Let $\{x_n\}_{n∈N},\ \{y_n\}_{n∈N}$ be two orthonormal sequences in an Hilbert space $H$. Assume that $\lim_{n→+∞} \langle x_n, y_n \rangle_H = 1$. Prove that $\lim_{n→+∞} ||x_n − y_n||_H = 0.$
From the text I suppose the sequences are in $\ell^2$, so:
$\begin{align*} ||x_n − y_n||_2 = \langle x_n − y_n,x_n − y_n \rangle &= \langle x_n,x_n \rangle - \langle x_n,y_n \rangle - \langle y_n,x_n \rangle + \langle y_n,y_n \rangle\\ &= \langle x_n,x_n \rangle - 2\langle x_n,y_n \rangle + \langle y_n,y_n \rangle \end{align*}$
$\big($in case the sequences are made of real numbers, does all these inner products be actually the usual products? i.e. $\langle x_n,y_n \rangle = x_ny_n?)$
Since the sequences are in $\ell^2$ they converge to $0$, i.e. $\lim_{n\to+\infty} x_n = 0$ and same for $y_n$, so I would say that:
$\lim_{n\to+\infty}\langle x_n,x_n \rangle = 0,\quad \lim_{n\to+\infty}\langle y_n,y_n \rangle = 0$.
Then I obtain $\lim_{n\to+\infty} ||x_n − y_n||_2 = -2 \lim_{n\to+\infty} \langle x_n,y_n \rangle = -2\ (\ne 0),$ since by hypothesis $\lim_{n→+∞} \langle x_n, y_n \rangle = 1.$
Moreover, I'm trying to figure out how can be that $\lim_{n→+∞} \langle x_n, y_n \rangle = 1$. I don't know if in this case we can apply the Bessel inequality because we have two orthonormal sequences, in case we can I would say : $\sum_{n=1}^{\infty} |\langle x_n,y_n \rangle|^2 \le ||x||_2^2=1$ since the sequence is orthonormal. Does not follow from this that the series converges and so its term tends to zero?
