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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 …

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 …

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 …

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 …

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 …

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{\ …

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 …

The answer is yes. First, since $\newcommand{E}{\mathcal{E}}\E$ is a Fréchet space, it is barrelled, and so any $\sigma(\E',\E)$-bounded subset of $\E'$ is equicontinuous, and therefore bounded in any …

The other answers very adequately explain why the norm topology is not Polish except for trivial cases, so this answer is about the weak-* topology. Also, most results in the literature are about the …

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- …

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 …

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 …

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 …

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 …

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 …