A minimising sequence need not converge strongly

Given: The Axiom of Choice; the real Hilbert space $H={\mathrm{\ell }}^2(\mathbb{N},\mathbb{R})$ (Square-summable families on an arbitrary index set and the space ${\mathrm{\ell }}^2(I)$), its coordinate vectors $e_k$, namely the family that is $1$ at $k$ and $0$ elsewhere, its closed unit ball $B=\{u\in H:\mathrm{\parallel }u\mathrm{\parallel }\le 1\}$, and the functional $I(u)={\sum }_{j\ge 0}{u}_j^2\mathrm{/}(j+1)$. The counting-measure dictionary ${\mathrm{\ell }}^p$ is the $L^p$ space of counting measure identifies $H$ with real $L^2(\mathrm{\#})$, which is reflexive (and thus Banach) by Reflexivity of Lp for one less p less infinity under Countable Choice, supplied here by AC; the series pairing is its inner product.

The series defining $I$ converges absolutely for every $u\in H$ because $0\le u_j^2\mathrm{/}(j+1)\le u_j^2$ and ${\sum }_j{u}_j^2=\mathrm{\parallel }u{\mathrm{\parallel }}^2<\mathrm{\infty }$; moreover $\mathrm{\mid }I(u+v)-I(u)\mathrm{\mid }\le 2\mathrm{\parallel }u\mathrm{\parallel }\mathrm{\parallel }v\mathrm{\parallel }+\mathrm{\parallel }v{\mathrm{\parallel }}^2$, so $I$ is continuous, and $I$ is convex and nonnegative (Square-summable families on an arbitrary index set and the space ${\mathrm{\ell }}^2(I)$).

Each coordinate vector satisfies $e_k\in H$, $\mathrm{\parallel }{e}_k\mathrm{\parallel }=1$ and $e_k\rightharpoonup 0$: by the duality of ${\mathrm{\ell }}^p$ and ${\mathrm{\ell }}^q$ every bounded linear functional on ${\mathrm{\ell }}^2$ is $\mathrm{\Lambda }(a)={\sum }_k{a}_k{b}_k$ for a unique $b\in {\mathrm{\ell }}^2$ (Counting measure specializes the representation theorem to ${\mathrm{\ell }}^p$ and ${\mathrm{\ell }}^q$), so $\mathrm{\Lambda }(e_k)=b_k$, and $b_k\to 0$ because a square-summable family has small tails, that is, for every $\epsilon >0$ there is a finite $F$ with ${\sum }_{k\mathrm{\notin }F}\mathrm{\mid }{b}_k{\mathrm{\mid }}^2<\epsilon$ (Square-summable families on an arbitrary index set and the space ${\mathrm{\ell }}^2(I)$), whence $\mathrm{\mid }{b}_k{\mathrm{\mid }}^2<\epsilon$ for every $k$ beyond all elements of $F$; weak convergence means convergence against every bounded linear functional (Weak convergence of nets and sequences).

In the direct method the compactness recovered is weak sequential compactness: a norm-bounded sequence in a reflexive Banach space has a weakly convergent subsequence (A bounded sequence in a reflexive Banach space has a weakly convergent subsequence), and the limit need not be a strong limit (The direct method in a reflexive Banach space, Proper, coercive and weakly lower semicontinuous extended-real functionals).