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 $$f_n$$.