I'm here to ask for some help in solving the following exercise:
Exercise. Suppose that $X$ is $\mathbb{R}^N$, a Hilbert space or more in general a $C^2$ complete Hilbert manifold. Suppose that $f:X\to \mathbb{R}$ is a $C^2$ function such that $m=\inf f>-\infty$ and $(PS)_m$ condition holds. Let $Z_m=\{u\in X: f(u)=m\}$, which is thanks to the $(PS)_m$ a non-empty compact set. Prove that if $V\supset Z_m$ is an open set then exists a $c>m$ such that $f^c=\{u\in X: f(u)\leq c\}$ is contained in $V$.
I'm able to prove that in the case of a finite-dimensional $X$, since in that case we can suppose wlog that $\overline{V}$ is compact and then $T=\inf f\mid_{\partial V}>m$. Then I can proceed by contradiction and obtain that exists a $c>m$ such that $\inf \{||{grad f(u)|| : u\in f^c\setminus V}\}>0$ and build an omotopy as in the well-known deformation lemma in order to let all the points of $f^c\setminus V$ reach the boundary of $V$. Since I can build an omotopy such that for every $u\in f^c \setminus V$ we have that $f(H_u(t))=f(H(t,u))$ is a decreasing function, then the only possibility is that $f^T\subset V$, which concludes.
I can't understand how to get rid of the hypothesis that $\inf f\vert_{\partial V}>m$ in the case of a Hilbert space or a Hilbert manifold.
Any help is appreciated!