Let $(x_n)_{n=1}^{\infty}$ and $(y_n)_{n=1}^{\infty}$ be sequences in the closed unit ball in a Hilbert space. Prove that if $\langle x_n,y_n \rangle→1$ then $\lVert x_n−y_n \rVert→0$.
(From the Fundamentos de análise funcional by Botelho Pellegrino.)
Can I say from the assumption that it is an orthogonal basis within the Hilbert space? Also, how does the other assumption about the closed unit ball help me?
Best Answer
Sine you always have $(\forall n\in\Bbb N):\langle x_n,y_n\rangle\leqslant\|x_n\|\|y_n\|\leqslant1$ and since $\lim_{n\to\infty}\langle x_n,y_n\rangle=1$, you have $\lim_{n\to\infty}\|x_n\|\|y_n\|=1$. It follows from this and from the fact that you always have $\|x_n\|,\|y_n\|\leqslant1$ that$$\lim_{n\to\infty}\|x_n\|=\lim_{n\to\infty}\|y_n\|=1.$$And then\begin{align}\lim_{n\to\infty}\|x_n-y_n\|^2&=\lim_{n\to\infty}\|x_n\|^2+\|y_n\|^2-2\langle x_n,y_n\rangle\\&=1+1-2\\&=0.\end{align}