3
$\begingroup$

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 $H$?

  • 1
    Yes: check that the sequence of partial sums is Cauchy, and use completeness.2012-11-05
  • 0
    @DavideGiraudo: Thanks for your comment. Does the orthogonality has anything to do with this?2012-11-05
  • 0
    Yes, otherwise take $f_n:=f/n$, where $f\neq 0$ is an element of $H$.2012-11-05
  • 2
    To 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
  • 0
    This 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
  • 0
    Yeah, getting the square inside the summation sign is called Pythagoras' Theorem and is works whenever the f_n are pairwise orthogonal2012-11-05

1 Answers 1