# Two theorems of existence of a minimizer in a Hilbert space

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!

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.

• Thank you for the answer, I fixed the mistake in Theorem 2 (sorry, I missed). Why you say that "C is weakly closed"? Do you have a reference for theorems stated above?
– user603537
Commented May 26, 2021 at 9:54
• Close convex sets are weakly closed or weakly sequentially closed (which is easier to proof). These results are folklore and should be mentioned in every book on functional analysis.
– daw
Commented May 26, 2021 at 10:04
• daw, sorry, I have one more comment. It makes sense to apply this result choosing $H=\mathbb{R}$? If I have a continuous function, is the weak lower semicontinuity automatically satisfied?
– user603537
Commented May 27, 2021 at 5:21
• yes to both questions.
– daw
Commented May 27, 2021 at 6:54
• @User1010 yes, it holds in inf-dim spaces
– daw
Commented May 30, 2021 at 11:27