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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3
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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 ...
2
votes
1answer
187 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 ...
0
votes
2answers
212 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
647 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
0answers
45 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: ...
2
votes
1answer
32 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
22 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
50 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 ...
3
votes
1answer
36 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
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0answers
20 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
74 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
14 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 ...
1
vote
0answers
23 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
42 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
15 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 ...
7
votes
2answers
264 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
26 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
87 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 ...
0
votes
1answer
25 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
48 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 ...
1
vote
1answer
42 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 ...
-2
votes
0answers
21 views

How to show the following vector field is not left invariant

Suppose we have a Lie group G=(R,+), which is just the real line with addition as the group operation. Is the vector field $x\partial_x$, where $x\in G$, is not left invariant? How to show that? ...
0
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0answers
47 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 space subspace ...
0
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0answers
14 views

What methods can I use to determine if the the algebraic and continuous duals of a space are equivalent?

This sort of a general question; I'm having a bit of trouble understanding how to show when a given infinite-dimensional space has equivalent algebraic and continuous duals, and in contrast, when it ...
0
votes
2answers
62 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 ...
7
votes
1answer
266 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 ...
0
votes
0answers
31 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 ...
2
votes
1answer
68 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 ...
0
votes
0answers
30 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
30 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 ...
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vote
0answers
11 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?
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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 $X$?
1
vote
1answer
30 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 ...
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
1answer
34 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
26 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 ...
0
votes
0answers
63 views

Linear Application that is open in a TVS

Let $T: E \to F$ be a linear map between topological vector spaces $E$, $F$. If for each nonempty open set $G$, the interior of $T(G)$ is non-empty, then, $T$ is open. Proof: $$\mathrm{Int}(T(G))= ...
1
vote
1answer
26 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
56 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 ...
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 $b_k$ ...
4
votes
1answer
81 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 ...
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 ...
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0answers
13 views

A question on Cauchy filters

Let $\mathfrak{F}$, $\mathfrak{G}$ and $\mathfrak{G_1}$ be Cauchy filters in a uniform vector space $(X, \mathfrak{U})$. Let $c\in \mathbb{C}$. (1) If $\lim (\mathfrak{G}- \mathfrak{G}_1) = 0$, can ...
3
votes
1answer
200 views

Continuous inclusions in locally convex spaces

Let $(X, \left \| \cdot \right \|_X )$, $(X, \left \| \cdot \right \|_Y)$ two normed vector spaces with $X \subset Y$, by definition we have $X \hookrightarrow Y$ if $\left \| x \right \|_Y \leq C ...
3
votes
0answers
49 views

What dimensions permit a cross product [duplicate]

A cross product is possible in a $3$D and in a $7$D system. What prevents a cross product from being possible in a vector system of higher number of dimensions? For instance $15$D or $2^n$$-1$ ...
3
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0answers
37 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 $\infty$ ...
0
votes
1answer
28 views

Show that $\{(x,y) \in \mathbb{R}^2 : |x|+|y|=2\} \subset \partial A$

Let $A= \{(x,y) \in \mathbb{R}^2 : |x|+|y|<2\}$. Show that $\{(x,y) \in \mathbb{R}^2 : |x|+|y|=2\} \subset \partial A$. As all norm are equivalents in $\mathbb{R}^2$, it is reasonable the use ...
0
votes
0answers
20 views

Distributions on compact and semi-open intervals

In the theory of distributions (aka generalized functions), one considers mostly distributions $T \in \mathcal{D}(\Omega)$ on an open subset $\Omega \subseteq \mathbb{R}^n$. Hereby, the space ...
0
votes
0answers
32 views

Is a strongly holomorphic function automatically continuous?

I haven't found any sources that state this explicitly for arbitrary topological vector spaces (most sources are concerned with Frechet spaces, where even weakly holomorphic implies continuous). I'm ...