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Let X be a Banach space and $J:X \rightarrow \mathbb{R}$ a functional. Show that J is weakly lower semi-continuous, if and only if the set $U_\alpha$ is weakly sequentially closed for any $\alpha$ $\in \mathbb R$ provided that it is non-empty.

$U_\alpha := \{w ∈ X : J(w) ≤ \alpha\}$


I am confused on how to use the definitions to proceed.

A functional $f : X \rightarrow \mathbb{R}$ is weakly lower semicontinuous on X if for all $x \in X$ and every sequence $x_n → x$ which converges weakly to x ∈ X, we have $$\liminf\limits_{x\rightarrow0} f(x_n) ≥ f(x)$$.


We need to use the following lemma:

Let $C\subset X$ be a closed and convex subset of a normed space $X$. Then C is weakly sequentially closed, i.e. for a sequence $(u^n )_n$ in $C$ with $u^n\rightharpoonup u$ in X for $n\rightarrow \infty$ we have that $u \in C$.

and also the fact that:

Let $X$ be a Banach space. If a functional $J : X \rightarrow \mathbb R$ is continuous and convex, then $U_\alpha$ is weakly sequentially closed for any $\alpha \in \mathbb R$.

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