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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Example of a locally-convex topological vector space which is not metrisable

I seek an example of a locally-convex topological vector space which is not a metric space. From google I found an example LF-Space. Does there exist other examples ?
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20 views

A Convex Function Inequality Problem

I would be grateful if someone can provide a proof of the following problem: Let $f(x)$ be convex function. Let $\lambda_1+\lambda_2+\lambda_3=1$, where $0\leq \lambda_k\leq 1$ for $k=1,2,3$. Let $\...
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0answers
160 views

Inductive limits of test function spaces and local convexity

This question is motivated by the test function spaces for distributions, but to be simple just let $K_i=[-i,i]\subseteq\mathbb R$, $i=1,2,\ldots$, and consider the inductive limit topology of the ...
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1answer
43 views

Semi-norms and Hausdorff compact convex set in locally convex space

Let $(X,\tau)$ be an infinite dimensionnal locally convex space (no necessarily Haussdorf) and let $K$ be a Haussdorf compact convex subset of $X$. Suppose that the topology of $X$ is generated by a ...
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1answer
33 views

Existence of Banach space in which nuclear space embeds densely

If $N$ is a nuclear space, does there exist a Banach space $X$, s.t. $N$ embeds densely in $X$?
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1answer
16 views

Seminorm continuity

I am reading functional analysis from an applied math textbook and I have to admit some of the exercises confuse me, in the sense I am not sure what is asked of me. "Show that each of the seminorms ...
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0answers
18 views

$c^1$-differentiability in locally convex spaces

Let $f: E \to F$ be a $\mathcal C^1$-map between locally convex spaces. Is the following correct $$ \mathrm{d}f(x)(h) = \langle f'(x),h\rangle\:\:? $$
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0answers
23 views

Convexity of min(max()) operation over a convex set and a hypercube

Suppose we have a hypercube $[0, 1]^n$, a convex set $V$, and another set $A = min\{max\{V, \mathbf{0}\}, \mathbf{1}\}$, what should be the convexity of $A$? I've found out that $A$ is not a convex ...
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0answers
61 views

$\ell^2$ is a projective limit in the Category of locally convex spaces?

Let $\mathbb{R}^d$ be the usual d-dimensional Hilbert space. I'm learning about projective limits in the category of LCS with continuous linear maps. How are the following different; and more ...
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1answer
72 views

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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1answer
23 views

Linear Combinations of a Dense Set

Let $X$ be a separable locally convex space and $D\subseteq X$ be dense. Then, is the set of all finite-linear combinations of $D$ equal to $X$? That is, when is $\operatorname{span}(D) = cl(D)$?
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17 views

Show the natural map $\pi: (X^*,wk^*) \to (X^*/M^\perp , \{\ \bar p_x\}_{x\in X})$ is open.

Let $M$ be a closed subspace of LCS(locally convex space) $X$ with base field $\mathbb F$. Define $$\bar p_x (x^* +M^\perp ) := \inf_{m^* \in M^\perp}|(x^*+m^*)(x)|$$ for $x^* \in X^*$. Then it can ...
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0answers
14 views

Confusion: Tensor norm between LCSs

Recently I have read that one may define a "tensor norm" on the projective tensor product $X\otimes_{\pi} Y$ between two locally convex spaces. If $X$ and $Y$ are not Banach spaces, how is this ...
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1answer
34 views

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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0answers
28 views

Hahn-Banach separation with point on boundary of a convex set

I want a separation result as follows; finite dimensions might not matter but is the case of present interest. Suppose $C\subset \mathbb R^n$ is convex (not necessarily open or closed) and $0\in \...
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1answer
28 views

Strongly Minkowski equivalence

Assume that $(X, \{ p_i \}_{i \in I})$ is a locally convex space. $A,B \subset X$ are said to be strongly Minkowski separated iff there exists $j \in I$ and $z \in X$ such that one of the following ...
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1answer
113 views

Are weakly compact sets bounded?

Let $X$ be a Hausdorff locally convex topological vector space, and let $X'$ denote its topological dual, that is, the vector space of all continuous linear functionals on $X$. If $A$ is a weakly ...
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0answers
30 views

Confusion: Tensor Norm on Lp spaces

In probability theory, I have often come across the identification $$ L^p_{\mathbb{P}}({\mathcal{F}})\otimes L^p_{\mathbb{P}}({\mathcal{F}}) \cong L^p_{\mathbb{P}\otimes \mathbb{P}}({\mathcal{F}\...
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1answer
55 views

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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1answer
22 views

Prove that a function is sequentially lower semicontinuous

Let be $(X, \{ p_i \}_{i \in I} )$ a locally convex space, $M_0\subset X$ a bounded and nonempty set and $f = l + I_{M_0}$ where l is a continuous function and \begin{equation*} I_{M_0}(x)= \begin{...
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1answer
18 views

Characterisation of the locally convex spaces which are sequentially dense in their completion

Is there a characterization of the locally convex spaces with the property that they are sequentially dense in their completion. In other words, under which conditions on a locally convex space $E$ is ...
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0answers
5 views

Generalization to $n$ dimensions of partition into convex subsets of a function’s domain?

Let $f:\mathbb R\to\mathbb R$ be a smooth and bounded function with a finite number of local minima. Then we can partition $\mathbb R$ into a finite number of sets $\{ A_i\}$, such that the ...
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1answer
42 views

A locally convex space is normable, iff there exists a bounded and open set.

I want to show, that a locally hausdorff space is seminormable, if and only if there exists a bounded and open set. My proof goes as follows: Let $B \subset V$ be an open and bounded 0-neighborhood, $...
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0answers
37 views

Epsilon tensor product of locally convex spaces

I want to understand the definition of the $\varepsilon$-tensor product of two locally convex vector spaces. (Mainly as a hobby.) Let $X,Y$ be locally convex vector spaces and let $B(X^{\ast},Y^{\...
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1answer
43 views

Equivalence of norms on Schwarz space

Consider the following norms on the Schwarz space, for $1\leq q\leq \infty$ $$\lVert f \rVert_{\alpha,\beta, p}=\lVert x^{\alpha}\partial^\beta f\lVert_{L^p}$$ I want to show that the norms $\lVert \...
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1answer
32 views

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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0answers
14 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 ...
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0answers
18 views

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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1answer
79 views

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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1answer
59 views

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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0answers
29 views

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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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 ...
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1answer
44 views

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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2answers
106 views

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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1answer
178 views

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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2answers
728 views

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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1answer
31 views

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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1answer
127 views

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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0answers
53 views

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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2answers
916 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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1answer
39 views

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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1answer
36 views

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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0answers
17 views

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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1answer
33 views

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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1answer
36 views

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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1answer
1k views

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 ...
3
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1answer
574 views

Tempered distributions and convergence

It is known that the Schwartz class $\mathcal{S}(\mathbb{R}^n)$ is a Fréchet space and also that the space of test functions $\mathcal{D}(\mathbb{R}^n)$ is dense in $\mathcal{S}(\mathbb{R}^n)$. Let $\...
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0answers
25 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} ...
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0answers
43 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. ...
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1answer
40 views

Is the space of bounded continuous functions on the reals with the topology of uniform convergence on compact sets fully barrelled?

Let us consider the space $X=C_b(\mathbb{R},\mathbb{R})$ endowed with the following topology: $f_n\rightarrow f$ iff $f_n\rightarrow f$ uniformly on compact subsets of $\mathbb{R}$. This space is a ...