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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1answer
26 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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26 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
25 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
102 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
27 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
20 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
15 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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4 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
35 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
42 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
30 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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13 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
17 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
56 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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1answer
53 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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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
43 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
86 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
125 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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1answer
30 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
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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0answers
51 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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1answer
32 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
16 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
32 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
35 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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0answers
22 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
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. ...
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1answer
31 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 ...
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1answer
93 views

What is the (strong) dual of a projective/inductive limit?

For a family of locally convex TVS $(E_\alpha)_{\alpha \in A}$ it is a well known fact that the dual of the product is the sum of the individual duals and vice versa, i.e. $$ \left(\prod_{\alpha \in A}...
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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 ...
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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 ...
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3answers
62 views

Is a linear functional on a locally convex space $X$ that continuous on a dense subset of X continuous?

Let $X$ be a locally convex space and let $M$ be a dense subset of $X$. Let $f$ be a linear functional on $X$ such that $f$ is continuous on $M$. Then is $f$ continuous on $X$? Thank you in advance!
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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 $x\in M$? ...
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0answers
47 views

How to extend $f\in M^*$ to $\overline{M}$? [closed]

Let $X$ be a locally convex space, let $M$ be a linear subspace of $X$, and let $f\in M^*$. If $M$ is closed, then $M=\overline{M}$. And we don't have to extend $f$. So how to extend $f$ to $\...
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1answer
113 views

One assumption in the proof of one result of Hahn-Banach theorem.

Theorem: Let $X$ be a locally convex space, let $M$ be a linear subspace of $X$, and let $f\in M^*$. Then there exists $h\in X^*$ such that $h(x)=f(x)$ for all $x\in M$. I saw one sentence in the ...
2
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2answers
44 views

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 $\sigma(V^*,V)$ ...
3
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1answer
103 views

How to show $X := \bigcup_{n \in \mathbb{N}}X_n$ is a locally convex space.

My textbook on functional analysis says as follows.(The book is written in Japanese, ISBN: 978-4-946552-18-2) Let $\{X_n\}$ be a sequence of locally convex spaces over $K (= \mathbb{R}$ or $\mathbb{C}...
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0answers
37 views

Continuity and sequential continuity of a linear functional

Let $E = C_c^0(\mathbb{R}^n;\mathbb{R}^m)$ be the space of compactly supported continuous functions on $\mathbb{R}^n$ with values on $\mathbb{R}^m$. There is a natural norm on this space: given $\...
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1answer
52 views

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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1answer
34 views

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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2answers
36 views

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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1answer
87 views

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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1answer
32 views

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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0answers
283 views

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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1answer
383 views

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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1answer
53 views

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 ...