Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 22H / c

Suppose first that $\Vert x_n-x_{\infty }\Vert \to 0$. The reverse triangle inequality givesso (i) implies (ii).

It also implies (iii). Given $ϵ>0$, choose $N$ so thatBy the Parseval identity for a Hilbertian basis, choose $I_0$ so that the basis tail of $x_{\infty }$ beyond $I_0$ is less than $ϵ/8$. For $n\ge N$ and $I\ge I_0$,There are only finitely many $n<N$, so increasing $I$ makes each of their tails less than $ϵ$. Thus one $I$ works for every $n$, proving (iii).

Assume (ii). Weak convergence givesUsing the inner product identity,Hence (ii) implies (i); this is the Radon-Riesz theorem in a Hilbert space.

Assume (iii), and let $\delta >0$. Choose $I$ such thatfor every $n$. Coordinate convergence implies, for every finite $J\ge I$,Letting $J\to \infty$ shows that the corresponding tail of $x_{\infty }$ is also at most $\delta$.

By the Parseval identity for a Hilbertian basis,The finite first sum tends to zero by weak coordinate convergence, while the second is at mostSince $\delta$ is arbitrary, $\Vert x_n-x_{\infty }\Vert \to 0$. Thus (iii) implies (i), completing the equivalence and proving the uniform basis-tail criterion for strong convergence.