Differentiable Functions on Normed Linear Spaces

Yasunari Shidama

Formalized Mathematics (2012)

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

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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Citations in EuDML Documents

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  1. Noboru Endou, Yasunari Shidama, Keiichi Miyajima, Partial Differentiation on Normed Linear Spaces R n
  2. Yuichi Futa, Noboru Endou, Yasunari Shidama, Isometric Differentiable Functions on Real Normed Space
  3. Noboru Endou, Yasunari Shidama, Differentiation in Normed Spaces
  4. Hiroyuki Okazaki, Noboru Endou, Keiko Narita, Yasunari Shidama, Differentiable Functions into Real Normed Spaces
  5. Takao Inoué, Adam Naumowicz, Noboru Endou, Yasunari Shidama, Partial Differentiation, Differentiation and Continuity on n -Dimensional Real Normed Linear Spaces
  6. Artur Korniłowicz, Mazur-Ulam Theorem
  7. Yasunari Shidama, Differentiable Functions on Normed Linear Spaces
  8. Keiko Narita, Noboru Endou, Yasunari Shidama, Differential Equations on Functions from R into Real Banach Space
  9. Takao Inoué, Noboru Endou, Yasunari Shidama, Differentiation of Vector-Valued Functions on n -Dimensional Real Normed Linear Spaces
  10. Takao Inoué, Adam Naumowicz, Noboru Endou, Yasunari Shidama, Partial Differentiation of Vector-Valued Functions on n -Dimensional Real Normed Linear Spaces

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