# 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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### 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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264 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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votes

**1**answer

70 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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88 views

### A sufficient condition for subdifferentiability

Corollary 9 in here (page 31) states that a proper convex function $g:Y\rightarrow \mathbb{R}\cup\{\infty\}$ (not necessarily continuous) on a locally convex space $Y$ is subdifferentiable on a point $...

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vote

**1**answer

66 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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vote

**1**answer

116 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 ...

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**3**answers

146 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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46 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**

votes

**1**answer

82 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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votes

**1**answer

137 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

154 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 ...

**4**

votes

**2**answers

190 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 ...

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84 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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211 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**

votes

**1**answer

210 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 ...

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**1**answer

140 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 ...

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107 views

### A point concerning Fremlin's example on Borel sets in non-separable Banach spaces

Let $E$ be a Banach space. Let us consider the following three sigma algebras on $E$.
$~~~~\mathcal{B}$= The $\sigma$-algebra coming from the norm topology.
$~~\mathcal{M}$= The sigma algebra ...

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votes

**1**answer

132 views

### A particular separation example

Q1. Does there exist a separable Banach space $X$ satisfying in the following property?
1- $X^*$ is non separable.
2- For every countable subset $F\subset X^*$ there exists $0\neq x_F\in X$ ...

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364 views

### Point-wise limit of finite valued functions

Let $X$ be a second countable topological vector space. Does there exist any sequence of finite valued functions $f_n\colon X\to X$ converging point-wise to the identity mapping on $X$?

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196 views

### Separable Lindelöf locally convex spaces that are not second-countable

A Lindelöf space is a topological space in which every open cover has a countable subcover.
Does there exists a Lindelöf locally convex space which is not second countable?
I am also looking for a ...

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votes

**1**answer

197 views

### A particular example of topological vector spaces

I am looking for a topological vector space $(X,\tau)$ enjoying the following conditions:
1- $(X,\tau)$ is not locally convex.
2- There exists a metric $d$ on $X$ and a sequence $\{X_n\}$ of ...

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votes

**3**answers

858 views

### Is the strong operator topology metrizable?

Let $X$ be a separable Banach space. Is the strong operator topology metrizable on $B(X)$, the space of all bounded operators on $X$?
SOT-$\lim T_i=0~$ if and only if $~\lim \|T_ix\|=0$ for every $x\...

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53 views

### Given projection PP is weak-star to weak-star continuous [closed]

Let $X$ be a Banach space, and $P$ be a projection of $X^*$ such that $\Vert P \Vert\leq 1$ and $\ker(P)$, ${\bf ran}(1-p)$ are weak-star closed. Show that $P$
is weak-star to weak-star continuous.
...

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314 views

### Does $End(V)$ remember $V$, where $V$ is a locally convex space?

Let $V$ be a locally convex topological vector space over $\mathbb C$, and let $A=\mathrm{End}(V)$ be its algebra of continuous linear endomorphisms (viewed just as a $\mathbb{C}$-algebra, not as a ...

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304 views

### Dual of representation is irreducible implies the representation is irreducible?

Suppose $V$ is an infinite dimensional $\mathbb{Q}_p$-representation of Lie algebra $\mathfrak{g}$ over $\mathbb{Q}_p$. If its dual representation $V^{\prime}$ is irreducible then, is it always true ...

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81 views

### On compactness of composition of a non-compact operator with a projection

Let $E$ be a locally convex space, and let $F$ be a complete locally convex space (which is not Banach, not separable), so we can consider $F$ as a projective limit of Banach spaces, that is, $F=\text{...

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votes

**1**answer

131 views

### Recognizing locally convex spaces on which all bounded linear functionals are continuous

Is it possible to characterize the Hausdorff locally convex spaces on which all bounded linear functionals are continuous?
It is known that a space is bornological if and only if the space is Mackey ...

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383 views

### Who first defined locally convex topological vector spaces?

Who first defined the class of locally convex topological vector spaces?

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90 views

### When is the strict topology bornological?

Let $X$ be a completely regular Hausdorff space. Are there known conditions under which the algebra of bounded continuous functions on $X$, endowed with the strict topology, is bornological?
(Of ...

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112 views

### Continuity under various topologies for positive linear functionals

It is known that if $\mathcal A$ is a unital $\mathbb C$-$*$-algebra and $A$ is a unital subalgebra closed under $*$, and if $f : A \to \mathbb C$ is linear, then $f$ is positive if and only if $f$ is ...

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315 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:
$$B(X)\to[0,\infty) : T\to|\langle Tx,x^*\rangle|$$
...

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56 views

### Working in coordinates with topologies on the algebra of continuous functions

Let $X$ be a Hausdorff completely regular topological space, and let $C_b (X)$ be its algebra of continuous bounded functions. Endow $C_b (X)$ with a topology given by some seminorms, that contains ...

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57 views

### An equivalent condition for second countable locally convex spaces

Let $(X,\tau)$ be a locally convex topological vector space.
Assume for any arbitrary topological base $\mathcal{E}$ of $\tau$ we have that: the Borel sigma algebras coming from $\mathcal{E}$ and $\...

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36 views

### Analogues of properties (DN) and (Ω) for more general locally convex spaces

In the structure theory of Fréchet spaces, especially results around splitting short exact sequences, the properties (DN) and (Ω) play a major rôle. There are many variants, but they are phrased in ...

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196 views

### Are separable F-spaces (completely metrizable topological vector space) homeomorphic to $l_2$?

An F-space is a completely metrizable topological vector space, i.e. the vector topology is induced by a complete metric. A Fréchet space is, by definition, a locally convex F-space.
It is known that ...

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votes

**1**answer

419 views

### Fixed point of a group action

Let $\mathbb{R}^\infty$ be the product of countably many real lines.
Assume that a finitely generated group $\Gamma$ acts on $\mathbb{R}^\infty$ (linearly and continuously) and there is a nonempty ...

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123 views

### A strict directed colimit of Hausdorff locally-convex spaces that is not Hausdorff

We work in the category of locally-convex spaces (morphisms are the continuous linear maps). Let $\Lambda$ be a directed set, for every $\lambda \in \Lambda$ let $V_{\lambda}$ be a locally-convex ...

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**1**answer

137 views

### The sheaf of generalized functions on compact subsets

For $K\subseteq \mathbb{R}^d$ compact, let $C_{\mathrm{c}}^{\infty}(K)$ denote the space of smooth functions on (an open neighborhood of) $K$ with compact support contained in $K$ with the usual ...

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votes

**1**answer

84 views

### Is there any dual relationship between quasi-completeness and barrelledness?

In the theory of stereotype spaces, it is known that for a locally convex space $X$,
If $X$ is pseudocomplete, then $X^{\star}$ is pseudosaturated, and
If $X$ is pseudosaturated, then $X^{\star}$ ...

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182 views

### Tensors and Nuclear/Fredholm Operators

For a locally convex Hausdorff spaces $E$, consider the canonical map
$$\overline{\psi}:E^\prime \hat{\otimes}_\pi E \longrightarrow L(E_\sigma)$$
that maps the projective tensor product to the space ...

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**1**answer

194 views

### Strictly finer bornological topology on Hilbert space

Question: Let $E$ be a Hilbert space. Can there exist a strictly finer bornological topology on $E$?
The background to my question is as follows. I am looking at locally complete, locally convex ...

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**0**answers

68 views

### Non-B-completeness of finest locally convex topology

For an index set $A$ consider the locally convex direct sum $X_A := \bigoplus_{\alpha \in A} \mathbb{R}_\alpha$ of $|A|$-many lines $\mathbb{R}_\alpha = \mathbb{R}$. Then $X_A$ is complete. It is ...

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votes

**1**answer

120 views

### Does the Banach algebra of jets have the approximation property?

To formulate my question I need the construction of the algebra $J^n_M(K)$ of jets of degree $n$ on a compact set $K$ of a smooth manifold $M$. I'll describe it for the simplest case of $M={\mathbb R}$...

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votes

**1**answer

115 views

### A variant of the approximation property?

The following looks like a strengthening of the approximation property, but I don't know, maybe this is equivalent. I would be grateful if somebody could explain this.
Let $X$ be a Banach space with ...

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**2**answers

360 views

### The topology of pointwise convergence with the adjoint operator on a von Neumann algebra

Let $H$ be a Hilbert space and ${\mathcal A}$ a von Neumann algebra in $B(H)$. Let us endow ${\mathcal A}$ with the topology of pointwise convergence with the Hermitian adjoint operator, i.e. a net $...

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225 views

### Admissible relations in a Banach algebra

Suppose that $\mathbb{C}\left\langle x, y \right\rangle = R$ is a free (associative and unital) algebra and $f \in R$. I wonder whether there exists a (unital) Banach algebra $A$ and a non-zero pair $...

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408 views

### Are bounded sets always weakly metrizable in reflexive separable spaces?

It is known that if a Banach space is reflexive and separable, its unit ball is weakly metrizable.
My question is about the generalization of this property :
1) Is it true that for all reflexive ...

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votes

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773 views

### Closed convex bounded sets are weakly compact for which spaces?

It is known that for all reflexive Banach spaces, closed convex bounded sets are weakly compact (compact for the weak topology).
What is the general class of topological vector spaces for which this ...

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**1**answer

217 views

### Is the equicontinuous weak-star topology locally convex on the dual of an LF-space?

The Banach-Dieudonné theorem states that if $X$ is a metrizable locally convex Hausdorff space then the equicontinuous weak-* topology $ew^*$ on $X'$ coincides with the topology of precompact ...

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849 views

### 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 satisfatory answer.
I was reading this post ...