Displaying similar documents to “Isometries of normed spaces”

Perturbations of isometries between Banach spaces

Rafał Górak (2011)

Studia Mathematica

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We prove a very general theorem concerning the estimation of the expression ||T((a+b)/2) - (Ta+Tb)/2|| for different kinds of maps T satisfying some general perturbed isometry condition. It can be seen as a quantitative generalization of the classical Mazur-Ulam theorem. The estimates improve the existing ones for bi-Lipschitz maps. As a consequence we also obtain a very simple proof of the result of Gevirtz which answers the Hyers-Ulam problem and we prove a non-linear generalization...

Bidual Spaces and Reflexivity of Real Normed Spaces

Keiko Narita, Noboru Endou, Yasunari Shidama (2014)

Formalized Mathematics

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In this article, we considered bidual spaces and reflexivity of real normed spaces. At first we proved some corollaries applying Hahn-Banach theorem and showed related theorems. In the second section, we proved the norm of dual spaces and defined the natural mapping, from real normed spaces to bidual spaces. We also proved some properties of this mapping. Next, we defined real normed space of R, real number spaces as real normed spaces and proved related theorems. We can regard linear...

A series whose sum range is an arbitrary finite set

Jakub Onufry Wojtaszczyk (2005)

Studia Mathematica

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In finite-dimensional spaces the sum range of a series has to be an affine subspace. It has long been known that this is not the case in infinite-dimensional Banach spaces. In particular in 1984 M. I. Kadets and K. Woźniakowski obtained an example of a series whose sum range consisted of two points, and asked whether it was possible to obtain more than two, but finitely many points. This paper answers this question affirmatively, by showing how to obtain an arbitrary finite set as the...

Mazur-Ulam Theorem

Artur Korniłowicz (2011)

Formalized Mathematics

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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.