Let $H$ be a Hilbert space and let the sequence $(x_n)_{n=1}^{\infty}\subset H$ and $x\in H$ such that $\lVert x_{n}\rVert=\lVert x\rVert=1$ for all $n\in\mathbb{N}$. Suppose also that $\lvert 1-<x,x_n>\rvert\leq\frac{1}{n}$ for all $n\in\mathbb{N}$. Now my question is if the sequence $(x_n)_{n=1}^{\infty}$ is an orthonormal basis in $H$, i.e if it is orthogonal and total? Does anyone have any ideas how to investigate this?
2026-04-23 21:15:22.1776978922
Orthonormal basis on a Hilbert space
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