Function space

In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in most applications, it is a topological space or/and a vector space. Function spaces appear in various areas of mathematics:

• 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 transformations 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; the best known examples include Hilbert spaces and Banach spaces.

• in functional analysis the set of all functions from the natural numbers to some set X is called a sequence space. It consists of the set of all possible sequences of elements of X.

• in topology, one may attempt to put a topology on the space of continuous functions from a topological space X to another one Y, with utility depending on the nature of the spaces. A commonly used example is the compact-open topology. Also available is the product topology on the space viewed as Y^X

• in algebraic topology, the study of homotopy theory is essentially that of discrete invariants of function 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 is called an exponential object. It appears in one way as the representation canonical bifunctor; but as (single) functor, of type [X, -], it appears as an adjoint functor to a functor of type (-×X) on objects;

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

• in domain theory, the basic idea is to find constructions from partial orders that can model lambda calculus, by creating a well-behaved cartesian closed category.