Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-26T00:32:40.333Z Has data issue: false hasContentIssue false
  • English
  • Français

Mean Value Theorems for Vector Valued Functions

Published online by Cambridge University Press:  20 January 2009

Robert M. McLeod
Robert M. McLeod
Affiliation:
American University of Beirut, Beirut, Lebanon
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The object of this paper is to give a generalisation to vector valued functions of the classical mean value theorem of differential calculus. In that theorem we have

for some c in the open interval a, b when f is a real valued function which is continuous on the closed interval a, b and differentiable on the open interval. The counterpart to (1) when f has values in an n-dimensional vector space turns out to be

where cka, b, 0 k, and .

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 14 , Issue 3 , June 1965 , pp. 197 - 209
Copyright
Copyright Edinburgh Mathematical Society 1965

References

REFERENCES

(1) Aumann, Georg, Reelle Funktionen (Berlin, 1954).CrossRefGoogle Scholar
(2) Aziz, A. K. and Diaz, J. B., On a mean value theorem of the differential calculus of vector-valued functions, and uniqueness theorems for ordinary differential equations in a linear-normed space, Contributions to Differential Equations, 1 (1963), 251269.Google Scholar
(3) Bourbaki, N., lments de Mathmatique, Livre IV, Fonctions d'une variable relle, Chaps. 1, 2, 3 (2nd ed., Paris, 1958).Google Scholar
(4) Dieudonn, J., Foundations of Modern Analysis (New York, 1960).Google Scholar
(5) Eggleston, H. G., Convexity (Cambridge, 1958).CrossRefGoogle Scholar
(6) Gl, Istvn S., On the fundamental theorems of the calculus, Trans. Amer. Math. Soc. 86 (1957), 309320.CrossRefGoogle Scholar
(7) Taylor, Angus E., Introduction to Functional Analysis (New York. 1958).Google Scholar