# Differentiable Functions on Normed Linear Spaces

Formalized Mathematics (2012)

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

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topYasunari 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 -

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## Citations in EuDML Documents

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