be an open neighbourhood of the origin. Does there always exist another open neighbourhood $U'$...
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### Complemented subspaces of $s$

Crossposted to MathOverflow Let $\mathcal s$ denote the space of rapidly decreasing sequences. It is well known that a space $X$ is isomorphic to a complemented subspace of $s$ if and only if it is ...
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### topological jargon confuses me - weak/strong topology and convergence

I am confused about certain terms in topology that are used frequently, and that I can not find a precise explanation for. In this post somebody mentioned, that weak and strong are not used ...
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### Prove the compactness theorem for Radon measures by using Banach-Alaoglu theorem

I was reading the proof of the compactness theorem for Radon measures (Theorem 1.5.15) from Leon Simon's book: Geometric Measure Theory I was confused by the highlighted part. I hadn't learned the ...
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### closed convex hull, closure of convex hull and convex hull of closure

Let $X$ a topological vector space and $A\subseteq X$ a subspace. Let $co(A)$ the convex hull of $A$ (the smallest convex subspace containing $A$) and $\overline{co}(A)$ the closed convex hull of $A$...
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### A topological vector space is separated if and only if $\{0\}$ is closed

I am new to the study of topological vector space and I would like to show the result stated in the title above but I have some doubt, hence the question. For the first implication, I don’t know how ...
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### $\mathcal{O}(U)$ as a projective limit of Hilbert spaces

It is well-known that the space of holomorphic functions $\mathcal{O}(U)$ (with the standard topology of compact-uniform convergence) on an open set $U \subset \mathbb{C}$ is a projective limit of ...
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