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Results tagged with fa.functional-analysis
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user 61785
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
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
12
votes
Accepted
Unconditionally convergent series in some functional spaces
A good resource for these things is Section IV.10 of Schaefer's Topological Vector Spaces, so you should look there for the proofs of the following statements. For $E$ a locally convex space, let $\el …
- 3,768
12
votes
Accepted
Is $X\times X$ homeomorphic to $X$ for a space of probability measures?
The answer is yes, following from Klee's extension of Keller's theorem on homeomorphisms of infinite dimensional compact convex sets:
Klee, V. L., Some topological properties of convex sets, Trans. Am …
- 3,768
10
votes
Accepted
Banach space with uncountable basis
I will pull together the comments into a community wiki answer with some of my own remarks so that the question isn't left on the unanswered questions list.
If you're willing to accept that it is cons …
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
9
votes
A characterization of $L_1(\mu)$ in $L_\infty(\mu)^*$
The criterion suggested in the question works fine for $\sigma$-finite spaces, and Michael Greinecker's answer is correct under this assumption.
However, the suggested criterion is not (provably) suf …
- 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
6
votes
About the category of von neumann algebras
I agree with Dmitri Pavlov that separability is not so important in the modern theory of von Neumann algebras, and this answers the second question. However, an example answering the first question ha …
- 3,768
6
votes
Accepted
Examples of non-isomorphic $C^\ast$ algebras with isomorphic quasi-state spaces
For any C$^*$-algebra $A$, we can define its opposite algebra $A^{\mathrm{op}}$, which is the algebra where $ab$ is defined to be $ba$, as calculated in $A$. Let's restrict to unital algebras for simp …
- 3,768
5
votes
Set of w*-continuous operators closed for the weak* topology or not?
The answer is no. I know that for some people here, saying "It's false for $X = \ell^1$" would be a good enough hint, but I also know that this question originated on Math StackExchange, so I've inclu …
- 3,768
5
votes
Duality between Banach spaces and compact convex spaces
Judging by what you say in the question, I think you are referring to what Świrszcz called "Saks spaces" in this article:
Monadic functors and convexity, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astro …
- 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
4
votes
Accepted
Sequential Continuity in dual spaces of separable Banach Spaces
This is true. To show it, in the following I will use $\langle \mbox{-}, \mbox{-} \rangle$ for the pairing between a space and its dual (with the vectors from the space on the left, the dual on the ri …
- 3,768
4
votes
Accepted
Baire category theorem for uncountable unions
The hyperstonean case can be dealt with using a result from Fremlin's Measure Theory. For every hyperstonean space $X$, we can find a semi-finite measure $\mu$ defined on the sets with the Baire prope …
- 3,768
3
votes
Non-separable metric probability space
Iosif Pinelis has given an answer to question 1 and partial answers to 2 and 3. Since he advised me to turn my comments into an answer, here it is.
I will deal with the case where the axiom of choice …
- 3,768