Let $X$ a topological space.My book define : A functional $\varphi: X \rightarrow \Bbb{R}$ is lower-semicontinuous (l.s.c) if $\varphi^{-1}(a, + \infty)$ is open in $X$ for any $a \in \Bbb{R}.$ (1)
And the book tell : Given a Hilbert space $E$ and a functional $\varphi: E \rightarrow \Bbb{R}$, we recall that $\varphi$ is weakly l.s.c if it is l.s.c considering $E$ with its weak topology: in other words, $\varphi(\hat{u}) \leq \liminf \varphi(u_n)$ whenever $u_n$ converges weakly to $\hat{u}$. (2)
I was trying to use this : If $X$ satisfies the first axiom of countability then $\varphi: X \rightarrow \Bbb{R}$ is l.s.c if and only if $\varphi(\hat{u}) \leq \liminf \varphi(u_n)$ whenever $u_n$ converges weakly to $\hat{u}$.
but in this notes of De Figueiredo http://www.math.tifr.res.in/~publ/ln/tifr81.pdf he says in page 5 :
infinite dimensional Banach spaces X (even separable Hilbert spaces) do not satisfy the First Axiom of Countability under the weak topology.
Now i dont know what to do to prove (2) from (1). Someone can give me a help to prove? thanks in advance
The conclusion (1) $\implies$ (2) is already in Figueiredo's notes (Prop 1.2a), the converse result is not true in general and needs more assumptions.