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 When are maps preserving semi-inner products linear?

2021 ◽  
Author(s):  
Paweł Wójcik
Keyword(s):  
Hilbert Space ◽  
Infinite Dimensions ◽  
Normed Spaces ◽  
Linear Isometry ◽  
Inner Products ◽  
Finite Dimensional ◽  
Uniformly Smooth ◽  
Non Linear ◽  
Extra Assumption ◽  
Continuous Injection

AbstractWe observe that every map between finite-dimensional normed spaces of the same dimension that respects fixed semi-inner products must be automatically a linear isometry. Moreover, we construct a uniformly smooth renorming of the Hilbert space $$\ell _2$$ ℓ 2 and a continuous injection acting thereon that respects the semi-inner products, yet it is non-linear. This demonstrates that there is no immediate extension of the former result to infinite dimensions, even under an extra assumption of uniform smoothness.

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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