Questions tagged [locally-convex-spaces]

Topological vector space with a locally convex topology, i.e. induced by a system of seminorms.

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16 votes
3 answers
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Why is multiplication on the space of smooth functions with compact support continuous?

I asked the question Why is multiplication on the space of smooth functions with compact support continuous? on M.SE sometime ago but I didn't receive a satisfactory answer. I was reading this ...
Hugo's user avatar
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14 votes
2 answers
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Are smooth functions tame?

I know the article of Hamilton on the inverse function theorem of Nash and Moser (with the same title) where he proves that $C^\infty(M)$ is a tame Fréchet space, when $M$ is closed or compact with ...
Matthias Ludewig's user avatar
9 votes
2 answers
988 views

Do Hausdorff locally convex inductive limits always exist?

The following is from Schaefer, "Topological Vector Spaces", 1999, p. 56/57: Let $(E_\alpha)_{\alpha \in A}$ be a family of locally convex spaces with $\alpha$ in a directed poset $A$ and $h_{\beta \...
yada's user avatar
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9 votes
2 answers
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Test functions with "wrong" topology not locally convex?

I didn't find it in any book, although it seems that this should be standard: Endow the space $C^\infty_c(\mathbb{R})$ of compactly supported functions with the inductive topology coming from the ...
Matthias Ludewig's user avatar
26 votes
2 answers
2k views

When is a locally convex topological vector space normal or paracompact?

All locally convex topological vector spaces (LCTVS) are completely regular, since their topology is given by a family of semi-norms. I'm interested in conditions that imply that a LCTVS is ...
Andrew Stacey's user avatar
18 votes
1 answer
2k views

Borel Lemma for vector-valued functions

The classical Borel Lemma states that for an arbitrary sequence $(v_n)_{n \in \mathbb{N}_0}$ of complex numbers there is a smooth function $f\colon \mathbb{R} \longrightarrow \mathbb{C}$ with Taylor ...
Stefan Waldmann's user avatar
11 votes
4 answers
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Example of noncomplete quotient of complete lcs mod closed subspace

The following statement is well-known: for a Fréchet space $V$ and a closed subspace $W \subseteq V$ the quotient $V / W$ is again complete and hence a Fréchet space. For the particular case of a ...
Stefan Waldmann's user avatar
10 votes
1 answer
639 views

Are functions of moderate growth a bornological space?

I was thinking a bit about distribution theory the last weeks and stumbled across the following question: There are two natural locally convex topologies on the space of smooth functions of moderate ...
Johannes Hahn's user avatar
8 votes
2 answers
350 views

Metrizability of a topological vector space where every sequence can be made to converge to zero

This is a follow-up to this answer. If $E$ is a (real or complex) topological vector space, we say that a sequence $\{x_n\}_{n=1}^\infty$ in $E$ can be made to converge to zero if there exists a ...
J. van Dobben de Bruyn's user avatar
7 votes
1 answer
741 views

An equivalent condition for separability of $X^*$

Let $X$ be a Banach space. By the weak operator topology on $B(X)$, we mean the locally convex topology implemented by the following semi-norms: $$B(X)\to[0,\infty) : T\to|\langle Tx,x^*\rangle|$$ ...
ABB's user avatar
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6 votes
1 answer
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Closed convex hull in infinite dimensions vs. continuous convex combinations

tl;dr: When is the closed convex hull of a set $K$ equal to the set of "continuous" convex combinations of $K$? I am essentially asking for the most general, infinite-dimensional analogue of ...
user163625's user avatar
4 votes
3 answers
644 views

$L^{\infty}$ as colimit

I recently read a result (in Jarchow's book) that any ultrabornological space can be expressed as a colimit (in the category LCS) of Banach spaces. My question is the following. Let $\mu$ be a ...
ABIM's user avatar
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4 votes
4 answers
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On Köthe sequence spaces

I asked this a week ago at math.stackexchange, but without success. As far as I understand, there are several meanings of the notion of the Köthe sequence space, in particular, Hans Jarchow in his "...
Sergei Akbarov's user avatar
4 votes
1 answer
238 views

Approximation of the identity by finite range functions in topological vector spaces

Let $X$ be a topological vector space. Assume that there exists a sequence $\phi_n:X\to X$ of finite range measurable functions with $\lim\phi_n(x)=x$ for every $x\in X$. Can we concluded there exists ...
ABB's user avatar
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4 votes
2 answers
524 views

Existence of a countable linear combination with positive coefficients

Consider a (Hausdorff and complete) locally convex topological vector space $V$ and a countable subset $(v_k)_{k=1}^\infty \subset V$ of non-zero vectors. $(*)$ Under what conditions on this subset ...
Igor Khavkine's user avatar
3 votes
2 answers
888 views

Topologies on space of compactly supported continuous functions

Let $X$ be a locally compact Hausdorff space. As far as I understand, the space $C_c(X) = C_c(X; \mathbb{C})$ of compactly supported continuous complex-valued functions on $X$ is (most?) often ...
epitaph's user avatar
  • 89
3 votes
0 answers
150 views

Elements of vector-valued $L^1$-spaces

Let $E$ be a complete locally convex space and let $(X, \Sigma, \mu)$ be a measure space where $\mu$ is a Radon measure. Then the space $L^{1}(X,E)$ is defined as a the completion of the space $S(X,E)$...
Christian's user avatar
  • 779
2 votes
1 answer
444 views

Weak convergence of probability measures on weak versus strong dual

The space of temperate distributions $S'(\mathbb{R}^d)$ is often equipped with the weak-$\ast$ or with the strong topology. When defining the notion of a probability measure on $S'(\mathbb{R}^d)$, ...
Abdelmalek Abdesselam's user avatar
2 votes
0 answers
70 views

Gluing together mixed normed vector spaces with mixed topologies

This is a variant of this question. Definitions/Facts $Ball_1(X)$ denotes the unit ball (about $0$) in a normed vector space $X$. MixTop of triples of pairs $(X,\tau)$ of normed vector spaces $X$ ...
ABIM's user avatar
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1 vote
0 answers
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Seminorms ported by a compact

Let $X$ be a Banach space, and $U\subset X$ an open balanced set. A seminorm $\rho$ in $\mathcal{H}(U)$ is said to be ported by a compact $K\subset U$ if for all open set $V$ such as $K\subset V \...
Nicolay Avendaño's user avatar
1 vote
0 answers
161 views

Example of locally convex space such that its weak and initial topology coincide

If $X$ is a normed space than it is well known that its norm topology and its weak topology coincide if and only if $X$ is finite-dimensional. Now I asked myself the same question about general ...
AlexE's user avatar
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1 vote
0 answers
96 views

Gluing together dense subset of Projective Limit in $Ban_1$

Let $(X_n,\pi_n^{m})$ be a countable projective system in the category Ban$_1$ of Banach spaces and short linear maps (is (continuous) linear constructions). Then (co)-completeness of this category ...
ABIM's user avatar
  • 5,019
0 votes
2 answers
403 views

Sufficient condition such that weak and initial topology coincide for a locally convex space

This is the opposite question to this one: Example of locally convex space such that its weak and initial topology coincide. If we have a normed vector space $X$ than its norm topology and weak ...
AlexE's user avatar
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