The isometric extension problem between unit spheres of two separable Banach spaces

2015 ◽  
Vol 31 (12) ◽  
pp. 1872-1878 ◽  
Author(s):  
Guang Gui Ding
Keyword(s):  
Banach Spaces ◽  
Extension Problem ◽  
Isometric Extension

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.

On the isometric extension problem

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

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.

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