Continuous functions, space of

A normed space  of bounded continuous functions  on a topological space  with the norm . Convergence of a sequence  in  means uniform convergence. The space  is a commutative Banach algebra with a unit element. If  is compact, then every continuous function  is bounded, consequently,  is the space of all continuous functions on .

When  is a closed interval of real numbers,  is denoted by . According to the Weierstrass theorem on the approximation of continuous functions, the set  of all non-negative integral powers forms a complete system in . (This means that the set of linear combinations of these powers, that is, polynomials, is everywhere-dense in .) Consequently,  is separable; it also has a basis, for example, the Faber–Schauder system of functions forms a basis in . A criterion for compactness in  is given by the corresponding theorem of Arzelá: For a certain family of functions  to be relatively compact in  it is necessary and sufficient that the family is uniformly bounded and equicontinuous. This theorem generalizes to the case of the metric space  of continuous mappings from one metric compactum  to another . For the compactness of a closed subset  of  it is necessary and sufficient that the mappings in  are equicontinuous. The distance between two mappings  and  in  is given by

#### References[edit]

[1] |  P.S. Aleksandrov, "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian)

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[2] |  A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)

#### Comments[edit]

The Arzelá theorem is also known as the Ascoli–Arzelá theorem in the setting of functions on a compact metric space . A sequence of functions  in  is relatively compact (i.e. the closure of the set  is compact), if the sequence is uniformly bounded (also called equibounded), i.e. , and is equicontinuous (in ), i.e.

#### References[edit]

[a1] |  N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) pp. Chapt. 10 (Translated from French)

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[a2] |  K. Yosida, "Functional analysis" , Springer (1978) pp. 68ff
[a3] |  B.Z. Vulikh, "Introduction to functional analysis" , Pergamon (1963)