Banach and Fréchet spaces of functions

Many familiar and useful spaces of continuous or differentiable functions are Hilbert or Banach spaces, with pleasant completeness properties, but many are not. Some are Fréchet spaces, thus still complete, but lacking some of the conveniences of Banach spaces. Some other important spaces are not Fréchet, either. Still, some of these important spaces are colimits of Fréchet spaces (or of Banach spaces), and the consequent quasicompleteness suffices for many subsequent applications. Here we look at some naturally occurring Banach and Fréchet spaces. Our main point will be to prove completeness with the natural metrics.