# 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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### 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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### 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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In probability theory, I have often come across the identification $$L^p_{\mathbb{P}}({\mathcal{F}})\otimes L^p_{\mathbb{P}}({\mathcal{F}}) \cong L^p_{\mathbb{P}\otimes \mathbb{P}}({\mathcal{F}\... 1answer 55 views ### Properties of topological vector spaces I'd like to understand better the significance of certain properties of topological vector spaces. It would be great if someone could give me an intuitive picture for what makes them "special", and/or ... 1answer 22 views ### Prove that a function is sequentially lower semicontinuous Let be (X, \{ p_i \}_{i \in I} ) a locally convex space, M_0\subset X a bounded and nonempty set and f = l + I_{M_0} where l is a continuous function and \begin{equation*} I_{M_0}(x)= \begin{... 1answer 18 views ### Characterisation of the locally convex spaces which are sequentially dense in their completion Is there a characterization of the locally convex spaces with the property that they are sequentially dense in their completion. In other words, under which conditions on a locally convex space E is ... 0answers 5 views ### Generalization to n dimensions of partition into convex subsets of a function’s domain? Let f:\mathbb R\to\mathbb R be a smooth and bounded function with a finite number of local minima. Then we can partition \mathbb R into a finite number of sets \{ A_i\}, such that the ... 1answer 42 views ### A locally convex space is normable, iff there exists a bounded and open set. I want to show, that a locally hausdorff space is seminormable, if and only if there exists a bounded and open set. My proof goes as follows: Let B \subset V be an open and bounded 0-neighborhood, ... 0answers 37 views ### Epsilon tensor product of locally convex spaces I want to understand the definition of the \varepsilon-tensor product of two locally convex vector spaces. (Mainly as a hobby.) Let X,Y be locally convex vector spaces and let B(X^{\ast},Y^{\... 1answer 43 views ### Equivalence of norms on Schwarz space Consider the following norms on the Schwarz space, for 1\leq q\leq \infty$$\lVert f \rVert_{\alpha,\beta, p}=\lVert x^{\alpha}\partial^\beta f\lVert_{L^p}$$I want to show that the norms \lVert \... 1answer 32 views ### Proving that a locally convex vector space is a topologial vector space. Let V be a vector space over \mathbb{C} endowed with a topology generated by a collection \{p_\alpha\}_{\alpha\in A} of semi-norms. I want to prove that this turns V into a topological ... 0answers 14 views ### Projective tensor product on projective LCM is exact I am reading the book "The Homology of Banach and Topological Algebras" by A.Y. Helemskii and couldn't understand one lemma on page 204 about complex splitting. I understood how to prove that complex ... 0answers 18 views ### direct limit in locally convex modules and continuous map Let we have short exact sequences of LCM over LC algebra A with continuous linear maps$$ 0\to B_j\;{\xrightarrow {\ f_j\ }}\;C_j\;{\xrightarrow {\ g_j\ }}\;D_j\to 0. $$We can take inductive limit (... 1answer 79 views ### convexity of a relatively open subset of a compact set I'm struggling with the following problem: it seems to be true but I'm not able to prove it! Let C be a compact convex subset of a locally convex metric vector space and \hat{C} be a relatively ... 1answer 59 views ### About the locally convex topology I know that if a locally convex space Hausdorff (X,S) is first numerable then for the \hat{0}\in X exists a countable local base \{V_n, n \in \mathbb{N}\} and to each V_n corresponds a ... 0answers 29 views ### Product of open maps open? (not the cartesian) Let A be a locally convex algebra, or even just a topological algebra, and let U_1,U_2\in A be open, is the product$$ U_1\cdot U_2=\left\{ a\cdot b\mid a\in U_{ 1} ,b\in U_{ 2} \right\} $$... 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 ... 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 \} ... 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 ... 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 ... 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, ... 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 ... 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 ... 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 ... 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 ... 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) ... 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 ... 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 ... 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\}} 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,... 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 ... 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 \... 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} ...
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 ...