Two theorems of existence of a minimizer in a Hilbert space

Question

I am looking for some results of existence of minimizers in infinite dimensional Hilbert spaces. I found (surfing the web) these two results:

${\bf Theorem 1.}$ Let $H$ be an Hilbert space and $C\subset H, C\neq\emptyset$ be a closed and convex subset. Let $J:H\to\mathbb{R}$ be a weakly lower semicontinuous functional. If $C$ is (also) bounded or $J$ is coercive, thus exists $\min_{u\in C} J(u)$.

${\bf Theorem 2.}$ Let $H$ be an Hilbert space and $C\subset H, C\neq\emptyset$ be a closed and convex subset. Let $J:C\to\mathbb{R}$ be a lower semicontinuous convex functional. If $C$ is (also) bounded or $J$ is coercive, thus $J$ is bounded from below and the minimum $\min_{u\in C}J(u)$ is attained.

I guess the Theorems states (almost) the same thing since:

$J$ lower semicontinuous and convex $\implies$ $J$ weakly lower semicontinuous.

The two differences I see are:

1. In the first case $J:H\to\mathbb{R}$, while $J:C\to\mathbb{R}$ in the second one, could anyone please explain me why?

2. In the second case, we are able to state that $J$ is bounded from below, what about the first case instead?

My question is: could someone please give me a detailed reference about these result? (better if it contains a proof).

Furthermore, there exists some conditions which ensure the convexity of a functional in a Hilbert space? I am interested in some characterizations.

I hope someone could help. Thank you in advance!

Answer 1

You forgot the assumption that $C$ is non-empty. In Theorem 2 some assumption is missing to get weak compactness ($C$ bounded or $J$ coercive).

1. It does not matter, how (or if) $J$ is defined outside of $C$. Since $C$ is weakly closed, everything happens in $C$.

2. If the minimum exists then $J$ is bounded below on $C$.

Examples of convex functions on Hilbert spaces: square of norm, functions with positive semi-definite second-order Frechet derivative, integrals of convex functions, etc.