Mathematics Stack Exchange
Mathematics Stack Exchange

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).

491 questions
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Dense locally convex subspace

$V$ is a locally convex space. I can't manage to prove that a subspace $M$ is necessarily dense if $\forall f\in V^*(f(M)=\lbrace{0\rbrace} \implies f(V)=\lbrace{0\rbrace})$. All I have is $\exists x_0 \exists I \subset \mathbb{N} \text{ finite}…
James Well
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If a line goes through the boundary of a convex set, does that line intersect with exactly two boundary points of the convex region?

I have found the following theorem that is often cited from the text Convex Figures by Yaglom and Boltyanskii: A bounded figure in $\mathbb{R}^2$ is convex iff every straight line passing through an arbitrary interior point of the figure…
Eager Student
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Counterexample for bounded subsets in a Frechet space

I do not know if anyone has asked about this before: Does anyone know an example of an unbounded subset of a Frechet space which have finite diameter? I saw this in Conway (A course in functional analysis, page 107, execrise 4), in which he suggests…
Bombyx mori
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$c^1$-differentiability in locally convex spaces

Let $f: E \to F$ be a $\mathcal C^1$-map between locally convex spaces. Is the following correct $$ \mathrm{d}f(x)(h) = \langle f'(x),h\rangle\:\:? $$
Richard Kim
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A problem with Theorem 6.4 in Rudin's Functional Analysis

I feel a bit uneasy about the proof of the following Theorem in Rudin's Functional Analysis, 2nd edition, p. 152-153. It says that for the space $\mathcal{D}(\Omega)$ and a certain systems $ \beta, \tau$ of its subsets, the following Theorem is…
Jorge.Squared
  • 537
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On the Space $C([a,b],X)$, where $X$ is LCS

Definition. A family $P$ of semi-norms on a vector space $X$ is called directed if for any $p_1,p_2\in P$ there exist $p\in P$ and $C>0$ such that $p_1(x)+p_2(x)\leq Cp(x)$ for all $x\in X$. Let $P$ be a family of semi-norms that generates the…
Juniven Acapulco
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Why locally convex topological vector spaces Hausdorff?

In A Course in Functional Analysis, John B. Conway, 100p, it is written that Definition. A locally convex topological vector space (LCTVS) is a TVS whose topology is defined by a family of seminorms $\mathcal{P}$ such that…
Jingeon An-Lacroix
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Unbounded element in $R^\infty$

Let $R^\infty$ be the vector space of all sequence $\{a_j\}$ of real numbers. Put $\|\{a_j\}\|_n:= \sum_{j=0}^n |a_j|$. This collection of semi norms make this as Frechet space. A set $B$ is bounded if every continuous seminorm is bounded on $B$. …
zapkm
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Unit vectors in locally convex spaces

Are there unit vectors in locally convex spaces. If yes, how can vectors be normalised in locally convex spaces?
Amanze
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