Weak convergence in a Hilbert space

A sequence $x_n$ converges weakly to $x$ when $⟨x_n,y⟩\to ⟨x,y⟩$ for every $y$ in the Hilbert space.

If $x_n\rightharpoonup x$ in a Hilbert space, some subsequence $(x_{n_k})$ has Cesaro means converging in norm to $x$. For $x=0$, choose the subsequence so successive vectors have summably small inner products; expansion of the squared norm of its averages then proves convergence.

For a Hilbertian basis $(e_i)$, a sequence $x_n$ converges weakly if and only if $(x_n)$ is norm bounded and every coordinate sequence $⟨x_n,e_i⟩$ converges.

Every bounded sequence in a separable Hilbert space has a weakly convergent subsequence. Successive subsequences make each basis coordinate converge, and a diagonal subsequence converges in every coordinate; the coordinate criterion for weak convergence in a separable Hilbert space finishes the proof.

If $x_n\rightharpoonup x$ in a Hilbert space, thenFor $x\ne 0$, take the inner product with $x$ and apply Cauchy-Schwarz before passing to the lower limit.

In a Hilbert space, weak convergence $x_n\rightharpoonup x$ together with $\Vert x_n\Vert \to \Vert x\Vert$ implies strong convergence $x_n\to x$.

Suppose $x_n\rightharpoonup x$ in a separable Hilbert space with Hilbertian basis $(e_i)$. Then $x_n\to x$ in norm exactly whenThe condition prevents norm from escaping to successively higher basis coordinates.

Every orthonormal sequence in a Hilbert space converges weakly to zero, by Bessel inequality, but no subsequence converges strongly because distinct terms remain distance $\sqrt{2}$ apart.

If $x_n$ converges weakly to $x$ in a Banach space, there are convex combinations of each tail $\{x_n,x_{n+1},\dots \}$ that converge in norm to $x$.

The weak closure and norm closure of a convex subset of a real or complex normed space coincide. If a point lies outside the norm closure, the Hahn-Banach separation theorem supplies a weakly continuous linear functional that separates it from the convex set.

A norm-closed convex subset of a Banach space contains every weak limit of its sequences. Apply Mazur lemma to obtain norm-convergent convex combinations that remain in the set.