# Differentiable Functions on Normed Linear Spaces

Yasunari Shidama 1
• 1 Shinshu University, Nagano, Japan

## 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].

If the inline PDF is not rendering correctly, you can download the PDF file here.

• Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.

• Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.

• Grzegorz Bancerek. König's theorem. Formalized Mathematics, 1(3):589-593, 1990.

• Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.

• Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.

• Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485-492, 1996.

• Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.

• Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.

• Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.

• Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.

• Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521-527, 1990.

• Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.

• Czesław Byliński. Introduction to real linear topological spaces. Formalized Mathematics, 13(1):99-107, 2005.

• Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.

• Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. The product space of real normed spaces and its properties. Formalized Mathematics, 15(3):81-85, 2007, doi:10.2478/v10037-007-0010-y.

• Hiroshi Imura, Morishige Kimura, and Yasunari Shidama. The differentiable functions on normed linear spaces. Formalized Mathematics, 12(3):321-327, 2004.

• Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.

• Anna Lango and Grzegorz Bancerek. Product of families of groups and vector spaces. Formalized Mathematics, 3(2):235-240, 1992.

• Hiroyuki Okazaki, Noboru Endou, Keiko Narita, and Yasunari Shidama. Differentiable functions into real normed spaces. Formalized Mathematics, 19(2):69-72, 2011, doi: 10.2478/v10037-011-0012-7.

• Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.

• Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.

• Konrad Raczkowski and Paweł Sadowski. Real function differentiability. Formalized Mathematics, 1(4):797-801, 1990.

• Laurent Schwartz. Cours d'analyse, vol. 1. Hermann Paris, 1967.

• Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39-48, 2004.

• Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.

• Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4):341-347, 2003.

• Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.

• Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.

• Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.

• Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.

• Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171-175, 1992.

OPEN ACCESS