Isometric Extension Problem Between Strictly Convex Two-dimensional Normed Spaces

2019 ◽  
Vol 35 (4) ◽  
pp. 513-518
Author(s):  
Guang Gui Ding ◽  
Jian Ze Li
Keyword(s):  
Extension Problem ◽  
Isometric Extension ◽  
Two Dimensional ◽  
Normed Spaces ◽  
Strictly Convex

scholarly journals ISOMETRIES BETWEEN UNIT SPHERES OF THE -SUM OF STRICTLY CONVEX NORMED SPACES

2013 ◽  
Vol 88 (3) ◽  
pp. 369-375 ◽  
Author(s):  
GUANG-GUI DING ◽  
JIAN-ZE LI
Keyword(s):  
Extension Problem ◽  
Isometric Extension ◽  
Normed Spaces ◽  
Linear Isometry ◽  
Strictly Convex ◽  
Surjective Isometry ◽  
Whole Space

AbstractWe prove that any surjective isometry between unit spheres of the ${\ell }^{\infty } $-sum of strictly convex normed spaces can be extended to a linear isometry on the whole space, and we solve the isometric extension problem affirmatively in this case.

MAZUR–ULAM PROPERTY OF THE SUM OF TWO STRICTLY CONVEX BANACH SPACES

2015 ◽  
Vol 93 (3) ◽  
pp. 473-485 ◽  
Author(s):  
JIAN-ZE LI
Keyword(s):  
Banach Spaces ◽  
Equivalent Form ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Extension Problem ◽  
Isometric Extension ◽  
Strictly Convex ◽  
Necessary And Sufficient ◽  
Strictly Convex Banach Spaces ◽  
Equivalent Conditions

In this article, we study the Mazur–Ulam property of the sum of two strictly convex Banach spaces. We give an equivalent form of the isometric extension problem and two equivalent conditions to decide whether all strictly convex Banach spaces admit the Mazur–Ulam property. We also find necessary and sufficient conditions under which the $\ell ^{1}$-sum and the $\ell ^{\infty }$-sum of two strictly convex Banach spaces admit the Mazur–Ulam property.

scholarly journals On normed spaces with the Wigner Property

2019 ◽  
Vol 11 (3) ◽  
pp. 523-539 ◽  
Author(s):  
Ruidong Wang ◽  
Dariusz Bugajewski
Keyword(s):  
Normed Space ◽  
Two Dimensional ◽  
Normed Spaces ◽  
Strictly Convex ◽  
Real Normed Space

AbstractThe aim of this paper is to generalize the Wigner Theorem to real normed spaces. A normed space is said to have the Wigner Property if the Wigner Theorem holds for it. We prove that every two-dimensional real normed space has the Wigner Property. We also study the Wigner Property of real normed spaces of dimension at least three. It is also shown that strictly convex real normed spaces possess the Wigner Property.

On the isometric extension problem

2013 ◽  
Vol 36 (3) ◽  
pp. 321-330
Author(s):  
Ruidong Wang
Keyword(s):  
Extension Problem ◽  
Isometric Extension

scholarly journals On the Aleksandrov-Rassias Problems on Linearn-Normed Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Yumei Ma
Keyword(s):  
Normed Spaces ◽  
Unit Distance ◽  
Strictly Convex ◽  
Surjective Mapping

This paper generalizes T. M. Rassias' results in 1993 ton-normed spaces. IfXandYare two realn-normed spaces andYisn-strictly convex, a surjective mappingf:X→Ypreserving unit distance in both directions and preserving any integer distance is ann-isometry.

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