# Differentiable Functions on Normed Linear Spaces

Formalized Mathematics (2012)

• Volume: 20, Issue: 1, page 31-40
• ISSN: 1426-2630

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## Abstract

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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].

## How to cite

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Yasunari Shidama. "Differentiable Functions on Normed Linear Spaces." Formalized Mathematics 20.1 (2012): 31-40. <http://eudml.org/doc/267628>.

@article{YasunariShidama2012,
abstract = {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].},
author = {Yasunari Shidama},
journal = {Formalized Mathematics},
language = {eng},
number = {1},
pages = {31-40},
title = {Differentiable Functions on Normed Linear Spaces},
url = {http://eudml.org/doc/267628},
volume = {20},
year = {2012},
}

TY - JOUR
AU - Yasunari Shidama
TI - Differentiable Functions on Normed Linear Spaces
JO - Formalized Mathematics
PY - 2012
VL - 20
IS - 1
SP - 31
EP - 40
AB - 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].
LA - eng
UR - http://eudml.org/doc/267628
ER -

## References

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