Past exam of the functional analysis II course of the University of Oxford / 2023 / 1 / b / ii

The orthogonal complement $x^{\perp }$ is infinite-dimensional: otherwise $X=\mathrm{span}\{x\}\oplus x^{\perp }$ would be finite-dimensional. Inductively choose an orthonormal sequence $(e_n)$ in $x^{\perp }$, each new vector being orthogonal to the preceding finite linear span. The Bessel inequality gives ${\sum }_n|⟨e_n,z⟩{|}^2\le \Vert z{\Vert }^2$ for every $z\in X$, so $⟨e_n,z⟩\to 0$. Thus $(e_n)$ is a weakly null orthonormal sequence.

SetBy orthogonality, $\Vert x_n{\Vert }^2=\Vert x{\Vert }^2+(1-\Vert x{\Vert }^2)=1$, while $⟨x_n-x,z⟩\to 0$ for every $z$. Hence $x_n\rightharpoonup x$. When $\Vert x\Vert =1$, this formula simply gives the constant sequence $x_n=x$. In particular, the weak closure of the unit sphere contains the whole closed unit ball.