So I start this by saying that I know very littler about topology and the related topics in this question, so this question may be trivial/ill-possed, etc.
I've read in many places that it is a well known result in functional analysis that a Banach space $X$ is reflexive if and only if the unit ball is weakly compact (compact in the weak topology).
So assume we have a real separable Hilbert space $H$ (this space is a reflexive Banach space), and consider the unit sphere $S=\{f\in H:\|f\|=1\}$, then $S$ is weakly compact.
I would like to use (some generalization of…) Weierstrass extreme value theorem to prove that a continuous function $F:S\to \mathbb R$ attains a maximum.
A little bit of background, here $H$ is a quantum Hilbert space, $S$ is the family of "quantum states". Then I wanted to prove that a (continuous) function of the quantum states will attain a maximum.
Is it possible to do something like this?
Thanks in advance!
Best Answer
If $F: S \to \Bbb R$ is continuous (where $S$ has the weak topology, and $\Bbb R$ the standard topology), a standard fact in topology is that $F[S]$ is also compact in $\Bbb R$ and by standard facts about the reals a compact set like $F[S]$ has a maximum and a mimumum. QED.
So a combination of a standard "compactness preserving" fact and properties of the reals. Both are easily shown from definitions.