Mazur-Ulam Theorem

Artur Korniłowicz

doi:10.2478/v10037-011-0020-7

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Mazur-Ulam Theorem

Formalized Mathematics

Volume 19 (2011): Issue 3 (September 2011)

By: Artur Korniłowicz

Open Access

|Apr 2012

Abstract

The Mazur-Ulam theorem [15] has been formulated as two registrations: cluster bijective isometric -> midpoints-preserving Function of E, F; and cluster isometric midpoints-preserving -> Affine Function of E, F; A proof given by Jussi Väisälä [23] has been formalized.

References

Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
Search in Google Scholar
Józef Białas. Infimum and supremum of the set of real numbers. Measure theory. Formalized Mathematics, 2(1):163-171, 1991.
Search in Google Scholar
Józef Białas and Yatsuka Nakamura. Dyadic numbers and T₄ topological spaces. Formalized Mathematics, 5(3):361-366, 1996.
Search in Google Scholar
Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
Search in Google Scholar
Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
Search in Google Scholar
Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
Search in Google Scholar
Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
Search in Google Scholar
Agata Darmochwał. Families of subsets, subspaces and mappings in topological spaces. Formalized Mathematics, 1(2):257-261, 1990.
Search in Google Scholar
Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces - fundamental concepts. Formalized Mathematics, 2(4):605-608, 1991.
Search in Google Scholar
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
Artur Korniłowicz. Collective operations on number-membered sets. Formalized Mathematics, 17(2):99-115, 2009, doi: 10.2478/v10037-009-0011-0.10.2478/v10037-009-0011-0
Search in Google Scholar
Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.
Search in Google Scholar
Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.
Search in Google Scholar
Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.
Search in Google Scholar
Stanisław Mazur and Stanisław Ulam. Sur les transformationes isométriques d'espaces vectoriels normés. C. R. Acad. Sci. Paris, (194):946-948, 1932.
Search in Google Scholar
Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.
Search in Google Scholar
Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.
Search in Google Scholar
Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.
Search in Google Scholar
Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4):535-545, 1991.
Search in Google Scholar
Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4):341-347, 2003.
Search in Google Scholar
Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.
Search in Google Scholar
Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
Search in Google Scholar
Jussi Väisälä. A proof of the Mazur-Ulam theorem. http://www.helsinki.fi/~jvaisala/mazurulam.pdf
Search in Google Scholar
Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.
Search in Google Scholar
Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.
Search in Google Scholar