# 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).

318 questions
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### Proving that a locally convex vector space is a topologial vector space.

Let $V$ be a vector space over $\mathbb{C}$ endowed with a topology generated by a collection $\{p_\alpha\}_{\alpha\in A}$ of semi-norms. I want to prove that this turns $V$ into a topological ...
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### Projective tensor product on projective LCM is exact

I am reading the book "The Homology of Banach and Topological Algebras" by A.Y. Helemskii and couldn't understand one lemma on page 204 about complex splitting. I understood how to prove that complex ...
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### direct limit in locally convex modules and continuous map

Let we have short exact sequences of LCM over LC algebra $A$ with continuous linear maps $$0\to B_j\;{\xrightarrow {\ f_j\ }}\;C_j\;{\xrightarrow {\ g_j\ }}\;D_j\to 0.$$ We can take inductive limit (...
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### Properties of topological vector spaces

I'd like to understand better the significance of certain properties of topological vector spaces. It would be great if someone could give me an intuitive picture for what makes them "special", and/or ...
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### convexity of a relatively open subset of a compact set

I'm struggling with the following problem: it seems to be true but I'm not able to prove it! Let $C$ be a compact convex subset of a locally convex metric vector space and $\hat{C}$ be a relatively ...
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### About the locally convex topology

I know that if a locally convex space Hausdorff $(X,S)$ is first numerable then for the $\hat{0}\in X$ exists a countable local base $\{V_n, n \in \mathbb{N}\}$ and to each $V_n$ corresponds a ...
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### Product of open maps open? (not the cartesian)

Let $A$ be a locally convex algebra, or even just a topological algebra, and let $U_1,U_2\in A$ be open, is the product $$U_1\cdot U_2=\left\{ a\cdot b\mid a\in U_{ 1} ,b\in U_{ 2} \right\}$$ ...
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### Can I construct a monoidal category on locally convex modules over algebras with approximate identiy

I am faced with the following problem: Let $A$ be a complete locally convex algebra with a uniform approximation of identity, that is a net $e_\lambda$, such that $p(e_\lambda a-a)\rightarrow 0$ for ...
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### This convex hull is balanced?

Let be (X,S) a locally convex space, and $B \subset X$ a nonempty sequentially closed bounded and convex set such that $\hat{0} \notin clB$,(the closure of B). Define the set T:=s-clco$\{B \cup-B \}$ ...
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### Convergence on locally convex spaces

I'm new on the locally convex spaces. I know that if $X$ is a vector space and $S$ an irreducible set of seminorms defined in $X$, $(X,S)$ is a locally convex vector space. The first question is, how ...
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### Convex function attains maximum at extreme point

So I am working on the proof that a convex continuous function $f$ on a convex and compact set $X$ attains its maximum at an extreme point of $X\subset\Omega$ where $\Omega$ is a locally convex space ...
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### Barrelled space

A locally convex space is called Barrelled if each closed absorbing convex set is 0-neighborhood See. But i doubt that every absorbing set contains zero. Then is every LCV is barreled. I think, ...
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### Two notions of boundedness in metrizable topological vector space.

In a metrizable topological vector space $X$ with the metric $d$, a subset A is said to be bounded if it can be absorbed by any neighbourhood of $0$ and a subset A is said to be d-bounded if its ...
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### Proof that a locally convex space $E$ is regular

Where a topological space $E$ is a regular space if, given any closed set $F$ and any point $x$ that does not belong to $F$, there exists a neighbourhood $U$ of $x$ and a neighbourhood $V$ of $F$ that ...
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### Rudin's functional analysis Theorem 3.18, second part.

Just a follow up to the following two questions: Rudin's functional analysis 3.18, every originally bounded subset of a locally convex space is weakly bounded. Theorem 3.18, Rudin's ...
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### Topology on the space of test functions

I try to read into the theory of distributions and there is one thing which bothers me. I read that a distribution is a linear, continuous functional from the space of test functions, which, depending ...
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### A theorem about convex function

Assume that function $h(x)=f(ax+b)$ is a convex function. What can we say about the convexity of function $f(x)$? My notes: By taking the second derivative from both sides of eqaution $h(x)=f(ax+b)$ ...
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### Rudin's functional analysis 3.18, every originally bounded subset of a locally convex space is weakly bounded.

Reading through the proof of the following In a locally convex space $X$, every weakly bounded set is originally bounded, and viceversa The trivial part of the proof. Since every weak ...
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### About Locally Convex Frèchet Space

I need to proof the following statements, having $\quad X = \{x_n : \mathbb{N} \rightarrow \mathbb{C}\}$ and $p_j = max_{k \le j}|x(k)|$ 1)$p_j$ is a countable family of seminorms which induces ...
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### Locally Convex tvs closure of $\{0\}$

Let $E$ be a topological vector space locally convex, defined by the family of seminorms $\mathcal{F}=(p_j)_{j\in J}$. I can't prove that $\underset{j\in J}\bigcap Ker(p_j)=\overline{\{0\}}$
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### On the completeness of topologically isomorphic spaces

Let $(E_1,\tau_1)$ be a locally convex space and let $(E_2,\tau_2)$ be a complete locally convex space. Suppose that $T:(E_1,\tau_1) \longrightarrow (E_2,\tau_2)$ is a topological isomorphism (that is,...
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### Convex Hull of Precompact Subset is Precompact

I'm trying to prove that, if $K$ is a precompact (I've also heard the phrase totally bounded used for this) subset of a Banach Space $X$, then its convex hull is also precompact. I've come across a ...
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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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### (Hausdorff ) Locally convex spaces and their “natural” metric

Today we were introduced to locally convex spaces, defined thusly: A vector space is locally convex iff it has a family of semi-norms $(p_i)$ such that $x=0$ if and only if $p_i(x)=0$ for all $i$. ...
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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 ...
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### Is the normal cone of a polyhedron a set of points or a set of vectors?

I looked for the normal cone to a polyhedron, and I found this definition from https://sites.math.washington.edu/~rtr/papers/rtr169-VarAnalysis-RockWets.pdf: But this is confusing to me. it seems to ...
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### Existence of continuous norm on C(X)

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 ...
Let $V$ be a complex vector space and let $\{0\} \subset V_1 \subset V_2 \subset \dots \subset V$ be an increasing sequences of subspaces of $V$, whose union is $V$. Suppose that each $V_n$ is a ...
I am trying to prove that the point to set distance when the set $S$ is a convex cone, is sub-additive: $K$ is a cone, and $x,y\in K$ $dist(x+y|K) \le dist(x|K) + dist(y|K)$ where \$dist(y|K)=\inf_{...