All rights reserved. If you wish to print a nicely formatted version of this chapter, you may download the rtf file , which will be interpreted and opened by Microsoft Word or the pdf file , which will be interpreted and opened by Adobe Acrobat Reader. Many of the calculations in this chapter are available in the form of a Mathematica notebook or Maple worksheet . Later, we will also speak of square-integrable functions on regions in two or three dimensions, in which case we have multiple integrals over those regions. Roughly speaking, a function on a finite interval is square integrable unless it is infinite somewhere.
Most familiar functions are square-integrable .
Definition III.2 . The sense in which this infinite sum converges is, of course, in mean square. This is often a practical way to show that a set is incomplete, because you just have to exhibit a nonzero function which is orthogonal to the entire set. Unfortunately, it is not as easy to prove completeness as it is to disprove it. Suppose we had left out some of the components.