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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3
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0answers
101 views

Why the space of section of a vector bundle is complete?

This page is about the space of sections: Let $E \stackrel{\mathrm{fb}}{\rightarrow} \Sigma$ be a smooth vector bundle. On its real vector space $\Gamma_{\Sigma}(E)$ of smooth sections consider the ...
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9 views

Does a family of seminorm induce by a vector space isomorpism turn one of this vector space into a frechet space?

Suppose $F$ is a Frechet space induced by a family of seminorms $P$. Now let $V$ be another vector space which is isomorphic to $F$ by $\tau: V \rightarrow F $. We define family of seminorms $K$ on $...
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1answer
34 views

Jordan decomposition functional $C^*$-algebra [closed]

Consider the following fragment from the thesis Injective and Semidiscrete von Neumann Algebras by Rasmus Sylvester Bryder: Why is the boxed equality true? In particular, I don't see why the right ...
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13 views

Equivalent topologies given by seminorm basis

Let $X$ be a set and $\tau_1,\tau_2$ two topologies whose's basis are familes of seminorms $p_s$ and $p'_k$ respectively. In order to proof that the topological spaces $(X,\tau_1)$ and $(X,\tau_1)$ ...
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1answer
48 views

In a finite-dimentional Hausdorff locally convex vector space, how to prove there exists a seminorm which is a norm?

Let E be a finite-dimensional Hausdorff locally convex vector space, and $e_1,\ldots, e_n$ is its basis. I know that from the Hausdorff and locally convexity, there exists a seminorm $p$ satisfying ...
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82 views

The relative interior of all faces of a closed and bounded set is a partition of that set.

Let $E$ be a Banach space, let $C$ be a closed bounded convex subset of $E$. For $x,y\in E$ denote $]x,y[=\{ \lambda x+(1-\lambda)y : \lambda\in ]0,1[ \}$ (in particular if $x=y$ this set is equal to $...
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62 views

Lemma of Farkas, as a an application of Separation Theorems of Hahn Banach

I got stuck in a math problem i recently got in functional analysis: It's some kind of Lemma of Farkas but i can't find it somewhere in the Internet: Let X be a real locally convex space and $\xi, \...
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1answer
28 views

What is the finest locally convex topology that coincides with the weak one on equicontinuous sets

In this book Perturbative Algebraic Quantum Field Theory at page 26 the author says We equip $E'$ with the finest locally convex topology $\gamma$ that coincides with the weak one on ...
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2answers
39 views

Explicit set of seminorms on $C^\infty_c(\mathbb R)$

Let $E_n = C_{[-n,n]}^\infty(\mathbb R)$ with the topology generated by the sets $$ \Gamma_n = \{q_{C^k, [-n,n]}|_{E_n}\} $$ of (restrictions of) seminorms $$ q_{C^k, L} : C^\infty_{c}(\mathbb R) \to [...
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1answer
244 views

How can I convert "norms" using the bijection between $\mathbb{N}$ and $\mathbb{Z}^{d}$?

Suppose $\varphi$ is a sequence $\varphi = \{\varphi(x)\}_{x\in \mathbb{Z}^{d}}$ satisfying the following condition. There exists $k \in \mathbb{N}$ and $C \ge 0$ such that: $$|\varphi(x)| \le C ||x||^...
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25 views

Why the linear span of a cylinder set is $\mu$-measurable?

I read Bogachev's book "Gaussian Measures" and did not understand the next moment (Lemma 2.1.4): why $f^{-1}(B)=A$?see here I think this is not true because $Ker f+A\subset f^{-1}(B)$, and ...
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63 views

Smooth vectors in closed invariant subspace

Suppose $G$ is a Lie group, and $(\pi, E)$ is a continuous representation of $G$ on a Fréchet space $E$. Let $E^\infty$ be the set of smooth vectors of $E$, which is known to be dense in $E$. In ...
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1answer
58 views

On a quasi-complete dense subspace of a complete locally convex space

Let $X$ be a complete locally convex (topological vector) space, and let $M$ be a dense subspace of $X$. If we suppose that $M$ is quasi-complete (i.e., every bounded closed subset of $M$ is complete),...
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30 views

K- or J-interpolation of uniformly convex Banach spaces

Let $X_0, X_1$ be two real Banach spaces and let $X = (X_0, X_1)_{s, q}$ be the Banach space obtained by the K- or J-method of interpolation. Here, $0 < s < 1$, $1 < q < \infty$. If one of ...
2
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1answer
75 views

Checking whether a collection of sets is a topology (Rudin Functional Analysis 1.37 Theorem)

Please help me escape from my tantalizing confusion. In the proof, it is claimed that $\mathcal{B}$ induces an translation-invariant topology $\tau$. The translation-invariance of the collection $\tau$...
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0answers
43 views

A question on (c) of Remark 1.38 of Rudin's Functional Analysis

I have some confusion on (c) of Remark 1.38 (page 29) of Rudin's Functional Analysis (second edition) . Let me recap the relevant content here. Let $X$ be a vector space and $\{p_k|k\in\mathbb{N}\}$ a ...
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17 views

About definition of Locally Convex Space

I'm reading"Methods of Modern Mathematical Physics" by Barry Simon and Michael C. Reed. But I can't understand the definition that they give of locally convex space: Let $V$ be a vector ...
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1answer
46 views

If a collection of seminorms on some point $x_0$ is zero then $x_0$ is zero?

The following is Example.1.7. page 101 Functional Analysis book of Conway: 1.7. Example. Let f be a normed space. For each $x^*$ in $X^*$, define $p_{x^*}(x) = |x^*(x)|$. Then $p_{x^*}$ is a seminorm ...
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1answer
43 views

Weak* Topology coarser than topology of uniform convergence on compact sets

Let $E$ be a locally convex topological space. Is the weak$^*$-topology on its topological dual space $\sigma(E', E)$ coarser than the topology of uniform convergence on compact sets? I know that the ...
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0answers
99 views

Implicit function theorem for Fréchet spaces and analyticity

Does there exist an implicit function theorem (IFT) covering the following setting: Consider $f\colon \mathbb{C} \times V \to V$ where $V$ is a Fréchet space, satisfying certain conditions. I wish to ...
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1answer
39 views

Seminorms generating the topology of the inductive limit of locally-convex spaces

Let $(E_n,\iota_n)$ be an inductive system of LCSs; i.e.: each $\iota_n:E_n\rightarrow E_{n+1}$ is a continuous linear map. Suppose that the topology on each $E_n$ is generated by a (countable) ...
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1answer
43 views

To what extent it is necessary to assume "complete regularity" on $X$ to induce this locally convex topology on $C(X)$?

I found the following example in Conway's Functional Analysis Book: Suppose $X$ is a completely regular space and let $C(X)=$ all continuous functions from $X$ into $\Bbb{C}$. If $K$ is a compact ...
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18 views

Relation between Frechet derivative and convex functions

I am studying Fréchet derivatives and I am trying to show that, with we let $(V,\|\cdot\|_V)$ be a normed vector space and $U\subset V$ an open and convex subset, if we define $f:U\rightarrow \mathbb{...
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1answer
24 views

Conditions for locally convex space to be normable

Let $X$ be a Hausdorff locally convex space and $P$ is a family of seminorms on $X$. How to show that $X$ is normable iff $P$ is equivalent to a finite subfamily $P_0 \subset P$? One implication is ...
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0answers
34 views

Discontinuous linear operator.

Does there exists a discontinuous linear operator $T$ between Hausdorff locally convex spaces $X$ and $Y$ such that $T$ is continuous with respect to the weak topologies on $X$ and $Y$?
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2answers
61 views

Topology of compact convergence on space of holomorphic functions

Consider $\mathbb{D}_{R} = \{z \in \mathbb{C}: |z| < R\}$. For given $f \in \mathscr{O}(\mathbb{D}_{R})$ - space of holomorphic functions on the open disk, denote by $c_n$ the $n$-th Taylor's ...
2
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1answer
65 views

Definition of countable dimension of vector space

It's a well known that Hausdorff locally convex space $(X, P)$, where $P$ is a family of seminorms which generate topology on $X$ is metrizable iff $P$ is equivalent to an at most countable subfamily $...
3
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1answer
69 views

$T: X \to Y$ is continuous if and only if for every continuous seminorm $p$ on $Y$, the map $p \circ T$ is continuous

I am reading a section from the book "A Course in Functional Analysis" by J. B. Conway, and I came across this exercise. Let $X$ and $Y$ be locally convex spaces and $T: X \rightarrow Y$ be ...
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1answer
111 views

Locally convex spaces are topological vector spaces?

I've come across the below definition of a 'locally convex space' and am trying to prove that addition and multiplication are continuous with respect to the locally convex topology generated by the ...
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1answer
56 views

Is the cartesian product of locally convex space again locally convex?

Let $E_{1},...,E_{n}$ be locally convex spaces. Is the cartesian product $E_{1}\times \cdots \times E_{n}$ locally convex when equipped with the product topology? More generally, is the infinite/...
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2answers
48 views

Why locally convex topological vector spaces Hausdorff?

In A Course in Functional Analysis, John B. Conway, 100p, it is written that Definition. A locally convex topological vector space (LCTVS) is a TVS whose topology is defined by a family of seminorms $...
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1answer
37 views

Smallest vector topology

This is with regard to this question: Topology induced by seminorms and initial topology I saw somwehere that topology $\mathcal{S}$ is the smallest topology with respect to which all the seminorms ...
2
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1answer
52 views

Equivalence of two seminorms

Let us consider the space $C^0([0,1])$ of real continuous functions over $[0,1]$. Let $A = \{a_n : n \in \mathbb{N}\}$ be a countable set of $[0,1]$ and $\alpha_n$ be strictly positive real numbers ...
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1answer
25 views

Every first contable locally convex space has a countable neighborhood basis of balanced and convex sets

Terminology: By a neighborhood of a point $x$ on a topological space, I mean any subset $V$ which contains an open set containing $x$. A set $B$ in a vector space $X$ is called balanced if $\lambda B \...
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2answers
50 views

Banach spaces admitting a coarser topology for which the closed unit ball is compact.

Let $X$ be a Banach space and let $B$ be its closed unit ball. It is well known that $B$ is compact in the weak topology provided $B$ is reflexive. Otherwise, if $X$ is at least a dual space, ...
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1answer
63 views

Subspace topology induced by Inductive Limit is coarser than original topology on subspace

Let $\{X,\{X_i,T_i\}_{i\in I}\}$ be an inductive system of TVS's and let $T$ be the inductive limit topology. I am trying to show that the relative topology on $X_i$ induced by $T$ is coarser than $...
3
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1answer
101 views

Linear functional on LCS is continuous iff it is bounded by finite linear combination of topology-defining semi-norms

Consider a linear functional f on a TVS, whose topology is generated by a family of semi-norms $\mathcal{P}$, such that the topology is Hausdorff. In functional analysis by Conway, it is said that f ...
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1answer
80 views

Closed linear span of a subset in LCS is equivalent to intersection of all closed hyperplanes containing the subset

I'm currently working through Conway's Functional Analysis, and have stumbled across a result that I can't seem to find the justification for myself: Let $\mathcal{X}$ be a Real Locally Convex Space ...
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2answers
78 views

Show that this family is equicontinuous at $0$

Let $E$ be a normed vector space, $$p_K(\varphi):=\sup_{x\in E}|\varphi(x)|\;\;\;\text{for }\varphi\in E'$$ for compact $K\subseteq E$ and $\sigma_c(E',E)$ denote the initial topology with respect to $...
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1answer
50 views

When is a set open in the topology of pointwise convergence?

If $E$ is a set, then the topology $\rho(E)$ generated by $$p_x(f):=|f(x)|\;\;\;\text{for }f:E\to\mathbb R$$ for $x\in E$ is called the topology of pointwise convergence on $\mathbb R^E$. Are we able ...
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1answer
37 views

Analytic subbasis for a locally convex topology

Let $E$ be a $\mathbb R$-vector space, $P$ be a nonempty family of seminorms on $E$, $$U_p:=\{x\in E:p(x)<1\}\;\;\;\text{for }p\in P$$ and $\tau$ denote the topology on $E$ generated by $P$. I was ...
2
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1answer
86 views

(Uniform) continuity of a nonlinear function on a locally convex topological vector space

Let $E_1$ be a (nontrivial) vector space, $P$ be a family of seminorms on $E_1$, $\tau_1$ denote the topology generated by $P$, $(E_2,\tau_2)$ be a topological space, $f:E_1\to E_2$ at $x\in E_1$. ...
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0answers
54 views

Analytic basis for a topology generated by seminorms

Let $(X,\tau)$ be a topological space, $Y$ be a normed vector space $$\overline p(f):=1\wedge\sup_{x\in X}\left\|f(x)\right\|_Y\;\;\;\text{for }f\in C(X,\tau;Y)$$ and $$p_K(f):=\sup_{x\in K}\left\|f(x)...
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0answers
56 views

Equivalent condition for a seminorm on a locally convex topological vector space to be continuous

Disclaimer: I'm new to general topology, so please bear with me. I've got a strong background in measure theory and the following seems to be complete analogous to $\sigma$-algebras generated by ...
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1answer
93 views

Relation between weak topology on space of signed measures on a metric space $E$ and the weak* topology on $C_b(E)'$

In general, if $E_i$ is a $\mathbb R$-vector space and $\langle\;\cdot\;,\;\cdot\;\rangle$ is a duality pairing between $E_1$ and $E_2$ and $$p_{x_2}(x_1):=q_{x_1}(x_2):=\left|\langle x_1,x_2\rangle\...
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1answer
39 views

Equivalent condition for $V\subset X$ to be a neighborhood in the inductive topology on $X$.

Let $\{X_i\}$ a family of topologcial vector space(equipped with topology $\mathcal{T_i}$) let $X = \bigcup_{i = 1}^\infty X_i$ equipped with inductive topology w.r.t $\{(X_i,\mathcal{T}_i,I_i)\}$(the ...
7
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1answer
142 views

Does a strictly convex and continuous function always exist?

Let $X$ be a locally convex topological vector space, and let $C$ be a nonempty convex subset of $X$. A real-valued function $f: C \to \mathbb R$ is strictly convex if for all $\lambda \in (0,1)$ and ...
0
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1answer
60 views

An application of the Baire category theorem

In the highlighted sentence, $ K_n$ is a closed subset of $V^o$, which is the polar of $V$ and so is compact in the weak-* topology by the Banach-Alaoglu theorem. Therefore, $K_n$ is weak-* compact. ...
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1answer
74 views

why every absolutely convex absorbing and closed set is a zero neighborhood?

I need to prove that "If $E=\lim_{n} E_n $, the inductive limit of Banach spaces $E_n$ and $A$ is absolutely convex, absorbing and closed set in $E$, then $id_n^{-1}(A)$ is a zero neighborhood in ...
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1answer
34 views

A separable family of semi-norms on $\mathcal{B(H)}$

Let $\mathcal{H}$ be a Hilbert space and $\mathcal{B(H)}$ be the space of bounded operators on $\mathcal{H}$. Let $p_x:\mathcal{B(H)}\rightarrow \mathbb{R}^+\cup \{0\}$ such that $p_x(u)=\Vert u(x)\...

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