The Online Encyclopedia and Dictionary






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 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 category theory the function space 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 (Xx -) on objects;

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 MN - or the subspace where no two positions were equal. To take account of both position and momenta one moves to the cotangent 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 "half" of (see lagrangian distribution ) 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.

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