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 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 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.

• 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.

Another related idea from physics is the configuration space. This has no single meaning, but for N particles moving in some manifold M it might be the space of positions M^N - or the subspace where no two positions were equal. To take account of both position and momenta one moves to the (co)tangent bundle. The configurations of a curve would be a function space of some kind. In quantum mechanics one formulation emphasises 'histories' as configurations. In short, a configuration space is typically a phase space that is constructed from a function space.

Configuration spaces are related to braid theory, also, since the condition on a string of not passing through itself is formulated by cutting diagonals out of function spaces.