Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-12-04T09:12:19.205Z Has data issue: false hasContentIssue false
  • English
  • Français

Abstract Analysis*

Published online by Cambridge University Press:  03 November 2016

F. Smithies
Get access Rights & Permissions [Opens in a new window]

Extract

The operations of elementary algebra are of finite character; that is to say, an algebraic expression depends, explicitly or implicitly, only on a finite number of variables. The step from algebra to analysis is taken when we begin to consider expressions depending on an infinite number of variables; for instance, the limit of a sequence (an) depends on the infinite set of variables a1, a2, …. Such an infinite set of variables is most conveniently thought of as the set of values of a function; thus the sequence (an) is a function of the positive integral variable n, the single term a3, for example, being the value of the function when n is given the particular value 3. Thus the characteristic features of analysis are (i) the systematic use of functions of a variable capable of an infinity of distinct values, and (ii) the consideration of expressions that involve the values of such a function for an infinity of values of the independent variable.

Type
Research Article
Information
The Mathematical Gazette , Volume 35 , Issue 311 , February 1951 , pp. 2 - 7
Copyright
Copyright © The Mathematical Association 1951

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

A paper given at the Annual Meeting of the Mathematical Association, January 1950.

References

1. Birkhoff, G.: Moore-Smith convergence in general topology Annals of Math. (2), 38 (1937), 39–56.Google Scholar
2. Bourbaki, N.: Topologie générale, Chaps. 1–2 (1940).Google Scholar
3. Cartan, H.: Théorie des filtres. CR. Acad. Sci. Paris, 205 (1937), 595–8.Google Scholar
4. Lefschetz, S.: Algebraic topology (1942).Google Scholar
5. Moore, E. H.: Definition of limit in general integral analysis. Proc. Nat. Acad. Sci. U.S.A., 1 (1915), 628–32.Google Scholar
6. Moore, E. H. and Smith, H. L.: A general theory of limits. Amer. J. Math., 44 (1922), 102–21.CrossRefGoogle Scholar
7. Tukey, J. W.: Convergence and uniformity in general topology (1940).Google Scholar