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).

113 questions
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State space of $C([0,1])$

Consider the $C^*$-algebra $A := C(X)$, where $X$ is a compact Hausdorff space. Denote by $S(A)$ the state space of $A$. In the weak$^*$ topology this is a convex compact set such that $S(A)$ is the ...
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Is $A = \{ (z_1 , z_2) \in \mathbb{C}^2 : |z_1| \leq |z_2|\}$ an absorbing, balanced and convex set?

Is $A = \{ (z_1 , z_2) \in \mathbb{C}^2 : |z_1| \leq |z_2|\}$ an absorbing, balanced and convex set? I am new in StackExchange I from Colombia, because I don't write English very well. Edited from ...
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The Minkowski gauge is always bounded by some semi-norm?

My problem sheet asks the following: Let $X$ be a locally convex space generated by the semi-norms $(p_i)_{i\in I}$. Let $A$ be some convex neighborhood of $0$, and $p$ the Minkowski gauge of $p$. ...
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Relations between different definitions of polar topologies

Let $\langle X, Y \rangle$ be a separated dual pair of (real) vector spaces. A non-empty family $\mathcal{A}$ of non-empty subsets of $Y$ is called polar if every $A \in \mathcal{A}$ is bounded (w.r....
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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 ...
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Direct limits of locally convex spaces and embeddings

I was thinking about whether this positive result would hold in the category of locally convex spaces also... Here is what I got so far: The direct limit of a locally convex system consists of the ...
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Topological modules with enough continuous linear functionals.

Context: I'm trying to find out which topological (unital) modules are "good enough" for generalizing results from real or complex functional analysis. For example, I say that a module, in order to be ...
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Convex hull of bounded set is bounded

I want to show that in a locally convex topological vector space $X$, the convex hull of a bounded set is bounded. Apparantly this does not hold if $X$ is not locally convex. So the fact that that ...
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Does $\mathsf{Man}$ possess countable products?

In our lecture we defined a category $\mathcal C$ to have arbitrary products, iff every diagram $A:\mathcal J \to \mathcal C$ with $\text{Morph}(\mathcal J)=\{\text{id}_j\}_{j \in Ob(\mathcal J)}$ has ...
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closed subspaces of locally convex inductive limits

Let $\left(V_{n},\phi_{n,n+1}\right)_{n\in\mathbb{N}}$ be an inductive sequence of LCTV spaces. A locally convex inductive limit of $\left(V_{n},\phi_{n,n+1}\right)_{n\in\mathbb{N}}$ is it's ...
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Is a single nontrivial convex set in a topological vector space enough to make it locally convex?

This is sort of a definition question. While a tvs is locally bounded if it contains a bounded neighborhood of the origin, a tvs is called locally convex if it contains a fundamental system of ...
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I am self-studying P. Lax's functional analysis book. Here is an exercise in p123. it is supposed to be very easy, but I really couldn't see it. Could you anyone help me out? Thanks. Let $\{ l_\... 0answers 12 views Projective tensor product on projective LCM is exact I am reading the book "The Homology of Banach and Topological Algebras" by A.Y. Helemskii and couldn't understand one lemma on page 204 about complex splitting. I understood how to prove that complex ... 0answers 15 views Can I construct a monoidal category on locally convex modules over algebras with approximate identiy I am faced with the following problem: Let$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$for ... 0answers 21 views Characterization of convex dual-like functions Let$f$be a proper, convex, lsc functional from, a locally-convex topological vector space,$X$to$\mathbb{R}$. Then by Fenchel-Monreau Theorem,$$f(x) =\sup \left \{ \left. \left\langle x^{\star} ... 0answers 41 views Does Grothendieck's theorem hold for Bounded borel functions? In the case of continuous functions on a compact Hausdorff space, we have that any bounded set is pointwise compact if and only if it is weakly compact and the two topologies coincide with this set. ... 0answers 42 views Is the extension in this answer well defined and linear? I ask this question One assumption in the proof of one result of Hahn-Banach theorem. before. But I have trouble understanding the answer. I am not quite sure the extension in the answer is well ... 0answers 23 views Is this extension continuous on$X$? Let$X$be a locally convex space. Let$M$be a dense subspace of$X$and let$f\in M^*$. And I am trying to show there exists$g\in X^*$such that$g|_M=f$. My attempts are: Let$x\in X$. Then ... 0answers 24 views Try to find a seminorm$p$on a locally convex space st$|f|\leq p$. Let$X$be a locally convex space. Let$M$be a linear subspace of a locally convex space$X$. Let$f\in M^*$. Then can we find a seminorm$p$on$X$such that$|f(x)|\leq p(x)$for all$x\in M$? ... 0answers 70 views 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, \...
$T$ is said to have local intersection property if for each $x\in X$ with $T(x)\neq\emptyset$, there exists an open neighborhood $N_{x}$ of $x$ such that $\cap_{z\in N_{x}}T(z)\neq\emptyset$. ...