Function space

In mathematics, a function space is some reification of a set of functions from a set X to a set Y, of a given kind. This apparently nebulous concept is of importance in numerous areas:

• in set theory, the power set of a set X may be identified with the set of all functions from X to {0,1};

• in linear algebra the set of all linear transformation from a vector space V to another one, W, over the same field, is itself a vector space;

• in functional analysis the same is seen for continuous linear transformations, including topologies on the vector spaces in the above, and many of the major examples are function spaces carrying a topology;

• in topology, one may attempt to put a topology on the continuous functions from a topological space X to another one Y, with utility depending on the nature of the spaces;

• in the theory of stochastic processes, the basic technical problem is how to construct a probability measure on a function space of paths of the process (functions of time);

• in category theory the function space appears as an adjoint functor, to a functor of type xX on objects;

• in lambda calculus and functional programming, function space types are used to express the idea of higher-order function.