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 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.

Contents

In linear algebra

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{\displaystyle (\sin +\exp )(x)=\sin(x)+\exp(x)} Example for addition of functions.svg
Addition of functions: The sum of the sine and the exponential function is with

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 a 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

Function spaces appear in various areas of mathematics:

Functional analysis

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

Norm

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]

is called the uniform norm or supremum norm ('sup norm').

Bibliography

See also

Footnotes

  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.

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In mathematics, any vector space has a corresponding dual vector space consisting of all linear forms on , together with the vector space structure of pointwise addition and scalar multiplication by constants.

Functional analysis Branch of mathematical analysis

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Normed vector space Vector space on which a distance is defined

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  1. It is nonnegative, that is for every vector x, one has
  2. It is positive on nonzero vectors, that is,
  3. For every vector x, and every scalar one has
  4. The triangle inequality holds; that is, for every vectors x and y, one has

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