http://books.google.at/books?id=9OSrV73a40gC&pg=PA45&lpg=PA45 gives a general Weierstrass theorem. Are there notable applications of this theorem, say in the calculus of variations? (I could not find any in that book.)
The theorem is as follows (slight variation):
Let $X$ be a topological space, let $f : X \to \mathbb{R}$. Assume that $f$ is i) lower semi-continuous and ii) lower semi-compact. Then $f$ assumes its minimum on $X$.
Here, i) means that the sublevel sets $\{ x \in X : f(x) \leq \alpha \}$, $\alpha \in \mathbb{R}$, are closed, and ii) means that they are compact.
Afaik, a standard way of showing existence of minimizers is via sequential closedness and sequential compactness, in the weak or weak-* topology, which is not a special case of the above theorem, or at least not obviously so to me.
