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
45 views

Nuclearity of $\mathscr{S}$

I have big problems proving that the Schwartz Space $\mathscr{S}(\mathbb{R}^n)$ together with the topology induced by the family $$ \|\varphi\|_{p}:=\sup_{x\in \mathbb{R^n}}\sup_{|\alpha|\leq ...
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1answer
29 views

Inductive Limit of directed locally convex Frechet Spaces

Let $\Phi=\bigcup_{i\in \mathbb{N}}\Phi_i$ be the inductive limit of an upwardly directed set of countably-seminormed spaces (i.e. the locally convex topology is given by a countable family of ...
3
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1answer
33 views

Local convexity of the topology of weak convergence of probability measures

Let $X$ be a Polish or standard Borel space, and $\mathcal P(X)$ be the space of all Borel probability measures on $X$ endowed with the topology of weak convergence. I am thinking of using ...
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17 views

A question Kolmogrov's generalized inequality for projection onto convex sets

Kolmogrov's inequality says that, if $C$ is a convex set, and $P_C(x)$ is an operator for projecting point $x$ into the convex set $C$, if $z = P_C(x)$, then for any $y \in C$ we have $$ (z - y).(x - ...
2
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1answer
87 views

Uniform convex space

Please I want to know if this space $$H^1_{0,p}([0,+\infty))=\lbrace u, u\in AC([0,+\infty)), u(0)=u(+\infty)=0,\sqrt{p}u'\in L^2\rbrace$$ where $p>0$, $p\in L^1((0,+\infty))$ ...
3
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1answer
51 views

Topology Book including specific aspects

I am looking for a basic book about Topology (maybe also a bit of Functional analysis but basically Topology) including the following points (in addition to the basic points): $\bullet$ Seminorms ...
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1answer
79 views

Is $\sqrt{x}$ concave?

I have function $f(x)= \sqrt(x)$. To check is it concave or convex i am checkin $f''(x). $ Which is $ -\frac{1}{4x^{\frac{3}{2}}} < 0$ So the $f(x)$ is concave. Is it correct ? And is is the same ...
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66 views

Strong Notion of Integral

Is there a strong(!) notion of integral that can face all of those issues: Singularities Decay Modes Oscillations Measure Spaces Locally Convex Spaces For example combining decay modes with ...
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14 views

Differential calculus on locally convex spaces

For real finite dimensional vector spaces $V,W,Z$, i know that a map $f:V \times W \to Z$ is smooth if the maps $f(v,.)$ and $f(.,w)$ are for every $v\in V, w \in W$. Does the same thing hold for ...
2
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1answer
43 views

Please verify this definition of locally convex vector space

A locally convex vector space is a vector space $V$ equipped with a family $P$ of separating semi-norms. Is this a correct definiton? So to determine whether a norm induces a locally convex topology ...
2
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1answer
57 views

Locally convex topological vector space

In $C(\mathbb{R})$ space we define two families of seminorms: $p_x(f)=|f(x)|$ where $ x\in\mathbb{Q}$ and $q_x(f)=|f(e^x)|$ where $ x \in \mathbb{R}$ I have to check if above families induce locally ...
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34 views

The closed graph theorem for Banach spaces isn't true. True?

I'm reading through some functional analysis lecture notes and there the closed graph theorem was stated in the following form: Let $X$ be a Baire locally convex space and $Y$ a Frechet space. If ...
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1answer
26 views

($\forall x\in E_1 \exists y\in E_2: f_1(x_1)=f_2(x_2))\ \Longrightarrow \ x \mapsto y$ continuous

I'm reading through some notes on functional analysis where on a couple of occasion we where in the following setting: $E_1$ and $E_2$ are two Frechet spaces and $f_i:E_i \rightarrow F$ two ...
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1answer
48 views

Projection onto a convex closed set

H, If $K$ is a non-empty convex and closed subset of a uniformly convex Banach space $X$ (Hilbert for example) and $v \notin K$, we know that there exists a unique $k_0\in K$ such that ...
4
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1answer
85 views

Caught in the net

I'm reading through some notes one locally convex spaces ("lcs" from now on) analysis and there the following version of the Banach-Steinhaus theorem is given Theorem (Banach-Steinhaus) $\quad$ ...
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45 views

Separability of nuclear spaces

I'm currently studying nuclear spaces and the nuclear spectral theorem with the books of Gelfand, Vilekin and Schilow, "Generalized functions", especially the second and fourth part. Currently, I'm ...
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2answers
100 views

On the convexity of a particular form of integrals

EDIT: I made some critical corrections below. Let $\mathcal{H}\colon\mathbf{w}\cdot\mathbf{x}+c=0$ be a hypeplane in $\mathbb{R}^n$. Also, let $g\colon\mathbb{R}^n\to\mathbb{R}_+$, be a non-negative, ...
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16 views

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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2answers
58 views

strong topology = inductive limit topology on duals of projective limits

I've been bothering with this for some time now, and can't find any source with an actual proof, the statement simply appears to be "well-known". If you know (a source with) a proof, I'd be happy :) ...
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1answer
42 views

Non-locally convex topologies on $\mathbb{R}^{n}$ compatible with the vector space structure

So I know that every locally convex topology on $\mathbb{R}^{n}$ is equivalent to the norm topology. Are there any non-trivial examples of non-locally convex topologies on $\mathbb{R}^{n}$ that still ...
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1answer
13 views

Detecting ray cross after hit on a convex object

I have a hw question im struggling to solve - Any guidance will be appreciated.
4
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1answer
76 views

Generalization of Hahn-Banach Theorem

Let $Y$ be a subspace of a locally convex topological vector space $X$. Suppose $T:Y\longrightarrow l^\infty$ is a continuous linear operator. Prove that $T$ can be extended to a continuous linear ...
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1answer
59 views

On convex hulls and intersections of chains of compact sets

Let $V$ be a topological vector space, let $\{ C_i \}_{i \in I}$ be a set of compact subsets of $V$ which forms a chain with respect to inclusion. For now, assume the following stronger properties: ...
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34 views

Do finite dimensional real spaces have a canonical structure of locally convex space? [duplicate]

Note that I intentionally didn't use "euclidean spaces" but "finite dimensional real spaces" so I do not want to consider any specific inner product or any specific norm on such spaces (just their ...
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1answer
40 views

When does a dense subspace destine the weak topology?

Let $E$ be a locally convex space, let $E^{\prime}$ be its continuous dual space and let $F$ be a subspace of $E^{\prime}$ which is dense with respect to the strong topology on $E^{\prime}$ (i.e. the ...
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$T: H^{-\infty}(R^n) \to H^\infty(R^n)$ continuous iff $T: H^{-r}(R^n) \to H^s(R^n)$ bounded for all $r,s>0$?

Denote by $H^s(\mathbb{R}^n)$ the Sobolev space on $\mathbb{R}^n$ of order $s \in \mathbb{R}$ and recall that we have $H^s(\mathbb{R}^n)^\ast \cong H^{-s}(\mathbb{R}^n)$ for the dual space of ...
3
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1answer
68 views

Extending continuous, densely-defined linear maps between locally convex spaces

Let $X$ and $Y$ be locally convex topological vector spaces, say over $\mathbb{C}$. To set the stage a bit, I'll say that the topology on $X$ is given by a separating family of semi-norms $(p_i)_{i ...
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1answer
40 views

How to detect reflexivity of the closure

Consider the space of continuous bounded functions on a bounded interval. Its closure for the Lebesgue $L_p$ norm is reflexive when $1 < p < \infty$, but it is not reflexive for $p = 1$. How ...
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84 views

Topology of pointwise convergence - open sets

Let $X$ be the vector space of all complex functions on $[0,1]$, topologized by the family of seminorms $p_{x}(f)=|f(x)|$, $0\le x\le1$. This topology is called the topology of pointwise convergence. ...
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1answer
54 views

About absolutely summing operators

The concept of absolute summing operator between two Banach spaces is already well-known. I just would like to know if one can extend the definition between general spaces, say locally convex spaces. ...
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1answer
47 views

In the proof the proving duality of topological vector space

I find hard to understand the proof for this theorem. Hope that some one can help me to clarify this. Thanks Suppose $X$ is a vector space and $X'$ is a separating space of linear functionals on ...
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68 views

Weak topology on C(X)

Let $X$ be a Tychonoff space. $C(X)$ is the space of scalar-valued continuous functions on $X$ endowed with the compact-open topology. Is the following true? $(C(X))^{*}$ is the set of all finite ...
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98 views

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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15 views

Seminorm topology on $C^{\infty}(\Omega)$ and proof of continuity [duplicate]

Let $\Omega\subset\mathbb{R}^n$ be open and $\Omega\neq\varnothing$ and suppose we have the Fréchet topology on $C^{\infty}(\Omega)$ (this can be obtained by the topology construction from out ...
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1answer
51 views

convex hulls of convex sets plus a point

In a real vector space, consider S and T to be disjoint, nonempty, convex subsets of the vector space and let a point x lie outside either set. How would I prove the following: co({x} union S) is ...
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74 views

Understanding bornologic spaces

The definition as it appears in Yosida's book: A locally convex space $X$ is called bornologic if it satisfies the condition: If a balanced convex set $M$ of $X$ absorbs every bounded set of $X$, ...
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113 views

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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144 views

The convex function proof.

Can anyone help me prove convexity conditions? I know convexity conditions. The second derivative function is greater 0 first order convexity conditions. convex function conditions Because my ...
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1answer
77 views

Can I decompose a compact set in a finite number of convex set?

My problem is in a finite-dimensional space. I look at $\mathcal{X}$ the support of a function $f$, that is continuous and has bounded support. \begin{eqnarray} \mathcal{X}_o & = & \{x \in ...
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0answers
51 views

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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1answer
128 views

Prove convexity of complicated rational function

Can anyone help me prove the convexity of this rational function? The man who proved the convexity of function used these facts. But I don't know this fact is correct or not. Here are the facts and ...
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1answer
43 views

Counterexample for bounded subsets in a Frechet space

I do not know if anyone has asked about this before: Does anyone know an example of an unbounded subset of a Frechet space which have finite diameter? I saw this in Conway (A course in functional ...
3
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1answer
181 views

Strict topology - what is the “importance” of the concept?

From Conway, A course in functional analysis, page 105. Problem 21. Let $X$ be a locally compact space and for each $\phi\in C_{0}(X)$, define $p_{\phi}(f)=|\phi f|_{\infty}$ for $f\in C_{b}(X)$. ...
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1answer
53 views

Different notions of differentiability

The following is somewhat unclear to me. Let $X$, $Y$ be locally convex vector spaces, let $f: X \supseteq U \longrightarrow Y$ be a (nonlinear) continuous map. Then one can say that $f$ is $C^1$ if ...
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1answer
526 views

How is the weak-star topology useful?

Today I learnt something about the weak-star topology, but I don't know what the use of weak-star topology is. I hope someone can tell me what we can do with the weak-star topology. Thanks in advance! ...
3
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1answer
118 views

An counterexample of Hahn-Banach theorem in a topological vector space

Problem : Give an example of a TVS $\mathcal{X}$ that is not locally convex and a subspace $\mathcal{Y}$ of $\mathcal{X}$ such that there is a continuous linear functional $f$ on $\mathcal{Y}$ with no ...
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76 views

Norm inequalities in a reflexive space

I am reading an article about reflexive spaces, with a specific example. The article mentions inequalities that I haven't been able to get around to. Here's the setup. The space $X = (\prod_n ...
2
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1answer
240 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 ...
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1answer
68 views

Quasi-Convexity

Can I get the conclusion that the function of matrix $P$ and $Q$ \begin{equation} \mathrm{tr}\left( PQ\right) \end{equation} is a quasi-concave function for $P>0$, and $Q>0$? It is true for ...