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

-1
votes
0answers
39 views

T/F: Properties of every non–trivial topological vector space over $\mathbb{R}$ or $\mathbb{C}$

Note: by "non–trivial" I mean "not discrete", which to the best of my knowledge is equivalent for a TVS over $\mathbb{R}$ or $\mathbb{C}$. Since any such space is over a local field, it is ...
2
votes
2answers
66 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 ...
3
votes
0answers
49 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 ...
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 ...
0
votes
0answers
69 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
1answer
22 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 ...
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_-\...
4
votes
1answer
22 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!
1
vote
1answer
21 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?
0
votes
0answers
20 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
58 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
51 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 ...
0
votes
1answer
16 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
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 ...
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
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
27 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 $\mathscr{D'}$ ...
3
votes
0answers
60 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: ...
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}:\...
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 ...
0
votes
0answers
15 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
26 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 $c_0$ ...
3
votes
1answer
44 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
1answer
48 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
19 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 \...
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?
0
votes
1answer
28 views

Show that, if we allow infinite intersection in the usual topology in R reduces to the discrete topology.

If X is a set and T is the collection of all the subsets of X, then all required of a topological space are automatically satisfied. This topology is called the discrete topology. Let X be the real ...
1
vote
1answer
53 views

Completeness of Schwartz space

I wanna to prove the completeness of Schwartz space $\mathscr{S}(R^{n})$ equipped with the induced topology from a set of seminnorms $$\|f(x)\|_{\alpha,\beta}=\sup_{x\in R^{n}}|x^{\beta}D^{\alpha}f(x)|...
1
vote
1answer
52 views

Topology :definition of neighbourhood of a point and basic questions

I am going through the basics of topology, in order to deal with topological vector space. I haven't taken any course of topology so I have some fundamental questions (I 've seen only some topological ...
0
votes
0answers
41 views

Definition of Banach Limits on $\ell^\infty$. Proof of Linearity and Continuity

I want to show that the Banach Limit $\Lambda$ on the set $\ell^\infty$ is a continuous linear functional in the dual space $(\ell^\infty)^\star$. I know that the Banach Limit exists, is left-...
0
votes
0answers
35 views

Minkowski functional on balance convex open neighborhood of 0

Let $X$ be a locally convex topological vector space, and suppose $C$ is a balanced, open, convex nbhd of $0$. I want to show that the Minkowski functional $\mu_C(x)=\inf(\lbrace t>0 \mid t^{-1}x \...
2
votes
0answers
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^{...
2
votes
1answer
70 views

Topology on the space of universally integrable functions

Let $X$ be a compact space. Let us call a function $f:X\to {\mathbb C}$ universally integrable if it is integrable with respect to each regular Borel measure $\mu$ on $X$ (i.e. a positive functional ...
1
vote
0answers
12 views

Nuclear Frechet space as inductive limit

Can a nuclear Frechet space also be defined as an countable inductive system of Banach spaces with nuclear maps?
1
vote
0answers
15 views

Existence of Banach space in which nuclear space embeds densely

If $N$ is a nuclear space, does there exist a Banach space $X$, s.t. $N$ embeds densely in $X$?
0
votes
0answers
10 views

Prove that $E ⊆ D_K$ is bounded if and only if $\{ \|\partial^\alpha\psi\|_{C(K)} : \psi\in E \}$ is bounded for every multiindex $\alpha$.

Let $K\in\mathbb{R^d}$ d be compact. Prove that $E ⊆ D_K$ is bounded if and only if $\{ \|\partial^\alpha\psi\|_{C(K)} : \psi\in E \}$ is bounded for every multiindex $\alpha$. $D_K$ is the space of ...
0
votes
2answers
92 views

Continuity of Minkowski functional in locally convex topological vector space

Let $X$ be a locally convex topological vector space over $\mathbb{R}$ or $\mathbb{C}$ and let $p_C(x)=\inf (\lbrace t>0 \mid t^{-1}x \in C\rbrace)$ be the Minkowski functional for an arbitrary ...
0
votes
1answer
35 views

Rudin's “Functional Analysis” theorem 6.5 [closed]

In the proof of part (a) of theorem 6.5 (pg. 139 of the first edition) he states that: Since $\mathcal{D}_k\cap W$ is open in $\mathcal{D}_k$, we have proved that $\mathcal{D}_k\cap V \in \tau_k$ ...
0
votes
0answers
27 views

The topology of $\mathbb{S}(\mathbb{R^d})$ induced by two different families of seminorms.

Let $\mathbb{S}(\mathbb{R^d})$ be the Schwartz class on $\mathbb{R^d}$ and define the following two families of seminorms.$$\rho_{\alpha\beta}(f):= \|x^\alpha\partial^\beta f\|_\infty \\\sigma_{\alpha\...
1
vote
1answer
33 views

A topology on $D(Ω)$ given by the seminorms $ρ_N (φ) := \max \{|∂^\alpha φ(x)| : x ∈ Ω\}$ is not complete.

Show that the topology on $D(Ω)$ given by the seminorms $ρ_N (φ) := \sup\{ |∂^ αφ(x)| : x ∈ Ω, |\alpha|\leq N\}$ is not complete for any nonempty open set $Ω ⊆ \mathbb R^d$. Where $\Omega\subseteq\...
1
vote
1answer
27 views

If $K$ and $K'$ are compact such that $K$ is contained in interior of $K'$, then show that $D_K$ is nowhere dense in $D_{K'}$.

Let $D_K$ be the space of all smooth functions on $\mathbb{R}^n$ which are compactly supported in $K$ for a compact $K$. If $K$ and $K'$ are compact subsets of $\mathbb{R}^n$ such that $K$ is ...
4
votes
1answer
59 views

The algebraic dual space of a TVS is complete

(Treves Exercise 5.4) Let $E$ be a TVS and $E^*$ its algebraic dual. Provide $E^*$ with the topology of pointwise convergence in $E$. A basis of neighborhoods of zero in this topology is provided by ...
2
votes
0answers
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}\...
7
votes
1answer
281 views

Why is multiplication on the space of smooth functions with compact support continuous?

I was reading Terence Tao post https://terrytao.wordpress.com/2009/04/19/245c-notes-3-distributions/ and i'm not able to prove the last item of exercise 4. I have a map $F:C_c^{\infty}(\mathbb R^d)\...
4
votes
1answer
85 views

Universal property of topology of uniform convergence

What kind of universal property does the strong dual topology on $X'$ have, for $X$ being a locally convex space. Is it possible to define $X'$ as the projective limit of the normed spaces $\mathcal{L}...
2
votes
1answer
40 views

For $W'\subset W$ with $W\in B,W'\in B'$ bases of topologies $T,T'$; we get $T\subset T'$

Let $A$ be a set with $T,T'$ topologies, and $B,B'$ bases for $T, T'$ respectively. 1 Suppose for all $a\in A$ and $W\in B$ with $a\in W$ there exists a $W'\in B'$ with $a\in W'$ and $W'\subset W$....