I am looking for a reference about some abstract theorems of existence of a minimum. I mean, in an abstract space (Banach or Hilbert), what are the theorems which guarantee the existence of a minimum for a function (functional)?
Could someone please give me some references?
Thank you in advance!
Hilbert and Banach spaces are particular topological spaces (where the topology is induced by the distance we obtain from the scalar product in the former case and from the norm in the latter). Hence any topological theorem that guarantees the existence of a minimizer works in Hilbert and Banach spaces:
(Extreme value theorem, topological version). Let ${X}$ be a compact topological space, and let $\mathcal{F}: X \rightarrow \overline{\mathbb{R}}$ be a lower semicontinuous functional. Then there exists $x \in X$ such that $$ \mathcal{F}(x)=\min _{X} \mathcal{F} $$ Note that in general topological spaces, the notion of compactness and sequential compactness are not related: you can find a compact set that is not sequentially compact and you can also find a sequentially compact set which is not compact. In any case, the previous theorem can be rewritten as follows, and the existence of the minimizer is still guaranteed:
(Extreme value theorem, sequential version). Let ${X}$ be a sequentially compact topological space, and let $\mathcal{F}: X \rightarrow \overline{\mathbb{R}}$ be a sequentially lower semicontinuous functional. Then there exists $x \in X$ such that $$ \mathcal{F}(x)=\min _{X} \mathcal{F} $$
In metric spaces (first-countable topological spaces is sufficient) the notions of compactness and sequential compactness coincide, as well as the notions of lower-semicontinuity and sequential lower-semicontinuity (henceforth lsc). In general this is not the case, but in any Hilbert and Banach space this is the case.
So, all boils down to prove the functional is lsc and the space is compact, which are two conditions that are "topologically in contrast": the easier for a functional to be lsc, the harder for the domain to be compact and viceversa (just think about open set definitions). What is usually done in Hilbert and Banach spaces is to show the functional is lsc with respect to the norm topology and the domain of the functional to be compact with respect to the weak topology (which is easier to prove since the topology is coarser, hence with fewer open sets). Then, all is about to prove, with some extra assumptions based on the problem, that the functional is also weakly-lcs (that is more difficult in general than proving strong lsc). Another condition that usually is stated to replace conditions on the domain $X$ is that the functional is coercive, giving what goes under the name of Tonelli's theorem.
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