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      All Questions

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      3
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      1answer
      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
      2answers
      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
      6answers
      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
      3answers
      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
      2answers
      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
      0answers
      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
      1answer
      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
      1answer
      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
      0answers
      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
      1answer
      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
      1answer
      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
      1answer
      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
      2answers
      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
      1answer
      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
      1answer
      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\...

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