Function space

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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 have 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.

In linear algebra[edit]

Let V be a vector space over a field F and let X be any set. The functions XV can be given the structure of a vector space over F where the operations are defined pointwise, that is, for any f, g : XV, 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 X is also vector space over F, the set of linear maps XV form a vector space over F with pointwise operations (often denoted Hom(X,V)). One such space is the dual space of V: the set of linear functionals VF with addition and scalar multiplication defined pointwise.

Examples[edit]

Function spaces appear in various areas of mathematics:

Functional analysis[edit]

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.

  • Schwartz space of smooth functions of rapid decrease and its dual, tempered distributions
  • Lp space
  • continuous functions endowed with the uniform norm topology
  • continuous functions with compact support
  • bounded functions
  • continuous functions which vanish at infinity
  • continuous functions that have continuous first r derivatives.
  • smooth functions
  • smooth functions with compact support
  • compact support in limit topology
  • Sobolev space
  • holomorphic functions
  • linear functions
  • piecewise linear functions
  • continuous functions, compact open topology
  • all functions, space of pointwise convergence
  • Hardy space
  • Hölder space
  • Càdlàg functions, also known as the Skorokhod space

Norm[edit]

If y is an element of the function space of all continuous functions that are defined on a closed interval [a,b], the norm defined on is the maximum absolute value of y (x) for axb,[2]

Bibliography[edit]

  • Kolmogorov, A. N., & Fomin, S. V. (1967). Elements of the theory of functions and functional analysis. Courier Dover Publications.
  • Stein, Elias; Shakarchi, R. (2011). Functional Analysis: An Introduction to Further Topics in Analysis. Princeton University Press.

See also[edit]

Footnotes[edit]

  1. ^ Fulton, William; Harris, Joe (1991). Representation Theory: A First Course. Springer Science & Business Media. p. 4. ISBN 9780387974958. 
  2. ^ Gelfand, I. M.; Fomin, S. V. (2000). Silverman, Richard A., ed. Calculus of variations (Unabridged repr. ed.). Mineola, New York: Dover Publications. p. 6. ISBN 978-0486414485.