Differentiable Functions on Normed Linear Spaces

• Autorzy: Yasunari Shidama
• Języki publikacji: EN

Abstrakty

EN

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].

• Wydawca: De Gruyter Open
• Czasopismo: Formalized Mathematics
• Rocznik: 2012
• Tom: 20
• Numer: 1
• Strony: 31-40

Daty

wydano

2012-01-01

online

2012-09-12

Twórcy

autor

Yasunari Shidama

• Shinshu University, Nagano, Japan

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Identyfikatory

DOI

10.2478/v10037-012-0005-1