# Questions tagged [locally-convex-spaces]

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

125
questions

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### Weaker version of the Borel lemma for vector-valued functions

Borel's lemma for Frechét-spaces $V$ says:
(i) For every $(v_j)_{j \in \mathbb{N}} \in V^\mathbb{N}$ there exists an interval $I \subseteq \mathbb{R}$ containing $0$ and a smooth $f: I \to V$ such ...

**3**

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**1**answer

96 views

### Compatibility of inductive and projective limits with dualization in functional analysis

Assume $(A_i)_{i \in I}$ is a family of locally convex topological
vector spaces which are all moreover assumed to be Banach spaces.
We assume moreover that $(A_i)_{i \in I}$ has additional
structure ...

**2**

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**1**answer

50 views

### Subspaces of quasi-complete locally convex spaces

Let $V$ be a quasi-complete Hausdorff locally convex space. (By quasi-complete, one means that every bounded closed subset of $V$ is complete.) For a bounded closed absolutely convex subset $B$, ...

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**1**answer

64 views

### Are bounded sets in second duals of locally convex spaces weak* pre-compact?

Let $X$ be a locally convex Hausdorff space. Then $X$ injects into $X^{**}$ via the canonical map $\kappa: X\to X^{**}$. Now, $X^{**}$ carries the weak* topology. Let $B$ be a bounded set in $X$. Is $\...

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50 views

### Good source for Jordan Fréchet algebras

Is there any good source for Jordan Fréchet (or more generally, Jordan locally convex) algebras?
I'm looking for something on the level similar to the level of the book "Banach and Locally Convex ...

**3**

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**1**answer

75 views

### Characterizing 'very homogeneous' finitely valued stochastic processes

Fix a positive integer $n$. Let $X = \{X_i\}_{i \in \mathbb{N}}$ be a discrete time stochastic process such that each $X_i$ is a $\{0,\dots,n-1\}$-valued random variable. Suppose that the joint ...

**5**

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**1**answer

269 views

### 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 ...

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70 views

### Is the Vietoris topology on compact subsets of $\mathbb R^n$ locally convex?

The title question says it all really.
If the question is negative for compact subsets of $\mathbb R^n$, is it affirmative for compact and convex subsets of $\mathbb R^n$? How about for all nonempty ...

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**1**answer

64 views

### Ultrabornological representation for the space of uniformly continuous functions?

Let $\{\omega_i\}_{i\in I}$ be a non-empty set of increasing (not necessarily strictly) continuous functions preserving $0$. Then, for each $i \in I$ define the space
$$
C_{\omega_i}(\mathbb{R}^n,\...

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**1**answer

228 views

### Known dense subset of Schwartz-like space and $C_c^{\infty}$?

After reading this question, which asked for some examples of commonly used (proper) dense subsets of $C_0^{\infty}(\mathbb{R}^n)$ with the $L^p$-norm I wonder. What are some "well-known" ...

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184 views

### Are linear continuous mappings open on totally bounded sets?

Let $X$ and $Y$ be locally convex spaces, and $\varphi: X\to Y$ a linear continuous mapping. Suppose first that $S$ is a compact set in $X$. Then $\varphi$, being considered as a mapping from $S$ to $\...

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**4**answers

254 views

### 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 "...

**6**

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**1**answer

107 views

### Infra-Pták space that is not Pták

From reading the literature of the 1970s heyday of locally convex spaces, it seems that it was an important open question whether there is an infra-Pták (i.e. $B_r$-complete) space that is not Pták (i....

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170 views

### Useful notion for locally convex spaces - well known?

In my current work the following property of maps between locally convex spaces showed up at several places and proved to be useful. It seems quite elementary to me, so I would like to know whether it ...

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**2**answers

229 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 ...

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**1**answer

123 views

### Density and the projective tensor product

Let $X$ be a locally convex space (over $\mathbb{R}$), $D\subset X$ be dense, $B$ be a Banach space (again over $\mathbb{R}$) with Schauder basis $\{b_i\}_{i =1}^{\infty}$. Is the set
$$
D^+\...

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63 views

### Associated barrelled topology of norm topology on $C_c(X)$

Let $X$ be a locally compact Hausdorff space, $C(X; K)$ the Banach space of continuous functions on $X$ with support in $K$, for compact $K \subseteq X$, and $C_c(X) = \lim_K C(X; K)$ the locally ...

**1**

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**1**answer

118 views

### Continuous function on colimit

Let $X$ be a Banach space and $f:X\rightarrow \mathbb{R}$ be continuous. Suppose that $\{X_n\}_{n \in \mathbb{N}}$ is a strictly nested sequence of sub-Banach spaces, for which $\cup_{n \in \mathbb{N}...

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**3**answers

504 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 ...

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49 views

### Refinement: Can $L^1_{loc}$ be represented as colimit?

Let $\mu$ be a $\sigma$-finite measure on a measure space $(\mathbb{R}^d,\Sigma)$. Can $L^1_{\mu,loc}$ be represented as an injective-limit in the category of LCS (locally convex spaces and ...

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67 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 $X$ ...

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69 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 ...

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96 views

### Does every locally convex space with a Schauder basis have the approximation property?

For Banach spaces, the existence of a Schauder basis implies that this space has the approximation property.
Since both the notion of Schauder bases and of the approximation property are well ...

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45 views

### A different kind of weighted Hardy space

Let $\mathbb{D}$ denote the open unit disk in $\mathbb{C}$, let $\mathcal{A}\left(\mathbb{D}\right)$ denote the vector space of all complex-valued functions which are holomorphic on $\mathbb{D}$, and ...

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128 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 $S(X,E)$...

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**3**answers

313 views

### Is the strong topology of a locally convex space always barrelled?

For a locally convex space $E$ let $E_\beta$ be the space $E$ endowed with the locally convex topology $\beta(E,E')$ whose neighborhood base at zero consists of barrels, i.e., closed absolutely convex ...

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106 views

### A new topology on the dual of a locally convex space?

Working with the separable quotient problem for locally convex spaces we (with Saak Gabriyelyan) arrived to an interesting topology on the dual of a locally convex space and we would like to know if ...

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**1**answer

153 views

### Inductive limit commutes with topological tensor product

Consider $H \left(U \right); U \subset \mathbb{C}$ - space of holomorphic functions with compact-open topology. In this topology, this space is Montel, nuclear and Frechet. I want to take the ...

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48 views

### direct limit in locally convex modules and continuous map

Let we have short exact sequences of LCM over LC algebra $A$ with continuous linear maps
$$
0\to B_j\;{\xrightarrow {\ f_j\ }}\;C_j\;{\xrightarrow {\ g_j\ }}\;D_j\to 0.
$$
We can take inductive limit (...

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**0**answers

144 views

### A characterization of nuclear functionals in terms of continuity with respect to some special topologies on $B(X)$?

I think, nuclear functionals on the space of operators $B(X)$ (on a Banach space $X$) must have a characterization in terms of some special continuity. I would be grateful if somebody could help me ...

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**1**answer

276 views

### When does the dual to the space $K(X)$ of compact operators consist of nuclear functionals?

Let $X$ be a Banach space and $B(X)$ be its space of all (bounded) operators. A nuclear functional on $B(X)$ is a linear functional $u:B(X)\to{\mathbb C}$ that can be represented in the form
$$
u(A)=\...

**4**

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**1**answer

187 views

### When is a totally bounded set of an inductive limit contained in a component of this limit?

A. P. Robertson and W. Robertson in their "Topological Vector Spaces" VII, 1.4, (and H.Jarchow in "Locally convex spaces", 4.6, Theorem 2) prove the following proposition:
Let $E=\lim_{n\to\infty}...

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**2**answers

246 views

### Is a set of nuclear functionals equicontinuous in compact-open topology if it is equicontinuous on each compact set?

Let $H$ be a Hilbert space and $B(H)$ be its space of all (bounded) operators. A nuclear functional on $B(H)$ is a linear functional $f:B(H)\to{\mathbb C}$ that can be represented in the form
$$
f(A)=\...

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**1**answer

57 views

### Two notions of boundedness in metrizable topological vector space [closed]

In a metrizable topological vector space X with the metric d, a subset A is said to be bounded if it can be absorbed by any neighbourhood of 0 and a subset A is said to be d-bounded if its diameter ...

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**1**answer

181 views

### Extension of Valdivia-Vogt isomorphism from $\mathscr{D}(K)$ to $\mathscr{E}'(K)$

Let $M$ be a $d$-dimensional (say, Hausdorff, paracompact, connected and oriented) smooth manifold, and $K\subset M$ compact with $\mathring{K}\neq\varnothing$. M. Valdivia has shown (based on ...

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106 views

### Is the Baireness a 3-space property of topological groups

It is known that the product of two Baire spaces can be meager.
On the other hand, by a recent result of Li and Zsilinszky the product of two Baire spaces is Baire if one of the spaces is countably ...

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308 views

### Can smoothness of curves into a convenient locally convex vector space be tested with just a dense subspace of the dual?

Let $E$ be a (Hausdorff) locally convex vector space (from now on just "lcs" for short). We say that $E$ is convenient (also called locally complete, Mackey-complete or $c^\infty$-complete) if, given ...

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**1**answer

81 views

### Openness of invertibility in Fréchet spaces for families parameterized by compact spaces

Consider the following setup. Let $K$ be a compact topological space, $X$ a Fréchet space and $T:K \times X \to X$ a continuous family of linear maps (i.e. $T$ is a continuous map and $T_k \equiv T(k, ...

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**1**answer

70 views

### A property on some unbounded metric spaces

Suppose that $(X,d)$ is a metric spaces. Which condition(s) can guaranties the following property:
$\forall x, \forall y \in X, \exists \{z_n\}$ such that $\lim_{n\to +\infty } d(x,z_n)=+\infty$ and $\...

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**1**answer

186 views

### The completeness of spaces of continuous functions with the compact-open topology

For a Tychonoff space $X$ let $C_k(X)$ denote the space of continuous real-valued functions on $X$, endowed with the compact-open topology.
Problem. Is the space $C_k(X)$ Polish if it is Polishable ...

**5**

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**3**answers

434 views

### Continuity and sequential continuity of a linear functional

Let $E = C_c^0(\mathbb{R}^n;\mathbb{R}^m)$ be the space of compactly supported continuous functions on $\mathbb{R}^n$ with values on $\mathbb{R}^m$. There is a natural norm on this space: given $\...

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**1**answer

128 views

### $L^p$-spaces for locally convex spaces

Let $(X,\sigma)$ be a locally convex space, say generated by a family of seminorms $\mathfrak{P}$. I know that there is the notion of the space of integrable functions $f:\Omega\rightarrow(X,\sigma)$ ...

**4**

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**1**answer

96 views

### Hereditary Lindelöfness in $C_p$-spaces

Let $X$ be a (infinite) separable topological space and consider $C_p(X)$, the space of continuous functions on $X$ endowed with the point-wise convergence topology.
Q. I am looking for ...

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**1**answer

224 views

### Does separability of the strong operator topology imply separability of the underlying space?

Let $X$ be a Banach space and $B(X)$ be the space of bounded operators on $X$.
Suppose that the strong operator topology on $B(X)$ is separable and that the cardinal number of $B(X)$ is continuum.
...

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votes

**1**answer

176 views

### Relation between the weak star topology and hereditary Lindelöfness

Let $X$ be a Banach space. Is the following implication valid?
$$ (X,w) \textrm{ is hereditarily Lindelöf}~ \Rightarrow X^*~ \textrm{is separable} $$
The converse is clearly true, since the ...

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votes

**2**answers

246 views

### pointwise convergence to the identity

Let $X$ be a separable topological vector space with size (cardinal number) no larger than $\mathfrak{c}$. Does there exist any sequence of finite rank linear maps $\phi_n:X\to X$ pointwise converging ...

**4**

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**0**answers

98 views

### point-wise approximation of the identity in hereditary Lindelof spaces

Let $X$ be a topological vector space. Assume that there exists a sequence of finite range measurable functions $\phi_n:X\to X$ with $\lim\phi_n(x)=x$.
Q. Can we concluded that $X$ is hereditery ...

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242 views

### The second dual of $C(X)$ with the compact-open topology

Let $X$ be a compact Hausdorff space. Then $C(X)$ is a Banach algebra with the supremum norm and so is $A=C(X)^{**}$ under either Arens product. Moreover, it is easy to verify that $A\cong C(Z)$ for ...

**4**

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**1**answer

218 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 ...

**3**

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**1**answer

161 views

### Are second-countable subsets of topological vector spaces metrizable?

Let $X$ be a topological vector space of size $\mathfrak{c}$. Assume that there exists a countable union $X=\cup X_n$ such that all subsets $X_n$'s are relatively second countable.
Q. Does there ...