- #1

gothlev

- 3

- 0

**1. Problem description**

Let [itex](e_n)_{n=1}^{\infty} [/itex] be an orthonormal(ON) basis for H (Hilbert Space). Assume that [itex] (f_n)_{n=1}^{\infty} [/itex] is an ON-sequence in H that satisfies [itex] \sum_{n=1}^{\infty} ||e_n-f_n|| < 1 [/itex]. Show that [itex](f_n)_{n=1}^{\infty}[/itex] is an ON-basis for H.

## Homework Equations

## The Attempt at a Solution

Somehow if it can be shown that [tex](f_n)_{n=1}^\infty [/tex] is an complete ON-sequence it can be concluded that [tex](f_n)_{n=1}^\infty [/tex] is a ON-basis for H. I tried to make use of Parseval's formula and also expanding the sum [tex] \sum_{n=1}^\infty ||e_n-f_n|| < 1 [/tex] with the rules for inner products, but it did not really get me anywhere. Since I can not really think of anything else I would need someone to point me in the right direction. I might be missing something really obvious, but can not really see it.