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Results tagged with topological-vector-spaces
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user 61785
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.
1
vote
Topology of ${\mathcal D}(\Omega)$ (space of test functions)
There's a general principle for proving that a topology on a vector space $E$ is not a weak topology (in the general sense, a topology of the form $\sigma(E,F)$ for some $F \subseteq E^*$).
For $\sigm …
- 3,768
7
votes
Accepted
Equivalence of σ-convex hull and closed convex hull
Wlod AA gave a good counterexample for the case when $K$ is not required to be compact, here I give a counterexample $K$ compact, first in a locally convex space, and then for a(n infinite-dimensional …
- 3,768
14
votes
A topological vector space $X$ is separable if its dual space $X^*$ is separable?
YCor has given a counterexample for topological vector spaces. The statement is still false for locally convex spaces. Consider the space $X$ defined to be a locally convex coproduct of $\newcommand{\ …
- 3,768
3
votes
Complete dual of bornological space
Jochen is quite right. I have another example, just using any irreflexive Banach space $A$. The space $E = (A^*,\mu(A^*,A))$ is Mackey, by definition. The bounded sets in $E$ are the same as the norm- …
- 3,768
5
votes
Accepted
Is the compact-open topology on the dual of a separable Frechet space sequential?
Yes. In the next paragraph I will show that if $X$ is a Fréchet space (without requiring separability) then $X'_c$ with the compact-open topology is a $k$-space. As you note, this implies sequentialit …
- 3,768
9
votes
Accepted
Is any dual metrizable locally convex space a Frechet space?
I am not 100% clear what you are asking, but I will answer according to two interpretations:
a) Suppose that $F$, a metrizable TVS, is the strong dual of $E$, a locally convex TVS. Need $F$ be comple …
- 3,768