In
mathematics, a function space is a
set of
functions between two fixed sets. Often, the
domain and/or
codomain will have additional
structure which is inherited by the function space. For example, the set of functions from any set X into a
vector space has a
natural vector space structure given by
pointwise addition and scalar multiplication. In other scenarios, the function space might inherit a
topological or
metric structure, hence the name function space.
Let F be a
field and let X be any set. The functions X → F can be given the structure of a vector space over F where the operations are defined pointwise, that is, for any f, g : X → F, any x in X, and any c in F, define
When the domain X has additional structure, one might consider instead the
subset (or
subspace) of all such functions which respect that structure. For example, if V and also X itself are vector spaces over F, the set of
linear mapsX → V form a vector space over F with pointwise operations (often denoted
Hom(X,V)). One such space is the
dual space of X: the set of
linear functionalsX → F with addition and scalar multiplication defined pointwise.
Examples
Function spaces appear in various areas of mathematics:
In
set theory, the set of functions from X to Y may be denoted {X → Y} or YX.
As a special case, the
power set of a set X may be identified with the set of all functions from X to {0, 1}, denoted 2X.
The set of
bijections from X to Y is denoted . The factorial notation X! may be used for permutations of a single set X.
In
topology, one may attempt to put a topology on the space of continuous functions from a
topological spaceX to another one Y, with utility depending on the nature of the spaces. A commonly used example is the
compact-open topology, e.g.
loop space. Also available is the
product topology on the space of set theoretic functions (i.e. not necessarily continuous functions) YX. In this context, this topology is also referred to as the
topology of pointwise convergence.
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);
Functional analysis is organized around adequate techniques to bring function spaces as
topological vector spaces within reach of the ideas that would apply to
normed spaces of finite dimension. Here we use the real line as an example domain, but the spaces below exist on suitable open subsets