Partial Differentiation, Differentiation and Continuity on n-Dimensional Real Normed Linear Spaces

Inoué, Takao; Naumowicz, Adam; Endou, Noboru; Shidama, Yasunari

doi:10.2478/v10037-011-0011-8

Abstract

In this article, we aim to prove the characterization of differentiation by means of partial differentiation for vector-valued functions on n-dimensional real normed linear spaces (refer to [15] and [16]).

References

[1] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
Search in Google Scholar
[2] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.
Search in Google Scholar
[3] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.
Search in Google Scholar
[4] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
Search in Google Scholar
[5] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
Search in Google Scholar
[6] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
Search in Google Scholar
[7] Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.
Search in Google Scholar
[8] Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577-580, 2005.
Search in Google Scholar
[9] Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. Partial differentiation on normed linear spaces Rⁿ. Formalized Mathematics, 15(2):65-72, 2007, doi:10.2478/v10037-007-0008-5.10.2478/v10037-007-0008-5
Search in Google Scholar
[10] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.
Search in Google Scholar
[11] Hiroshi Imura, Morishige Kimura, and Yasunari Shidama. The differentiable functions on normed linear spaces. Formalized Mathematics, 12(3):321-327, 2004.
Search in Google Scholar
[12] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.
Search in Google Scholar
[13] Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269-275, 2004.
Search in Google Scholar
[14] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.
Search in Google Scholar
[15] Walter Rudin. Principles of Mathematical Analysis. MacGraw-Hill, 1976.
Search in Google Scholar
[16] Laurent Schwartz. Cours d'analyse. Hermann, 1981.
Search in Google Scholar
[17] Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39-48, 2004.
Search in Google Scholar
[18] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
Search in Google Scholar

Partial Differentiation, Differentiation and Continuity on n-Dimensional Real Normed Linear Spaces

Abstract

Paradigm

My account