# All Questions

Tagged with fa.functional-analysis topological-vector-spaces

106
questions

**3**

votes

**1**answer

280 views

### Duality of Topological Vector Spaces

Let $K$ be a topological field. Let $\text{top-} K \text{-vect}$ be the category of topological $K$-vector spaces $V$, so that the maps $\cdot : K \times V \rightarrow V$ and $+ : V \times V \...

**0**

votes

**2**answers

218 views

### subspace topology and strong topology

Suppose $X$ is a locally convex space and $Y$ is a subspace of the strong dual of $X$, is the induced topology on Y equivalent to the strong topology $b(Y,Y')$ on $Y$? If this is not correct, then on ...

**8**

votes

**6**answers

400 views

### Open mapping theorem for complete non-metrizable spaces?

The classical open mapping theorem in functional analysis certainly holds in the Banach space setting, and this is where I first encountered it. Slightly more advanced textbooks (e.g. Rudin's ...

**2**

votes

**3**answers

156 views

### Every linear topological space embeds into the Tychonoff product of linear metric spaces

I need a reference to the following (known?)
Fact. Every topological vector space $X$ over the field of real numbers is topologically isomorphic to a linear subspace of the Tychonoff product of ...

**6**

votes

**2**answers

137 views

### Complete dual of bornological space

A bornologigal topological vector space is such that any bounded linear function on it is continuous. It is a standard result [Jarchow, Locally convex spaces, 1981] that if the dual $E'$ of a Mackey ...

**4**

votes

**0**answers

140 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 ...

**8**

votes

**1**answer

181 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**

votes

**1**answer

143 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}...

**1**

vote

**0**answers

100 views

### Mackey topology characterising property

Let $V$ be a topological $k$-vector space.
Let $V^{\star}$ denote the space of all linear functionals $V \rightarrow k$ and $V' \subset V^{\star}$ the subspace of all continuous linear functionals.
...

**3**

votes

**1**answer

71 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, ...

**3**

votes

**1**answer

89 views

### Is the compact-open topology on the dual of a separable Frechet space sequential?

Let $X$ be a separable Frechet space (= Polish locally convex linear metric space) and $X'_c$ be the space of linear continuous functionals on $X$, endowed with the compact-open topology (= the ...

**1**

vote

**1**answer

131 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 ...

**3**

votes

**2**answers

209 views

### On convergent sequences in locally convex topological vector spaces

Assume that a sequence $(x_n)_{n\in\omega}$ of points of a locally convex topological vector space converges to zero. Is it always possible to find increasing number sequences $(n_k)_{k\in\omega}$ and ...

**6**

votes

**1**answer

258 views

### Is restriction a closed map?

Originally asked on MSE.
Let $X$ be a normal (or even metrizable) topological space and let $Y$ be a closed subset of $X$. Let $C(X)$ be the linear space of all continuous scalar functions on $X$ ...

**1**

vote

**1**answer

110 views

### Compactness of operators and norming sets

Originally asked on MSE.
Let $T$ be a linear map from a normed space $E$ into a Banach space $F$.
Let $D\subset \overline{B}_{F^{\ast}}$ be norming, i.e., there is $r>0$ such that $\sup\limits_{v\...