The Ck Space

Katuhiko Kanazashi; Hiroyuki Okazaki; Yasunari Shidama

doi:10.2478/forma-2013-0002

.blurhash-client-img { display: none !important; }

The Ck Space

Formalized Mathematics

Volume 21 (2013): Issue 1 (January 2013)

By: Katuhiko Kanazashi, Hiroyuki Okazaki and Yasunari Shidama

Open Access

|Aug 2013

Abstract

In this article, we formalize continuous differentiability of realvalued functions on n-dimensional real normed linear spaces. Next, we give a definition of the C^k space according to [23].

References

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.
Search in Google Scholar
[2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
Search in Google Scholar
[3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
Search in Google Scholar
[4] Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.
Search in Google Scholar
[5] Czesław Bylinski. Finite sequences and tuples of elements of a non-empty sets. FormalizedMathematics, 1(3):529-536, 1990.
Search in Google Scholar
[6] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.
Search in Google Scholar
[7] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
Search in Google Scholar
[8] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
Search in Google Scholar
[9] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
Search in Google Scholar
[10] Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.
Search in Google Scholar
[11] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.
Search in Google Scholar
[12] Noboru Endou, Hiroyuki Okazaki, and Yasunari Shidama. Higher-order partial differentiation. Formalized Mathematics, 20(2):113-124, 2012. doi:10.2478/v10037-012-0015-z.10.2478/v10037-012-0015-z
Search in Google Scholar
[13] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1): 35-40, 1990.
Search in Google Scholar
[14] Takao Inou´e, Adam Naumowicz, Noboru Endou, and Yasunari Shidama. Partial differentiation of vector-valued functions on n-dimensional real normed linear spaces. FormalizedMathematics, 19(1):1-9, 2011. doi:10.2478/v10037-011-0001-x.10.2478/v10037-011-0001-x
Search in Google Scholar
[15] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.
Search in Google Scholar
[16] Beata Perkowska. Functional sequence from a domain to a domain. Formalized Mathematics, 3(1):17-21, 1992.
Search in Google Scholar
[17] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.
Search in Google Scholar
[18] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.
Search in Google Scholar
[19] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.
Search in Google Scholar
[20] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
Search in Google Scholar
[21] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.
Search in Google Scholar
[22] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.
Search in Google Scholar
[23] Kosaku Yosida. Functional Analysis. Springer Classics in Mathematics, 1996. 10.1007/978-3-642-61859-8
Search in Google Scholar

Articles in this issue

DOI: https://doi.org/10.2478/forma-2013-0002 | Journal eISSN: 1898-9934 | Journal ISSN: 1426-2630

Journal RSS Feed

Language: English

Page range: 25 - 31

Published on: Aug 23, 2013

Published by: University of Białystok

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Related subjects:

Mathematics,

Logic and set theory,

Computer sciences,

Artificial intelligence

© 2013 Katuhiko Kanazashi, Hiroyuki Okazaki, Yasunari Shidama, published by University of Białystok
This work is licensed under the Creative Commons License.

Volume 21 (2013): Issue 1 (January 2013)