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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

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1898-9934
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Volume 20, Issue 1

Issues

Differentiable Functions on Normed Linear Spaces

Yasunari Shidama
Published Online: 2012-09-12 | DOI: https://doi.org/10.2478/v10037-012-0005-1

Differentiable Functions on Normed Linear Spaces

In this article, we formalize differentiability of functions on normed linear spaces. Partial derivative, mean value theorem for vector-valued functions, continuous differentiability, etc. are formalized. As it is well known, there is no exact analog of the mean value theorem for vector-valued functions. However a certain type of generalization of the mean value theorem for vector-valued functions is obtained as follows: If ||ƒ'(x + t · h)|| is bounded for t between 0 and 1 by some constant M, then ||ƒ(x + t · h) - ƒ(x)|| ≤ M · ||h||. This theorem is called the mean value theorem for vector-valued functions. By this theorem, the relation between the (total) derivative and the partial derivatives of a function is derived [23].

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About the article


Published Online: 2012-09-12

Published in Print: 2012-01-01


Citation Information: Formalized Mathematics, Volume 20, Issue 1, Pages 31–40, ISSN (Online) 1898-9934, ISSN (Print) 1426-2630, DOI: https://doi.org/10.2478/v10037-012-0005-1.

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