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.

learn more… | top users | synonyms

0
votes
0answers
22 views

Generalizing norms: leaving out absolute homogeneity

Given a function $\rho:X\to\mathbb{R}$ on a vector space $X$ which satisfies the following properties: $\rho(x)=0$ if and only if $x=0$ $\rho(x+y)\leq\rho(x)+\rho(y)$ $\rho(-x)=\rho(x)$ for any $...
2
votes
1answer
29 views

Why isn't $\ell^p$ locally convex for $0<p<1$?

I believe we have to distinguish the finite-dimensional from the infinite dimensional case. Regardless, if $0<p<1$, $\|x\|_p := (\sum |x_i|^p)^{\frac 1 p}$ is not a norm as it fails to satisfy ...
2
votes
1answer
33 views

Closure of an operator

I am wondering what is the closure of the domain of the operator $A_0:D(A_0)(\subset H)\to H$in $H=L^2(0,1)$ $$A_0= f^{(4)}-f^{(6)}$$ $$D(A_0)=\big\{ f\in H^6(0,1)\cap H_0^3(0,1) |f^{(3)}(1)=f^{(4)}(...
1
vote
1answer
65 views

Continuous semi-norms on subspace

Suppose $X$ is a locally convex topological vector space, let $P$ be the set of all continuous semi-norms on $X$. Suppose $M$ is a subspace of $X$, denote $P|_M$ as the set of semi-norms in $P$ ...
2
votes
0answers
41 views

Continuity and measurability of differential operator [closed]

Let $\mathcal{C}^k(\mathbb{R},\mathbb{R}^n)$ denote the space of $k$-times differentiable $\mathbb{R}^n$-valued functions on $\mathbb{R}$ equipped with the topology of uniform convergence of ...
0
votes
0answers
11 views

Hausdorffness of projective tensor product of locally convex space

I do not see why the following is true: Given two locally convex t.v.s. E and F, if the projective tensor product is Hausdorff then both E and F are Hausdorff. The converse holds and the proof is ...
4
votes
2answers
480 views

Example of a topological vector space such that $E = M \oplus N$ algebraically, but not topologically

I have the following question: Give an example of a topological vector space $E$ with subspace $M$ and $N$, such that $E = M \oplus N$ algebraically, but not topologically (so $E \ncong M \sqcup N$...
3
votes
0answers
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 ...
2
votes
2answers
69 views

A doubt about the vectorial topology on $\mathcal{D}(\Omega)$

We denote with $\mathcal{U}_0$ the family of all subsets $U \in \mathcal{D}(\Omega)$ convex and balanced such that $U \cap \mathcal{D}_K(\Omega) \in \mathcal{T}_K$, where $\mathcal{T}_K$ is the ...
0
votes
0answers
78 views

Dual space $E'$ is metrizable iff $E$ has a countable basis

I saw that it was already asked, but the book where I'm studying is slightly different. Recall some definition, if $E$ it's $\mathbb{K}$-vector space and let $\mathcal{E}$ be a vector subspace of the ...
0
votes
0answers
33 views

Constructing vector topologies (TVS's)

Consider the following theorem extracted from "An Introduction to Functional Analysis" by Charles Swartz (1992): Theorem 1: Let $X$ be a vector space. Let $\mathcal{U}$ be a family of subsets of $...
0
votes
0answers
62 views

When is it true that $V \subseteq \overline V \subseteq U$ will hold for open sets?

Let $(X, \mathfrak{T})$ be a topological space Let $V, U \in \mathfrak{T}$, and suppose that $V \subseteq U$ Then when it is true that $V \subseteq \overline V \subseteq U$, where $\overline V$ is ...
5
votes
1answer
198 views

locally convex space generated by a countable family of seminorms is metrizable

Let $X$ be a locally convex Hausdorff topological space, whose topology if generated by the countable family of seminorms $\{p_i:\space i\in\mathbb{N}\}$. I'd like to prove that $X$ is metrizable. So, ...
0
votes
1answer
26 views

Let $X$ be a finite dimensional normed space. Does the algebraic dual space $X^*$ and the dual space $X'$ coincide?

I am currently studying for my Functional Analysis test and then started thinking about the following and figured it is true (if it is not true, please do tell me - I am just thinking about this to ...
3
votes
2answers
70 views

Can every Hausdorff topological space be homeomorphically embedded in a topological vector space?

It's true that for any metric space, we can isometrically embed it in a Banach space, so that the image of the metric space by that embedding is a linearly independent set. Is the analogous theorem ...
0
votes
0answers
27 views

Complemented Spaces: Continuity vs. Closedness

Given a topological vector space $X$. (Not necessarily Hausdorff!) Consider subspaces: $$U_\pm\leq X:\quad X=U_++U_-\quad U_+\cap U_-=(0)$$ Equivalently an isomorphism: $$\Phi:U_+\oplus U_-\...
5
votes
1answer
133 views

Part (a) 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 (a) of Exercise 13 of first chapter of Walter Rudin's book "Functional Analysis": Let $C$ be the vector space of all ...
1
vote
1answer
23 views

Topological vector spaces with $0$-neighbourhood base of von Neumann-bounded sets

Let $X$ be a topological vector space, s.t. there exists a $0$-neighbourhood (sub-)base of von Neumann-bounded subsets. Do such spaces have a name?
4
votes
1answer
23 views

Weak and weak star topologies are locally convex

Why weak and weak star topologies are locally convex? I searched for a basis that the open sets at the origin consisting of convex set but I did'nt reach any result!
0
votes
0answers
21 views

Detecting (FH)-spaces

Let $H$ be a Hausdorff topological vector space. A completely metrizable topological vector space $X \subseteq H$ is an (FH)-space if the topology of $X$ is finer than that induced by $H$ (that is the ...
0
votes
1answer
59 views

Lebesgue measure as $\sup$ of measures of contained compact sets

I know, from Kolmogorov-Fomin's Элементы теории функций и функционального анализа, the definition of external measure of a bounded set $A\subset \mathbb{R}^n$ as $$\mu^{\ast}(A):=\inf_{A\subset \...
2
votes
2answers
53 views

A characterization of tempered distributions

The Schwartz space on $\mathbb{R}^n$ is the function space $$ S \left(\mathbf{R}^n\right) = \left \{ f \in C^\infty(\mathbf{R}^n) : \|f\|_{\alpha,\beta} < \infty,\, \forall \alpha, \beta\in\mathbb{...
1
vote
0answers
17 views

Conditions for non-normability of (nontrivial proper and nondense) subspace of non-normable space.

Lets $X$ be a non-normable topological vector space and let $Y\subset X$ be a proper subspace. Clearly if $Y$ is dense in $X$ then $Y$ must be non-normable too. Can we have that conclusion with weaker ...
3
votes
1answer
328 views

How can this theorem about weakly measurable functions on $\sigma$-finite measure spaces be deduced from the finite measure space case?

I am reading a theorem about measurability of vector-valued functions in a note on functional analysis: Theorem 3.6.1. If $X$ is a separable, metrizable locally convex space, $(\Omega, \Sigma, \mu)...
3
votes
0answers
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: ...
5
votes
1answer
93 views

Topological vector spaces book recommendation

I'm currently taking a class covering the theory of topological vector spaces using the book Topological Vector Spaces, Distributions, and Kernels by Francois Treves. I find the subject to be very ...
0
votes
1answer
17 views

Is the weak-* topology on a topological vector space Hausdorff?

Let $V$ be a topological vector space and $V^*$ be the space of linear functionals induced with the weak-* topology. Can we say that $V^*$ is Hausdorff? Here is my attempt: Let $\lambda\ne\lambda'\in ...
2
votes
1answer
64 views

Space of test functions defined by norms

This is the problem assigned: So I know that a locally convex Hausdorff space is defined by a vector space and a family of seminorms. So is part $a$ just wanting me to show that $\|\phi\|_m$ is in ...
2
votes
0answers
477 views

Convex hull of bounded set is bounded

I want to show that in a locally convex topological vector space $X$, the convex hull of a bounded set is bounded. Apparantly this does not hold if $X$ is not locally convex. So the fact that that ...
2
votes
0answers
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 ...
5
votes
1answer
128 views

Are countable strict inductive limits of Fréchet spaces always LF-spaces?

I would like to work with a slightly loser definition of an LF-space but am unsure what niceties I'm throwing away in the process. Let me provide a comparison of the conventional definition and my own ...
0
votes
1answer
16 views

Convex subsets of a topological vector space

I'm trying to prove: Let $X$ be a topological vector space and $A \subseteq X$. $A$ is convex if and only if $$\forall s,t \in \mathbb{R}_{+}, (s+t)A = sA + tA $$ Let $sx + ty \in sA + tA$. How ...
3
votes
1answer
64 views

Is the 3d Schwartz space isomorphic to a subspace of the 1d Schwartz space?

Are the Schwartz-spaces $\mathscr{S}(\mathbb{R})$ and $\mathscr{S}(\mathbb{R}^3)$ isomorphic (as topological vector spaces)? Is $\mathscr{S}(\mathbb{R}^3)$ at least isomorphic to a subspace of $\...
3
votes
1answer
195 views

On Absolutely Continuous Functions

I would like to know if we can extend the concept of absolute continuity to functions $f:[a,b]\to X$, where $X$ is a topological vector space. I browsed some books on Topological Vector Spaces but can'...
0
votes
2answers
215 views

Does “two topological vector spaces in duality” means they are each other's continuous dual?

When saying two topological vector spaces $E$ and $F$ are in duality, does it mean that they are each other's continuous dual, i.e. $E = F^*$ and $F=E^*$, or just that one is the other's continuous ...
8
votes
1answer
682 views

Sequential and topological duals of test function spaces

Given a test function space, in particular $\mathcal{S}=\mathcal{S}(\mathbb{R}^n)$ (the Schwartz space) or $\mathcal{D}=\mathcal{D}(\mathbb{R}^n)$ (the space of compactly supported smooth test ...
3
votes
1answer
35 views

Linear algebra with 2-dim. functions instead of matrices

I just thought about what would happen if we try to do matrix calculus with functions $\mathbb R^2 \to \mathbb R$ instead of matrices. The matrix multiplication would be something like $$ (f \times ...
0
votes
0answers
30 views

Difference between a Möbius Strip and a Simple Surface

I am trying to distinguish between a Möbius strip and a surface that has no separations, holes and a connected boundary (homeomorphic to a disk or a half-sphere). Since a Möbius strip also has all the ...
0
votes
1answer
26 views

$r:\mathscr{S'}\rightarrow\mathscr{D'}$, $u\mapsto u|_{\mathscr{D}}$ is not a topological embedding

Show that $r:\mathscr{S'}\rightarrow\mathscr{D'}$, $u\mapsto u|_{\mathscr{D}}$ is not a topological embedding. For this problem, would it suffice to construct a sequence $\{u_n\}$ in $\mathscr{D'}$ ...
8
votes
1answer
52 views

Does the operation of completion preserve injectivity?

It seems to me I saw a counterexample somewhere, but I can't find it, can anybody help me? Let $\varphi:X\to Y$ be a linear continuous map of locally convex spaces, and $\widetilde{\varphi}:\...
3
votes
1answer
45 views

Do tempered distributions form a topological subspace of the space of distributions?

I'm learning about distributions and tempered distributions. From what I understand, by "enlarging" the space of test functions $\mathcal{D}$ to the Schwarz space $\mathcal{S}$ and correspondingly "...
1
vote
0answers
24 views

Show that $\left.\dfrac{d}{d s}\right|_{s=0}X_\eta (m)(f\circ \varphi_{\exp(s\xi )})=(X_\eta f)_{*,m}(X_\xi (m))$

Let $G$ be a Lie group, $M$ be a manifold, $\varphi:G\times M \rightarrow M$ smooth action of $G$ over $M$ where $\varphi(g,m)=g\cdot m$. For each $\xi \in \mathfrak{g}=T_e G$ we difine the ...
1
vote
1answer
81 views

In the first countable TVS, if every Cauchy sequence convergence then every Cauchy net convergent

Let $X$ be a topological vector space with the first countable topology(that is, every point has a countable neighborhood basis).If every Cauchy sequence convergence, we want to show that every Cauchy ...
0
votes
0answers
17 views

Binary operations being continuous under a topology?

For a set $S$ and some function $f: S \rightarrow S$ and $a\in S$, $f$ is continuous at $a$ under a topology $N$ if for all neighborhoods $N_1(f(a))$ there exists a neighborhood $N_2(a)$ such that $f(...
1
vote
0answers
27 views

Show that $\|\cdot\|_Q$ is a seminorm on $l_{\infty}$

On the space $l^{\infty}$, define $||.||_Q=\limsup |x_n|,$ $x=(x_n)$ belongs to $l^{\infty}.$ Show that $||.||_Q$ is a seminorm on $l^{\infty}$ and that its null space is precisely $c_0$, where $c_0$ ...
1
vote
1answer
54 views

Intuition for separable spaces?

What should I think of when I read that a space is separable? Is it a 'nice' property? Why would I prefer a separable space over a non-separable space or vice versa? When I think of compact spaces I ...
1
vote
1answer
21 views

Close graph of multivalued function

Assume that $X,Y$ are two closed sets in $R^n$ (with the induced euclidian topology) and let $f:X \rightarrow Y$ be a multivalued function. Assume also that the graph of $f$ is a closed subset in $X \...
8
votes
2answers
290 views

Topological vector space with discrete topology is the zero space

Hello i have a question about topological vector spaces. To remind the definition of such a space: A topological vector space is a pair $(X,\tau)$ with $X$ a vector space and $\tau$ a topology on $...
0
votes
1answer
29 views

Weak and weak* topologies

If X is a locally convex vectorspace, does the weak and weak* topologies on X* coinside? If so how to prove it?
2
votes
1answer
103 views

Closed unit ball of $B(H)$ with wot topology is compact

The following is a Theorem of Conway's operator theory: I can not understand how he proves it. I think $\phi(\text{ ball B(H)})$ is compact if $\phi(\text{ ball B(H)})$ is closed subset of compact ...