For something to be a basis each element needs to be the finite linear combination. Of course there are notions for different kinds of choices of basis, where y

For something to be a basis each element needs to be the finite linear combination. Of course there are notions for different kinds of choices of basis, where you have countable linear combinations. The article (maybe for good reason) completely ignores the (usually quite hard) choice in functional analysis which vectorspace to use. And the one outlined here, where functions are defined pointwise, is amost always the most useless one to consider. Although I get that it was done to teach the general outline of the subject, which is quite valuable on its own. >Clearly there is no Fourier transformation possible on all functions? You don't even get a useful metric for such a space.