I want to show that if $\,T$ is not compact then there exists an orthonormal sequence $x_{n}$ and $R>0$ such that $ \forall n\in \mathbb{N}\,\,\,\,\|T(x_{n})\|\geq R$. It is obvious by the definition of compact operator that we can find a sequence satisfying the above condition but how can we find an orthonormal one? Thank you.
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http://math.stackexchange.com/questions/73069/show-u-n-orthonormal-a-compact-implies-au-n-to-0?rq=1 – daw Oct 21 '14 at 06:34
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this question is not answered – Oct 21 '14 at 13:23