Every Hilbert space with the inner product is one, although the converse is not true, and the simplest example is an n-dimensional Euclidean space with a distance function.
A Besov Space is one of these if the factors p and q are between one and infinity. The Schauder fixed point theorem involves convex subsets of them, and they are called prime if each of their complemented subspaces are isomorphic to them. Every Hilbert space with the inner product is one, although the converse is not true, and the simplest example is an n-dimensional Euclidean space with a distance function. For 10 points -- identify these complete vector spaces with a norm, named for a mathematician who did work on cutting spheres with Alfred Tarski.
A Besov Space is one of these if the factors p and q are between one and infinity.
Banach space