Continuous functions, space of

A normed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c0256701.png" /> of bounded continuous functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c0256702.png" /> on a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c0256703.png" /> with the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c0256704.png" />. Convergence of a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c0256705.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c0256706.png" /> means uniform convergence. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c0256707.png" /> is a commutative Banach algebra with a unit element. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c0256708.png" /> is compact, then every continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c0256709.png" /> is bounded, consequently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c02567010.png" /> is the space of all continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c02567011.png" />.

When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c02567012.png" /> is a closed interval of real numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c02567013.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c02567014.png" />. According to the Weierstrass theorem on the approximation of continuous functions, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c02567015.png" /> of all non-negative integral powers forms a complete system in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c02567016.png" />. (This means that the set of linear combinations of these powers, that is, polynomials, is everywhere-dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c02567017.png" />.) Consequently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c02567018.png" /> is separable; it also has a basis, for example, the Faber–Schauder system of functions forms a basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c02567019.png" />. A criterion for compactness in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c02567020.png" /> is given by the corresponding theorem of Arzelá: For a certain family of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c02567021.png" /> to be relatively compact in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c02567022.png" /> it is necessary and sufficient that the family is uniformly bounded and equicontinuous. This theorem generalizes to the case of the metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c02567023.png" /> of continuous mappings from one metric compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c02567024.png" /> to another <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c02567025.png" />. For the compactness of a closed subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c02567026.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c02567027.png" /> it is necessary and sufficient that the mappings in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c02567028.png" /> are equicontinuous. The distance between two mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c02567029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c02567030.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c02567031.png" /> is given by

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c02567032.png" />

References

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

The Arzelá theorem is also known as the Ascoli–Arzelá theorem in the setting of functions on a compact metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c02567033.png" />. A sequence of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c02567034.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c02567035.png" /> is relatively compact (i.e. the closure of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c02567036.png" /> is compact), if the sequence is uniformly bounded (also called equibounded), i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c02567037.png" />, and is equicontinuous (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c02567038.png" />), i.e.

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025670/c02567039.png" />

References

[a1]	N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) pp. Chapt. 10 (Translated from French)
[a2]	K. Yosida, "Functional analysis" , Springer (1978) pp. 68ff
[a3]	B.Z. Vulikh, "Introduction to functional analysis" , Pergamon (1963)