Re:Truth, Math, and Models (Part 8 in a series on the scient...
by btilly on Wednesday May 29, 2024 @05:50PM
(#40517456)
Complete does not mean uncountable. Complete means that any sequence that looks like it is converging, actually is converging.
Related to Hilbert spaces, consider the set of analytic functions with period 1. Under the L2 norm, this is an inner product space. It is also uncountable. However as Fourier showed, the Fourier series for the square wave does not converge within the set of analytic functions. In fact it is trying to converge to a function that not only fails to be analytic, it fails to be continuous!
Related to Hilbert spaces, consider the set of analytic functions with period 1. Under the L2 norm, this is an inner product space. It is also uncountable. However as Fourier showed, the Fourier series for the square wave does not converge within the set of analytic functions. In fact it is trying to converge to a function that not only fails to be analytic, it fails to be continuous!