The study of vector space with a topology which makes the maps which sums two vectors and which multiply a vector by a scalar continuous. It's a natural generalization of normed spaces.

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452 views

Differential forms on fuzzy manifolds

This post will take a bit to set up properly, but it is an easy read (and most likely easy to answer); in any event, please bear with me. Question In the usual setting of open subsets of ...
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135 views

Isomorphism between spaces of sections.

Let $M$ be a compact manifold and let $E_i \xrightarrow{\Large \pi_i} M$, $i = 1, 2$, be two (real or complex) vector bundles of the same rank $k$ over $M$. Assume we have metrics $g_1, g_2$ on $E_1$, ...
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29 views

Closed Monoidal Structures On The Category Of Complete Topological Vector Spaces

Context: The category of Banach spaces, with the projective tensor product is a closed monoidal category. Question 1: Is there a tensor product on the category of complete topological vector spaces, ...
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149 views

Evaluation map is not continuous always.

Let $E$ be a not normable locally convex space, define $$F: E'\times E\to \mathbb R$$ $$(f,e)\to f(e)$$ I have to show that $F$ is not continuous when $E'\times E$ is given product topology. I was ...
4
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28 views

How do they call the topological tensor product that classifies operators from Hilbert space?

Let $V$ and $W$ be topological vector spaces. There are different ways to complete the tensor product $V \otimes W$, and the only ones that are usually discussed in introductory literature are the ...
4
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116 views

Curiosities about the content of a rare book: Topological Vector Spaces by A. Grothendieck

The book is a celebrated and highly influential book by A. Grothendeck, which was published in 1954, in French and for various reasons, it has been out of print since 1973. I am very much interested ...
4
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149 views

The converse of James's Theorem

The famous James theorem states that: Theorem. Let $X$ be a (Hausdorff separated) locally convex space (LCS for short) with topological dual $X^*$ and let $B\subset X$ be weakly-closed. If $X$ is ...
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52 views

How general is the convergence of the exponential function's power series?

Let $\mathbf{V}$ be a Fréchet space whose underlying set is $V$. Let $\;\; \beta \: : \: V\times V \: \to \: V \;\;$ be a continuous bilinear map that has an identity element and is ...
3
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40 views

Every locally connected non separating plane continuum is an AR

A continuum is a compact connected metric space. A plane continuum is a continuum contained in $R^2$. We say that a plane continuum does not separate the plane provided that $R^2\setminus X$ is ...
3
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100 views

Which (endo)functors of the category of finite-dimensional real vector spaces induce continuous maps between Hom-sets?

Let $\operatorname{Vect-fin}$ be a category of finite-dimensional vector spaces over $\mathbb{R}$. In this category Hom-sets $\operatorname{Hom}(V,W)$ are themselves finite-dimensional vector spaces ...
3
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101 views

Density of operators

I am interested in operators on non-reflexive Banach space. Let $X$ be a Banach space and let $L(X)$ be the algebra of operators acting on $X$. We may embed $L(X)$ into $L(X^{**})$ by ...
3
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351 views

Translation invariant metric

Under what conditions can a metric vector space be given an equivalent metric that is translation invariant? I was wondering if the probability measures on real line can be embedded in vector space ...
3
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207 views

Definition of a topological module

A topological universal algebra of type $\Omega$ is a universal algebra $A$ of type $\Omega$ that is also a topological space, such that for any $n\!\in\!\mathbb{N}$ and any operation ...
3
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99 views

Self-absorbing subsets in a vector space

From planetmath Let $V$ be a vector space over a field $F$ equipped with a non-discrete valuation $|\cdot|:F\to \mathbb{R}$ . Let $A$ and $B$ be two subsets of $V$. Then $A$ is said to absorb ...
3
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327 views

Understanding examples - metric spaces, Minkowski functionals and topologies

I'm teaching myself a course on functional analysis but having trouble understanding the notes I've been using. I was hoping I could write out a section of the content and you might be able to help me ...
2
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8 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 ...
2
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0answers
52 views

Is there a non-complete and non-separate metric space?

Is there a (non-trival) non-complete and non-separate metric space? Some notions are here: math.stackexchange.com/questions/182316.
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45 views
2
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57 views

What do we call a Schauder-like basis that is uncountable?

In a topological vector space, every Schauder basis is assumed countable, by definition. Supposing we drop the countability condition, we call this a [what goes here?] basis?
2
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97 views

Functional analysis exercise

I would really appreciate it if you could give me some advice on the following exercise in Rudin. Put $K=[-1,1]$; define $\mathcal{D}_K$ as the set of all smooth functionals supported in $K$. ...
2
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69 views

Pseudo norm-exercice

Let $f$ be a measurable function with finite values almost everywhere. We put $$N_0(f) = \displaystyle\int \dfrac{|f|}{1 + |f|} d \mu.$$ We denoted by $L^0$ the set of measurable functions $f$ such ...
2
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93 views

Which are nontrivial examples of analytical functions on Frechet spaces?

Let $X$ be a real linear topological space, which is a (separable) Frechet space, such that the topology on $X$ is generated by the countable family $\{p_n:n\in\omega\}$ of norms . A real-valued ...
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37 views

Density problem

$U$ is any open set of $\mathbb{R}$. We known that $C_0^\infty(U)$ is dense in $C^k(U)$. But what about, say $C_0^\infty((0,1))$ in $C^k([0,1])$?
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51 views

Topology of $(\mathcal{A},*)$ determined by $\mathcal{A}_{sa}$?

Let $(\mathcal{A},*)$ be a $*$-algebra, we have the following observation: Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be two norms on $\mathcal{A}$ such that the involution is an isometry with respect to ...
2
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169 views

Hairy ball theorem: references to applications

I'm looking for references to applications of the Hairy ball theorem. I already visited wikipedia and cited references, but I need a little more explanation in both meteorology and applications in ...
2
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84 views

Consequence of metrizability proof - disregard, the question is an error

In Marian Fabian et al's Functional Analysis and Infinite-Dimensional Geometry, Proposition 3.22 states/proves that if $X$ is a separable Banach space, then the (closed) unit ball, $B_{X^{*}}$ of ...
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48 views

When is a subset of {0,1} valued borel functions on a standard borel space (polish space) complete (see *) under the pointwise convergence topology?

*In other words, what restrictions on a family F of {0,1} valued borel functions will tell us that the pointwise limit of any net in F is borel. I feel like there must be lots known about this but I ...
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38 views

Open Convex Subsets of Dense Spaces

So I asked this question yesterday, Existence of Non-Trivial, Convex, Open Set in $C_{\mathbb{C}}[0,1]$ Under $L^{0}$ Metric, and it made my start wondering the following... Suppose the following: ...
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28 views

Does this topology on the dual have a name

Let $X$ be a topological vector space. Let $X^\ast$ denote its continuous dual. It is possible to endow $X^\ast$ with the weak star topology: Def.1: If $e_x: X^\ast \to \mathbb C$ is the map $\varphi ...
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28 views

Direct limit, density and norm

Let $E$ be a Banach space, $A_n$ be an increasing sequence of finite dimensional subspaces of $E$, $B_n$ be an increasing sequence of subspaces of $A_n$ and let $C_n = A_n/B_n$. Assume that the ...
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73 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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47 views

Differential Operators over the space of Analytic Functions

Let $\mathcal{A}(-a,a)$ be the vector space of functions that are analytic on the interval $(-a,a)$ Is there a common topology to place on this space, if yes what is the topology and is it induced ...
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53 views

On Some Locally Convex Topologies of a Vector Space(Update)

This is an update of my previous question in here. Suppose that $(X,\tau)$ is already a locally convex TVS. Let us denote by $X'$, the space of all $\tau$-continuous linear functionals on $X$, the ...
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46 views

A separation lemma in a real vector space

A lattice $N$ is a free $\mathbb{Z}$-module of finite rank. Let $V$ be the real vector space $N\otimes_\mathbb{Z} \mathbb{R}.$ A cone is a set $\sigma = \{ r_1 v_1 + \ldots + r_k v_k \in V : r_i\geq 0 ...
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48 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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35 views

Bounded Semivariation and Bounded Variation in locally convex TVS

Let $(X,\tau)$ be a Hausdorff locally convex TVS and let $P(X)$ be a family of seminorms on $X$ that generates $\tau$. We consider the following definitions. Definition 1. A function $f:[a,b]\to X$ ...
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48 views

Show that the norm of the derivitive of a $C^1$ function over a vector space is non-negative, homogeneous and satisfies the triangle ineq

For $f$ in $C^1[a,b]$, define $p(f)= \parallel f'\parallel _{\infty}$. Show that $p$ is non-negative, homogeneous, and satisfies the triangle inequality. Why is it not a norm? -I can easily show the ...
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49 views

Two different opinions on whether a topological vector space is a uniform space

Van de Vel's Theory on Convexity Structures says a TVS is uniform iff it is locally convex: 3.10.1. Proposition. Let $X$ be a topological vector space, equipped with the standard convexity and ...
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51 views

Determining Similarity of Unit Vectors

I'm seeking for an injective piecewise continuous function $f:\mathbb S^n\rightarrow[0,1]$ where $\mathbb S^N$ is the set of vectors with $L_2$ norm equals $1$. The piecewise continuity requirement ...
1
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0answers
65 views

Inductive limits

Let $E_n$ be a family of Banach spaces. Under which conditions imposed on $(E_n)$ can we represent the $\ell_\infty$-sum $(\bigoplus_{n\in \mathbb{N}} E_n)_{\ell_\infty}$ as a complemented subspace of ...
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64 views

When is the orbit of a vector a minimal sequence? When does an operator have a minimal orbit vector?

Let $X$ be a banach space. We say a sequence $(x_n)$ is minimal if for each $k$, $x_k\notin [x_n]_{n\neq k}$, where $[x,y,z,\cdots]$ is the closed linear span of the vectors. For ...
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151 views

Are topological vector spaces completely regular?

Every uniformizable space is a completely regular topological space. topological vector spaces are uniform spaces. every Hausdorff topological vector space is completely regular. From ...
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94 views

Are nuclear Montel spaces projective?

The well known fact about $\ell^1$ says that the Schauder basis of $\ell^1(I)$ behaves more-less like a Hamel basis, namely if $X$ is any Banach space and $\mathcal{E}=(e_i)_{i\in I}$ is a basis for ...
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20 views

Linearly compact subspace is closed

Let $V$ be a vector space with a linear topology, meaning open linear subspaces form a basis of open neighborhoods of $0$. Recall that a subspace $W$ is called linearly compact if for any open ...
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25 views

Some fundamental relations in topology

Are the following relations correct? $\ \{ Normed\, Vector\, Spaces\} \subset \{Topological\, Vector\, Spaces\} \subset \{Uniform \,Spaces\} \subset \{Topological\, Spaces\}$ Then $\ \{Normed\, ...
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20 views

Convex in $ \mathbb{R^n}$

Prove that: [A be a convexe part $(A\subseteq \mathbb{R^n})] \implies [\forall x_1,x_2,...x_n\in A ,\forall\alpha_1,\alpha_2,...\alpha_n\ge0 $ $with$ $ \ \alpha_1+\alpha_2+...+\alpha_n=1 ...
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46 views

topological vector space of measure functions

Let $(X, \mathcal X, \mu )$ be a measure space, and let $ L(X)$ be the space of measurable functions $f: X \to \mathbb C$. Show that the sets $B(f, \epsilon ,r ): = \{ g \in L(X) : \mu( \{ x : | f(x) ...
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33 views

A vector space with topology generated by a family of typologies each makes it a topological vector space is a topological vector space

Let $V$ be a vector space, and let $(\mathcal F_ \alpha ) _{ \alpha \in A}$ be a family of topologies on V, each of which turning $V$ into a topological vector space. Let $\mathcal F$ be the vector ...
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0answers
17 views

A question on a function of bounded semivariation(part 2)

Let $(X,\tau)$ be a Hausdorff locally convex topological vector space and let $P(X)$ be a family of continuous seminorms on $X$ that generates the topology $\tau$. I got the following definition from ...
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25 views

On function of bounded semi-variation

The following ideas were taken from the article of DUCHON, entitled "On Vector Measures and Distributions." Definition. Let $X$ be a locally convex space whose topology is generated by a family $P$ ...