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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179 views

Nuclear spaces vs Banach spaces

The Wikipedia article on nuclear spaces say the following: "There are no Banach spaces that are nuclear, except for the finite-dimensional ones. In practice a sort of converse to this is often true: ...
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65 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, ...
6
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168 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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142 views

Connections and dependences between topological and algebraic basis in topological vector space

On my last functional analysis exam, one of the tasks was to show that if normed vector space $X$ have countable Hamel basis, then $X$ is separable space (over field $\mathbb{R}$). I am not sure if ...
5
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87 views

Infinite Dimensional Topological Vector Space

Let $V$ is a finite-dimensional vector space over $\mathbb{R}$ (or $\mathbb{C}$). To make $V$ a topological space, we may choose the sets $f^{-1}(U)$ as a sub-basis, where $f$ ranges over all linear ...
5
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152 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 ...
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39 views

When is a topology is defined for this special kind of uniform convergence?

Question: Let $\phi:X\to X$ be bijective and continuous. Does a topology $\tau$ exist such that $$f_n\overset\tau\to f\Leftrightarrow \phi\circ f_n\overset\infty\to\phi\circ f$$ where $\infty$ ...
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95 views

Are uncountable “Schauder-like” bases studied/used?

We could define the following notion of basis in a way analogous to unconditional Schauder basis: If $X$ is a topological vector space over $\mathbb R$ and $B=\{b_i; i\in I\}$ be a subset of $X$. ...
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133 views

The topology of $GL(V)$

Let $V$ be a topological vector space (not necessarily finite-dimensional) over a field $K$, and let $GL(V)$ be the group of invertible linear maps $V\to V$ under composition. There are two obvious ...
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72 views

Schwartz space of functions versus Schwartz space in a more general sense?

Part of me is afraid that this isn't a well-formed question, but try as I might, I can't seem to figure out anything reasonable on this topic. I'm hoping someone here can help. In functional analysis,...
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51 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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252 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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50 views

Extending the topology on a set to the group it generates

The multiplicative group $\Bbb Q^+$ can be viewed as a $\Bbb Z$-module. To see this, note that any rational can be decomposed into the form $2^{n_2} \cdot 3^{n_3} \cdot 5^{n_5} \cdot ...$ The tuple ...
3
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61 views

Reference request for Grothendieck's work on “Integration with values in a topological group”

Recently I was reading the available part of the second part of W. Scharlau's book on Alexandre Grothendieck (see here). There I found, An anecdote survives about Grothendieck's arrival in Nancy: ...
3
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80 views

When weakly compactness implies compactness?

Let $A$ be a Banach space. The weak topology on $A$ is a topology which produced by the following family of seminorms: $~~~~~~~~~~~~~~~~~~~~P_f(x)=|f(x)|,\qquad$ where $f\in A^*$ and $A^*$ is dual ...
3
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69 views

Closedness of convex sets in a locally convex space

Let $C$ be a convex subset of a locally convex topological vector space. Consider the properties: a) $C$ is closed. b) $C$ is weakly closed. c) $C$ is weakly sequentially closed. d) $C$ is ...
3
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143 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?
3
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69 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 power-...
3
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68 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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83 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 ...
3
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144 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 ...
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111 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 $\Phi(T)=T^{**}$;...
3
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719 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 ...
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345 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 $\omega\!\in\!\...
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221 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 ...
3
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103 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 $X^{*...
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113 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 $...
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527 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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19 views

TVS on the reals which or which not induces convergence in norm

Recently I wondered, if convergence in some given metric $d$ on $\mathbb R^n$ induces convergence in norm. Of course, if $d(x,y) = \|f(x)-f(y)\|$, where $f$ is a bijection on $\mathbb R^n$, then this ...
2
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38 views

Topology generated by Minkowski functionals

Let $X$ be a locally convex topological vector space and let $\gamma$ be a local base of convex sets. Associate to each $C \in \gamma$ the Minkowski functional $\mu_C(x)= \inf(\lbrace t > 0 \mid t^{...
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27 views

Interchanging Limits: $a_{n,k}\to C$ implies $\sum_k^n a_{n,k}b_k\to C\sum_k^\infty b_k$?

Consider a doubly-indexed sequence $(a_{n,k})$ which, for each $k$, converges as $n\to\infty$ to an absolute constant $C$. In what generality can we say that for a sequence $b_k$ $$\lim_{n\to\infty}\...
2
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0answers
37 views

$W^{1,p}$ is separable for $1\leq p<\infty$

I've been asked to prove that the Sobolev spaces $W^{1,p}(\Omega)$, $\Omega$ open in $\mathbb R^n$, are separable for $1\leq p <\infty$ using the map $$i\colon W^{1,p}(\Omega)\to L^p(\Omega)\times ...
2
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0answers
76 views

Differential operators acting on the Schwartz space

I was asking me the following question and cannot find any answer to it. Any help/suggestion is most than welcome! Let $D$ be a linear differential operator with polynomial coefficients on $\mathbb{R}^...
2
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0answers
32 views

Linear finite-dimensional topological vector space is closed.

Suppose $X$ is a topological vector space and let $Y \subset X$ be its subspace with $\dim Y < \infty$. The goal is to prove that $Y$ is closed in $X$. I know how to prove this fact when $X$ is ...
2
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0answers
67 views

Show that $df_x: TM_x\rightarrow TN_{f(x)}$ is well-defined

Let $f\colon M\rightarrow N$ be a $C^\infty$ function between $C^\infty$ manifolds. Show that $$df_x: TM_x\rightarrow TN_{f(x)}$$ defined by $$df_x([c_0]) := [f \circ c_0]$$ is a well-defined map. ...
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91 views

Are Lusin and Souslin spaces sequential or even Fréchet-Urysohn?

First some definitions: A Polish space is a separable and completely metrizable topological space. A Hausdorff space is Lusin if it is the image of a Polish space under a bijective continuous map. A ...
2
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0answers
50 views

If $A$ is a balanced and convex set, then $\lambda A + \mu A = (|\lambda| + |\mu|)A$?

I have proven that $\lambda A + \mu A \subset (|\lambda| + |\mu|)A$. How do I prove the reverse inclusion?
2
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100 views

Part (b) of Exercise 13 of first chapter of Rudin's book “Functional Analysis”

I would really appreciate it if you could give me some advice on the part (b) of Exercise 13 of first chapter of Walter Rudin's book "Functional Analysis": Let $C$ be the vector space of all ...
2
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0answers
76 views

Show that continuous linear maps on the space of test functions take $C_K^\infty(\Omega)$ into some $C_{K_N}^\infty(\Omega)$

Let $\Omega$ be a nonempty open subset of $\mathbb{R}^n$, and let $\cup_{n=1}^\infty K_n = \Omega$ be an exhaustion of $\Omega$ by compact sets. Let $\mathcal{D}(\Omega) = \mathcal{D}$ be the standard ...
2
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0answers
18 views

Is the unit ball of $H^\infty(\mathbb{D})$ a metrizable topological semigroup under multiplication?

The space $H^\infty(\mathbb{D})$ of all bounded holomorphic functions on the open unit disc carries many different topologies. One such topology is given by uniform convergence on compact subsets; ...
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32 views

Existence of particular functionals in a family of linear functionals

Let $U\subset B$ be a subset of a Banach space $B$, and let $D$ be a complete topological vector space. I have given a family $\mathcal L(U)=\{L_u\ |\ u\in U\}$ of linear functionals $L_u:D\to\mathbb ...
2
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0answers
232 views

Continuity of implicit function

I find a proof of the following theorem in A.N. Kolmogorov and S.V. Fomin's Элементы теории функций и функционального анализа (pp. 492-493 here): Let $X,Y,Z$ be Banach spaces, $U$ a neighbourhood ...
2
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0answers
63 views

Limit topology of a sequence of topological vector spaces

Under which circumstances is the limit topology of an increasing sequence $E_0\subseteq E_1\subseteq E_2\subseteq\cdots$ of topological vector spaces, where the inclusion maps are linear and ...
2
votes
0answers
57 views

Topological vector space question

$C[0,1]=$ space of all continuous complex valued function over $[0,1]$. Define metric, $d(f,g)={\int_{0}^{1} \frac {|f(x)-g(x)|}{1+|f(x)-g(x)|}}$, for all $f,g\in C[0,1] .$ Let $(C[0,1],\sigma)$ ...
2
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0answers
47 views

Metric space with measure and a special property

Let $R$ be a metric space endowed with a (complete) measure $\mu$ satisfying the following condition: all the open and closed sets of $R$ are measurable and for any measurable set $M\subset R$ and any ...
2
votes
0answers
43 views

Extension of function with values in a Banach space

I want to prove the following Let $E,X$ be Banach spaces, and $Y\subset E$ a closed subspace with codimension $1$. Let $T:Y \to X$ be a continuous linear function. Then there exists a continuous ...
2
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0answers
25 views

T1 axiom in dual space

My books talks about the conjugated space, but does it mean dual space? Are not the same thing? I don't understand why in the dual space $E^\ast$ of $E$, the separation axiom $T_1$ is satisfied and ...
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28 views

When a sort of weak topology is enough to generate vector space topology

Consider a vector space $V$, and some functions $f_\alpha: V \rightarrow \mathbb{C}$ where $\alpha$ ranges over some index set $A$. We can think about the coarsest topology which: a) makes the $f_\...
2
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0answers
40 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
votes
0answers
82 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.