Show that an orthonormal(ON) sequence is also a ON-basis in a Hilbert Space

[gothlev]

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

Homework Equations

The Attempt at a Solution

Somehow if it can be shown that (f_n)_{n=1}^\infty is an complete ON-sequence it can be concluded that (f_n)_{n=1}^\infty is a ON-basis for H. I tried to make use of Parseval's formula and also expanding the sum \sum_{n=1}^\infty ||e_n-f_n|| < 1 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.

 

[ystael]

You know that (f_n) is an orthonormal sequence, so the only way it can fail to be an orthonormal basis is if the closed subspace V generated by (f_n) is not the entire space H. Think about how vectors in the orthogonal complement V^\perp = H \ominus V expand in terms of (e_n) and how that relates to the f_n.