 In addition they can be composed, whence they form a vector space equipped with a bilinear associative operation, namely composition. In the finite-dimensional case, composition is just the usual matrix product. Vector spaces furnished with such a product constitute associative algebras. Up to isomorphism, all associative algebras arise in this way whether of finite or infinite dimension, providing a satisfactory and insightful characterization of the notion in lieu of an axiomatic characterization, not given here.