While reading the well known book *Minimax Methods in Critical Point Theory with Applications to Differential Equations* by Paul Rabinowitz, in the proof of a generalisation of the Mountain Pass Theorem (Theorem 5.29 in the book), I encountered the following abstract result:
>Let $E$ be a real Hilbert space and consider $b\in C^1(E, \mathbb{R}$) such that $b'$ is compact. Then, $b$ is weakly continuous, i.e. if $(u_n)_n\subseteq E$ converges weakly to $u\in E$, then $b(u_n)\to b(u)$ as $n\to \infty$.

The reference for this result given in the book is 
> M. A. Krasnoselski, *Topological methods in the theory of nonlinear integral
equations*, Macmillan, New York, 1964.

However, even if this is a well renowned book in the field of nonlinear analysis, I do not have access to it. Does anyone know a modern reference for this result? The proof looks nontrivial to me, at least I do not know how to approach it. 

EDIT: The fact that $b'$ is compact means that if $A\subseteq E$ is bounded, then the closure of $b'(A)$ is compact. Of course, $b'$ denotes the mapping $x\mapsto b'(x)$, i.e. $b'$ associates with each $x\in E$ the Frechet derivative of $b$ at $x$, which we denote by $b'(x)\in E^{*}$.