A norm-closed nonconvex set need not be weakly closed

Given: The Axiom of Choice; the real Hilbert space $H={\mathrm{\ell }}^2(\mathbb{N},\mathbb{R})$ (Hilbert space, Square-summable families on an arbitrary index set and the space ${\mathrm{\ell }}^2(I)$), its coordinate vectors $e_j$, the families that are $1$ at $j$ and $0$ elsewhere, and its unit sphere $S=\{u\in H:\mathrm{\parallel }u\mathrm{\parallel }=1\}$. 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 unit sphere of an infinite-dimensional Hilbert space is norm closed and norm bounded; its weak closure is the closed unit ball (Weak closure of the unit sphere is the closed unit ball).

Each coordinate vector satisfies $e_j\in H$, $\mathrm{\parallel }{e}_j\mathrm{\parallel }=1$ and $e_j\rightharpoonup 0$: by the duality of ${\mathrm{\ell }}^p$ and ${\mathrm{\ell }}^q$ every bounded linear functional on ${\mathrm{\ell }}^2$ is $\mathrm{\Lambda }(a)={\sum }_j{a}_j{b}_j$ for a unique $b\in {\mathrm{\ell }}^2$ (Counting measure specializes the representation theorem to ${\mathrm{\ell }}^p$ and ${\mathrm{\ell }}^q$), so $\mathrm{\Lambda }(e_j)=b_j$, and $b_j\to 0$ because a square-summable family has small tails, that is, for every $\epsilon >0$ there is a finite $F$ with ${\sum }_{j\mathrm{\notin }F}\mathrm{\mid }{b}_j{\mathrm{\mid }}^2<\epsilon$ (Square-summable families on an arbitrary index set and the space ${\mathrm{\ell }}^2(I)$), whence $\mathrm{\mid }{b}_j{\mathrm{\mid }}^2<\epsilon$ for every $j$ beyond all elements of $F$; weak convergence means convergence against every bounded linear functional (Weak convergence of nets and sequences). Hence the sequence $(e_j)$ lies in $S$ and converges weakly to the origin, which is not in $S$.

A norm-closed convex set is weakly closed under the Axiom of Choice (A norm-closed convex set is weakly sequentially closed); the direct method's admissibility requirement is weak sequential closedness (The direct method in a reflexive Banach space).