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Concept
Function space
Summary
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 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.
Vector space#Function spaces
Let be a vector space over a field and let be any set. The functions → can be given the structure of a vector space over where the operations are defined pointwise, that is, for any , : → , any in , and any in , define
When the domain has additional structure, one might consider instead the subset (or subspace) of all such functions which respect that structure. For example, if is also a vector space over , the set of linear maps → form a vector space over with pointwise operations (often denoted Hom(,)). One such space is the dual space of : the set of linear functionals → with addition and scalar multiplication defined pointwise.
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 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.
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