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
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
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 ...
1
vote
1answer
54 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)|...
0
votes
2answers
93 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
284 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)\...
0
votes
0answers
45 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-...
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 ...
0
votes
0answers
36 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^{...
1
vote
0answers
13 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 $X$?
1
vote
1answer
34 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\...
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
37 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\...
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
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
61 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}\...
4
votes
1answer
87 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 $W'\subset W$....
0
votes
0answers
14 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
215 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$ ...
4
votes
0answers
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 $\infty$ ...
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
34 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 ...
3
votes
1answer
39 views

Infinite dimensional topological vectorspaces with dense finite dimensional subspaces

Consider $\mathbb R$ as a $\mathbb Q$ vector space. Using the usual metric on $\mathbb R$, we find: $\mathbb Q \subset \mathbb R$ is dense and one dimensional (indeed every non-zero subspace appears ...
1
vote
1answer
37 views

Closed subsets of $\mathbb{C}^*$ proper for multiplication

Let $S_1$ and $S_2$ be two proper closed subsets of $\mathbb{C}^*$. Let's denote by $\overline{S_1}$ and $\overline{S_2}$ their closure in $\mathbb{C}_{\infty}.$ (Alexandrov compactification) ...
0
votes
0answers
21 views

Dense convex set in $*$-weak topology

Let $X$ be a Hausdorff topological vector space over $\mathbb{K}$. Suppose $W$ is a convex subset of its topological dual $X'$. How to prove that if for any $x\in X\setminus\{0\}$ set $\{f(x):f\in W\}$...
1
vote
1answer
19 views

Bounded neighbourhood of zero in TVS

Is is true that in any topological vector space, which is $T_1$ there exists bounded neighbourhood of zero ? Is is still true if we omit $T_1$ axiom ?
0
votes
0answers
17 views

Space of polynomials as a continuous image of F-space

Let $X=\mathbb{R}[a,b]$. Is there any norm $\|\cdot\|$ on $X$ s.th. $X$ is a continuous image of some $F$-space. ($F$-space means that there exists complete metric s.th. $d(x+z,y+z)=d(x,y)$) ? My ...
9
votes
1answer
240 views

Why do we give $C_c^\infty(\mathbb{R}^d)$ the topology induced by all good seminorms?

Briefly, my question boils down to the following: What benefits do we gain from considering the space of test functions in the topology induced by all "good" seminorms, as opposed to other topologies ...
0
votes
0answers
80 views

When is a diffeomorphism analytic?

I've read somewhere "a class $C^\infty$ diffeomorphism is said to be analytic" but I forgot to write down where I read this and now I can not find it, which makes me wonder if it's true? I'm working ...
0
votes
0answers
50 views

Looking for an example of a bounded set.

Consider the local base over the space of complex continuous functions over $[0,1]$ (denoted by $\mathcal{C}[0,1]$) defined for each fixed $x\in [0,1]$ and $\epsilon>0$ by $$\mathcal{B}_{\epsilon,...
1
vote
1answer
65 views

6.5 theorem in Functional Analysis by Rudin about topological vector space and $\mathscr{D(\Omega)}$

I am reading the proof of 6.5 Theorem in Rudin's book. For part (c), he wants to prove that a bounded subset $E \subset \Omega$ must be contained in a $\mathscr{D}_{k}$ for some compact subset $K \...
0
votes
1answer
56 views

Is a sequentially compact (non-metrizable) uniform space totally bounded?

First some topological definitions in terms of nets and sequences: A topological space $(X, \tau)$ is compact iff every net has a convergent subnet sequentially compact iff every sequence has a ...
1
vote
1answer
38 views

Open sets in the unitary group $ U(\mathcal{H}) $ of a Hilbert space $ \mathcal{H} $.

Let $H$ be an infinite dimensional Hilbert space and let $(x_i)_1^\infty$ be an orthonormal basis for $H$. Consider $U(H)$ the unitary group of the continuous unitary operators on $H$. Equip $U(H)$ ...
0
votes
0answers
38 views

embeddings of TVS are continuous

How to show that the two embeddings of TVS are continuous: $C^{\infty}_c\subset S\subset C^{\infty}$? According to Wikipedia, the definition of "Continuously embedded" is that " one normed vector ...
0
votes
0answers
35 views

Check proof that the embedding of the unit ball $B\subset X$ into $X^{**}$ in weak-* dense

I have to prove the following theorem: Let $X$ be a (real) Banach space, and let $B$ denote its closed unit ball, and let $\tau (B)$ denote its canonical embedding into $B^{**}$, the closed unit ...
1
vote
5answers
658 views

Nonconstant linear functional on a topological vector space is an open mapping

In the middle of another proof (Theorem 3.4, p. 60) in his Functional Analysis book, Rudin says that "every nonconstant linear functional on $X$ (topological vector space) is an open mapping." Is ...
1
vote
0answers
44 views

Existence of linear continuous functional on locally convex TVS

Let $c>1$ and $X$ be a locally convex TVS. Take $a,b\in X$ and a closed subspace $Y$ of $X$ such that $Y\cap \{(1-t)a+bt \ | \ t\in\mathbb{R}\}=\emptyset$. How to prove that there exist $f\in X'$ ...
1
vote
1answer
120 views

$V$ is finite dimensional iff $V'$ with the weak topology is normable

Why is the following statement valid? Note, $V$ is locally convex Hausdorff topological vector space over $\mathbb{C}$ and $V'$ is the space of all continuous linear maps from $V \to \mathbb{C}$. $V$...
1
vote
1answer
48 views

Locally bounded topological vector spaces

Let ‎$ X $ ‎‎‎be a topological vector space ‎such that every neighborhood of zero contains an infinite dimensional subspace. Then ‎$ X $ is not locally bounded. I do not know, why? We know ‎$ X $ is ...
0
votes
2answers
79 views

Is Topological Space a Metric Space?

What's the correct relationship between these two spaces? I think that topological space is a metric space, since the open is defined by a metric such that $d(x, a) < \epsilon$.
3
votes
1answer
43 views

The weak topology on $H$ is the weak* topology on $H^*$ pulled back via $\Phi$

I'm reading the following in Analysis Now by Pedersen: The map, $H$ a Hilbert space $$\Phi:H\to H^*: x\mapsto(\cdot\mid x)=[y\mapsto (y,x)]$$ is a conjugate linear isometry. Then define the weak ...
1
vote
1answer
21 views

Does $\frac{1}{n}x_0$ converge to the origin in any topological vector-space?

Let $X$ be a topological $\mathbf{R}$-vector-space (not necessarily Hausdorff) and $x_0 \neq 0$ a non-zero element of $X$. Then intuitively the sequence $(\frac{1}{n} x_0)_{n \in \mathbf{N}}$ ...
1
vote
1answer
52 views

Complete as a semimetric space but not as a topological group

I shall begin with some definitions. 1) Suppose that $X$ is a topological (additive) group and $(x_{s})\subseteq X$ is a net, we said that $(x_{s})$ is Cauchy whenever $U$ is a neighbourhood of $0$, ...
0
votes
1answer
46 views

Proof of a property regarding weak-star topology

I am reading a book in which a theorem is only proven analogously in the situation of weak topology. The proof does not seem to work in the case of weak$^{*}$ topology. Let $X$ be a normed linear ...