# Questions tagged [locally-convex-spaces]

For questions about topological vector spaces whose topology is locally convex, that is, there is a basis of neighborhoods of the origin which consists of convex open sets. This tag has to be used with (topological-vector-spaces) and often with (functional-analysis).

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### Balancing convex absorbing neighborhoods

Given a convex absorbing neighborhood of 0 in a topological vector space is it always possible to construct a subset that is also convex absorbing neighborhood of 0 but balanced as well? A locally ...
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### One assumption in the proof of one result of Hahn-Banach theorem.

Theorem: Let $X$ be a locally convex space, let $M$ be a linear subspace of $X$, and let $f\in M^*$. Then there exists $h\in X^*$ such that $h(x)=f(x)$ for all $x\in M$. I saw one sentence in the ...
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### Joint continuity of bilinear pairing (with unusual topology)

Let $V$ be a complex vector space, which we endow with the finest linear topology. Then continuous dual $V'$ coincides with the algebraic dual $V^*$. We choose the weak-star topology $\sigma(V^*,V)$ ...
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### A particular TVS

I am looking for a topological vector space $X$ satisfying in the following properties: (1) Cardinal number of $X$ is at most of continuum. (2) $X$ is not a hereditary Lindelof space. (3) $X$ ...
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### Reference request: Locally convex space is Hausdorff if and only if …

I was given the following theorem by my professor: Let $A$ be a locally convex space. Then $A$ is Hausdorff if and only if, for every seminorm $p$, we have that $p(x)=0$ implies that $x=0$. If I ...
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### Is a countable product locally convex?

Let $X$ be a countable set. Consider the space $\mathbb{R}^X$ of real-valued functions on $X$ equipped with the product topology. Is $\mathbb{R}^X$ locally convex? If not, is the space of real-...
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### A linear subspace $Y$ is dense iff there is no trivial funcitonal vanishing on $Y$

So I was reading Conway's book "A course in functional analysis" and stumbled upon the following corollary of the Hahn-Banach separation theorem: If $X$ is a locally convex space and $Y$ is a ...
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### Locally convex vector space: Unified polars of zero neighbourhoods are the dual space

This is a statement which I found without proof and maybe it's obvious, but I can't understand why it should be true: Let $X$ be a locally convex vector space, $\mathcal{W}$ a neighbourhood basis ...
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### a list of known convex functions?

I am working on the development of novel optimization algorithm where it DOES NOT suffer from the non-differentiability (cf. subgradient), which also works well in differentiable class of the ...
Let $X$ be a metrizable topological space, and $C(X)$ the space of continuous functions. Is there a continuous norm (as function to $\mathbb{R}$) on $C(X)$? The topology is given by the family of ...