$\def\|{\mmlToken{mo}[stretchy="false"]‖}$ Let $X$ be a real Hilbert space with the inner product $⟨⋅, ⋅⟩$.

Let $K$ be a non-empty closed convex subset of $X$ and suppose $F: K → ℝ$ is a nonlinear functional that satisfies the following conditions: Prove that the infimum of the function $F$ over the set $K$ is attained, i.e., there is a point $x_0$ in $K$ such that $F(x_0) ⩽ F(x)$ for all $x ∈ K$.

We let $d=\inf_{x ∈ K} F(x)$. Let $x_n ∈ K$ be a minimising sequence, i.e, $F(x_n) → d$. By the property (iii), $\sup\|x_n\|<∞$. In a Hilbert space, a bounded set is sequentially weakly compact and there is a subsequence, denoted again by $x_n$, that converges weakly to $x_0$ in $X$. By the result of question (a), we can find a subsequence $x_{n_k}$ such that \[ y_m=\frac{1}{m} \sum_{k=1}^m x_{n_k} → x_0 . \] By convexity of $K, y_m ∈ K$ and thus $x_0 ∈ K$. By convexity of $F$, \[ d ⩽ F(y_m) ⩽ \frac{1}{m} \sum_{k=1}^m F(x_{n_k}) → d . \] So, we have $F(y_m) → d$ as $m → ∞$. By continuity of $F$, we find $F(y_0)=d$.