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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Example of Topological Vector Space

Is there a topological vector space such that, for every $x\in X$, there is a proper neighbourhood $V$ of $x$ in $X$ which is convex, but the whole space is not locally convex (i.e. $X$ has a local ...
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1answer
13 views

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?
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37 views

Prove that the set of extreme points of a compact convex set is not empty.

The Krein–Milman theorem states that if $S$ is convex and compact in a locally convex space, then $S$ is the closed convex hull of its extreme points. In particular, such a set has extreme points. Is ...
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1answer
24 views

Where can I find the nowhere subdifferentiable example of rockafellar?

I'm told that Rockafellar gave an example of a real extended function defined on a locally convex space, whose subdifferential is empty at each point of its domain. The function is proper, lower ...
2
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1answer
29 views

Dual of continuous functions in various topologies

Let $S$ be compact and Hausdorff and $C(S)$ be its space of continuous complex functions. When $C(S)$ is endowed with the $\sup$ norm, its dual is well known. Since this topology is too strong for my ...
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1answer
58 views

Topology on compactly supported smooth functions

I'm confused by a set of lecture notes I'm reading and would like help in understanding what's going on. First, there is the following nice theorem. Theorem. The topology of a locally convex space is ...
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6 views

Suppose $P = \text{conv}(\text{ext}(P))$. Is it possible to find $S \subset P$ of cardinality $< \text{ext}(P)$ s.t $\text{conv}(S) =P$?

Suppose that $P$ is a polyhedron and that $P = \text{conv}(\text{ext}(P))$. Suppose further that $ \text{ext}(P)$ has cardinality $r$. How do I see that I cannot find a set $S \subset P$ of ...
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38 views

Relation between the number of halfspaces and the number of vertices of a convex polytope

Suppose we have an $n$-dimensional convex polytope $\mathbf{P}$ represented by an intersection of half-spaces as the following: \begin{equation} \mathbf{P} = \{ x \in \mathbb{R}^n \mid x \in ...
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35 views

Show that continuous linear maps on the space of test functions take $C_K^\infty(\Omega)$ into some $C_{K_N}^\infty(\Omega)$

Let $\Omega$ be a nonempty open subset of $\mathbb{R}^n$, and let $\cup_{n=1}^\infty K_n = \Omega$ be an exhaustion of $\Omega$ by compact sets. Let $\mathcal{D}(\Omega) = \mathcal{D}$ be the standard ...
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1answer
57 views

Characterize polytopes resulting from cutting a convex polytope by a hyperplane

We have a convex polytope $P$ for which we know its set of vertices. Using this set we characterize the H-representation of $P$ as: $\mathbf{A}\mathbf{x} < \mathbf{b}$. If a hyperplane defined by ...
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62 views

Half space representation of a convex polytope

We know that the half space representation of a polytope is given by: $Ax<b$. Consider a convex polytope in $\mathbb{R}^3_+$ with vertices given by the following set of points: ...
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11 views

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 ...
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0answers
81 views

Find the intersection between two convex hulls, in this specific case

We work over $\mathbb{R}^K$. Let $V$ the set of vectors whose coordinates take values $0$ or $1$, or equivalently the corners of the unit cube $[0,1]^K$, . Let $d:\{0, \ldots, K\} \to \mathbb{R}_+$ ...
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1answer
48 views

Approximate model of a convex/concave surface

I have a set of measurements in 3d that yields a concave surface of a function $f(x,y)$ that I don't know its expression. I am thinking to approximate the function to a model where any point from the ...
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0answers
23 views

The element presentation of convex hull of the union of compact sets

I want to show that the convex hull of the union of compact convex sets $k_{1}$ and $k_{2}$ in a locally convex topology linear space, consist of the points of the form $$ay_1+(1-a)y_2,\ \ \ \ \ ...
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1answer
90 views

Reference request: infinite-dimensional manifolds

The following books and/or notes develop various aspects of the theory of infinite-dimensional manifolds: Lang, Fundamentals of Differential Geometry. Kriegl & Michor, The Convenient Setting of ...
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1answer
43 views

$x\in X$ LCS, $f\in X^\ast$ s.t. $f(x)=1$, $f|_Y=0$

Let $X$ be a locally convex space (topology induced by a family of seminorms $P$ which separates points) and $Y\subset X$ a closed subspace. Assume $x\in X\setminus Y$. Show that there exists a $f\in ...
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28 views

Are not all neighborhoods of $0$ in a locally convex space absorbent?

A locally convex space (LCS) can be defined as a topological vector space (i.e. scalar product and sum are continuous) whose topology is generated by translation of a family of balanced and absorbent ...
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1answer
28 views

Proof of convexity of a quadratic function

I have the next problem: If $f(x)$ is a quadratic function with n variables: $f(x) = 0.5$$\mathbf{x}^T$$A$$\mathbf{x}$$+$$\mathbf{b}$$^T$$\mathbf{x}$$+$$\mathbf{c}$ were $A$ is a symmetric matrix ...
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29 views

Can a non-convex set be partitioned into a set of nearly convex subsets? [closed]

Consider a non-convex bounded subset $S \subseteq \mathbb{R}^{n}$. Is it always possible to partition this set into a finite set of disjoint subsets \begin{equation} S = \bigcup_{i=1}^{n}s_i, \quad ...
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1answer
45 views

Convex open neighborhood of compact convex subset

I'm stuck on what ought to be a straightforward topology problem. Say $X$ is a compact convex subset of a locally convex space (everything in sight is assumed Hausdorff). Say $Y\subseteq X$ is a ...
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57 views

Continuity of a function in a locally convex topological space

I endow the space of bounded sequences with a locally convex topology $\tau$ such that $\tau$ is strictly finer than the product topology (the topology of pointwise convergence), $\tau_p$, and ...
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10 views

Decomposing continuous linear functionals on a locally convex space with 2 seminorms

Let $X$ be a locally convex topological vector space whose topology is defined by the seminorms $\rho_1$ and $\rho_2$. (Let us require that topological vector spaces be Hausdorff by definition.) If ...
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2answers
23 views

Does every LCS has a convex balanced local base?

Does every LCS--locally convex (topological vector) space has a convex balanced local base? Then it implies every LCS is topologized with a countable number of separating seminorms. So there seems ...
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1answer
65 views

Topological modules and relative homological algebra.

This question might be a bit dumb but I'm tired right now and this is just going over my head at the moment, in "The homology of Banach and topological algebras" Helemskii said that relative ...
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0answers
44 views

Weak* topology on Hilbert space

I am a little confused about the weak* topology on Hilbert space $H$. Beyond doubt, the weak* topology on $H^{**}$ is $\sigma(H^{**},H^*)$. Suppose $\tau$ is the natural embedding from $H$ onto ...
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1answer
169 views

Generalization of inner product spaces (analogue to uniform spaces/locally convex spaces)

In the following I am going to devise a chart of topological spaces that contains inner product spaces, normed vector spaces, metric spaces and other related spaces. In the end there will be a gap in ...
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35 views

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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1answer
66 views

Preserved properties through continuous linear maps

I just looked at the fact (at least according to Definition 2.8.1. in Distribution Theory by Friedlander et al.) that for $K_0\subseteq{\bf R}^{n(0)}$ compact, $\Omega_1\subseteq{\bf R}^{n(1)}$ open ...
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1answer
77 views

About locally convex Hausdorff topological vector space

Let $E$ be a locally convex Hausdorff topological vector space. Show that $E$ is isomorphic to a subspace of a product of normed spaces. All I know is that, if $E$ is locally convex Hausdorff, then ...
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35 views

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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35 views

Books on convex sets?

I'm looking for good books on convex sets. Idealy I'd like an introductory text AND a more advanced one. Appart from basic definitions and the like I have no background on the topic.
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1answer
140 views

Is the dual space of all Radon measures the space of signed measures on a $\delta$-ring?

Consider the Banach space $C_c(\mathbb{R})$ of continuous functions with compact support equipped with the uniform norm $||f||_\infty := \sup_{x \in \mathbb{R}} |f(x)|$. Then it is known (Riesz ...
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1answer
34 views

Geometrical meaning of a face

Let $(X,P)$ be a locally convex space, $K$ a compact, convex subset of $X$. A face $F$ of K is a nonempty, compact, convex subset of $K$ s.t. $$\forall y,z\in K \,\forall t\in(0,1) \left[ (1-t)y + tz ...
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40 views

Continuity of the dual product reloaded

Let $X$ be a Banach space with topological dual $X^*$. Then the dual product $(x,x^*)\in X\times X^*\to x^*(x)\in\mathbb{R}$ is strongly$\times$strongly continuous in $X\times X^*$. That does not ...
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1answer
59 views

A Problem on Locally Convex Spaces

In the book A Course in Functional Analysis by Conway, there is the following problem: Problem. Let $ X $ be a completely regular topological space, and let $ C(X) $ denote the set of all ...
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1answer
31 views

Continuity of the dual product

Let $X$ be a Banach space with topological dual $X^*$. Then the dual product $$ (x,x^*)\in X\times X^*\to x^*(x)\in\mathbb{R} $$ is strongly$\times$strongly continuous on $X\times X^*$, mainly because ...
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6answers
637 views

Is there such thing as an unnormed vector space?

I learned about Banach spaces a few weeks ago. A Banach space is a complete normed vector space. This of course made me wonder: are there unnormed vector spaces? If there are, can anyone please ...
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1answer
21 views

Regarding embeddings of locally convex spaces

If $f:E\rightarrow E'$ is a linear embedding of locally convex topological vector spaces, and $A\subseteq E$ open and convex, can we always find $A'\subseteq E'$ open and convex sucht that ...
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60 views

Closedness of convex sets in a locally convex space

Let $C$ be a convex subset of a locally convex topological vector space. Consider the properties: a) $C$ is closed. b) $C$ is weakly closed. c) $C$ is weakly sequentially closed. d) $C$ is ...
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2answers
192 views

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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2answers
59 views

Is the algebraic interior relatively open in a closed convex set?

I am struggling with proving or disproving the following: Let $X$ be a locally convex space and let $C\subset X$ be closed convex with non-empty algebraic interior which we denote by $C^i$ (recall ...
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1answer
57 views

Strong convexity on sets?

Consider the definition of convex functions: $$ f(tx+(1-t)y) \le t f(x)+(1-t)f(y) $$ It is easy to show the definition of the convexity on sets with respect to the above definition (Specifically for ...
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38 views

How to show that the vertices of a convex hull are given by these specific subsets…

We work over $\mathbb{R}^N$. Let $V$ be the corners of the unit cube $[0,1]^N$, or equivalently the set of vectors whose coordinates take values $0$ or $1$. Let $d:\{0, \ldots, N\} \to \mathbb{R}_+$ ...
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35 views

When is the normal cone to a closed convex set in a locally convex set maximal monotone?

Let $X$ be a locally convex set with the following property: (P) $\forall C\subset X$ closed convex, the normal cone $N_C$ is maximal monotone as a multi-valued operator from $X$ to its topological ...
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1answer
72 views

Existence of a Frechet topology on the dual of a barreled space

I have a Hausdorff separated locally convex barreled space $(X,\tau)$ with topological dual $X^*$. My questions are: $Q_1$ Is there a topology $\tau^*$ on $X^*$ that is finer than the weak-star ...
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1answer
77 views

Uniqueness of projection in a Banach space

Let $X$ be a Banach space, $M$ be a subspace of $X$ and $x \in X$ be any vector in $X$. Consider $\displaystyle \hat{x}_M=\arg \inf_{m\in M}\|x - m\|$. Under what conditions for $l_p$ norms $p = ...
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1answer
53 views

Proving a point is a local minimum

I have a rather basic question. I have a function $f:R \rightarrow R$, and I want to show a point, $x^*$, is local minimum, i.e., $f(x^*+\delta) \geq f(x^*), \ \delta \to 0$. I can show that: $f(x^* ...
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40 views

Direct sum decomposition of vector spaces and their tensor powers

Let $V$ be a locally convex vector space and let $U$ be a finite-dimensional subspace of $V$. The Hahn-Banach theorem guarantees that there exists a closed subspace $W$ of $V$ such that $$V=U\oplus ...
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1answer
52 views

Relation between $\text{Hom}_{\mathsf{Alg}_{\mathbb{R}}}(\mathcal{C}^\infty(M),A) $ and $ X \otimes_\mathbb{R} A$?

This question is a little bit of a shot in the dark, but maybe someone stumbled over it before... Let $M$ be a (simply connected) smooth manifold modelled on a locally convex space $X$ over ...