If $\{f_{n}\}$ is a sequence of orthogonal functions in a Hilbert space $H$, such that $\sum_{n}\|f_{n}\|_{H}^{2}<\infty$. Does this imply that the series $\sum_{n}f_{n}$ belongs to the space $H$?
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0Yeah, getting the square inside the summation sign is called Pythagoras' Theorem and is works whenever the f_n are pairwise orthogonal – 2012-11-05
1 Answers
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Let $s_n:=\sum_{j=1}^nf_n$. We have, using orthogonality, $\left\lVert s_{m+n}-s_n\right\rVert^2=\left\lVert \sum_{j=n+1}^{m+n}f_j\right\rVert^2=\sum_{j=n+1}^{m+n}\left\lVert f_j\right\rVert^2\leqslant \sum_{j=n+1}^{+\infty}\left\lVert f_j\right\rVert^2.$ As the series $\sum_{j=1}^{+\infty}\left\lVert f_j\right\rVert^2$ is convergent, the sequence $\{s_n\}$ is Cauchy in $H$. We conclude by completeness.
If $\{f_n\}$ is not assumed orthogonal, the result may fail. For example, if $f\in H$, $f\neq 0$, then $f_n:=\frac 1nf$ is such that $\sum_{j=1}^{+\infty}\left\lVert f_j\right\rVert^2<\infty$ holds, but the sequence $\{s_n\}=\{\sum_{j=1}^n\frac 1jf\}$ is not convergent in $H$, as it's not bounded.