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 $H$?
Series of functions
3
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real-analysis
sequences-and-series
functional-analysis
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1Yes: check that the sequence of partial sums is Cauchy, and use completeness. – 2012-11-05
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0@DavideGiraudo: Thanks for your comment. Does the orthogonality has anything to do with this? – 2012-11-05
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0Yes, otherwise take $f_n:=f/n$, where $f\neq 0$ is an element of $H$. – 2012-11-05
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2To expand on @DavideGiraudo's comment, orthogonality gives you $\|\sum f_n\| = \sqrt{\sum \|f_n\|^2}$, but $\sum \|f_n\|^2 < \infty$ does not imply $\sum \|f_n\| < \infty$. – 2012-11-05
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0This is exactly what I was looking for, I was wondering how to make the square inside the summation sign, now I know why! Thanks all! – 2012-11-05
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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