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Results tagged with topological-vector-spaces
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user 18943
A topological vector space is a vector space $V$ over a topological field $\mathbb{K}$ (typically $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$), together with a topology on $V$ such that vector addition and scalar multiplication are both continuous. Hilbert spaces and Banach spaces are examples of topological vector spaces.
8
votes
Closed vector subspaces of large powers of R
Yes, that's true. This is called a space of minimal type (see. H.H.Schaefer p.191):
A locally convex space $X$ over $\mathbb R$ is isomorphic to some ${\mathbb R}^I$ ($I$ being a cardinal number) …
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3
votes
0
answers
114
views
Approximation of a linear functional by linear continuous functionals
Let $X$ be a locally convex space, $T$ a balanced convex compact set in $X$, and $f:X\to\mathbb{C}$ a linear functional which is (not necessarily continuous on $X$, but) continuous on $T$. It is not d …
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2
votes
Topological vector space textbook with enough applications
I would recommend you the book by Yu.I.Lyubich (an unfavourable Zentralblatt review is here, see comments below). It's a good introduction to functional analysis for people who are interested in appli …
4
votes
1
answer
289
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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5
votes
1
answer
209
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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3
votes
1
answer
156
views
$\varepsilon$-product in Bierstedt's paper
I am reading K.D.Bierstedt's paper Gewichtete Räume stetiger vektorwertiger Funktionen und das injektive Tensorprodukt. I. Journal für die reine und angewandte Mathematik 259 (1973): 186-210. It is wr …
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3
votes
1
answer
225
views
Is compact-open topology stable with respect to injective limits?
Let $X$ be a locally convex space, and $\{Y_i;\ i\in I\}$ a covariant system of locally convex spaces over a partially ordered set $I$, i.e. a system of linear continuous mappings is given $\sigma^j_i …
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4
votes
4
answers
745
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 "L …
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5
votes
2
answers
270
views
Is a set of nuclear functionals equicontinuous in compact-open topology if it is equicontinu...
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
$$
f(A)= …
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8
votes
1
answer
655
views
When does the dual to the space $K(X)$ of compact operators consist of nuclear functionals?
Let $X$ be a Banach space and $B(X)$ be its space of all (bounded) operators. A nuclear functional on $B(X)$ is a linear functional $u:B(X)\to{\mathbb C}$ that can be represented in the form
$$
u(A)=\ …
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4
votes
0
answers
147
views
A characterization of nuclear functionals in terms of continuity with respect to some specia...
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 wi …
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2
votes
Pontryagin-reflexivity of spaces of continuous functions
I would say that this is not well-known:
It is well-known that a Banach space $V$ is always Pontryagin-reflexive, i.e. the natural map $V\to \text{Hom}_\mathbb{R}(\text{Hom}_\mathbb{R}(V, \mathbb{R}) …
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