scholarly journals On the semigroup of differentiable mappings (II)

1972 ◽  
Vol 13 (2) ◽  
pp. 122-128
Author(s):  
G. R. Wood ◽  
Sadayuki Yamamuro
Keyword(s):  
Banach Spaces ◽  
Real Variable ◽  
Differentiable Functions ◽  
Finite Dimensional ◽  
Differentiable Mappings

In [2], K. D. Magill, Jr. has proved that every automorphism of the semigroup (with respect to composition) of all real-valued differentiable functions of a real variable is inner. The purpose of this paper is to generalize this fact to arbitrary finite-dimensional real Banach spaces.

scholarly journals General Norm Inequalities of Trapezoid Type for Fréchet Differentiable Functions in Banach Spaces

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1288
Author(s):  
Silvestru Sever Dragomir
Keyword(s):  
Banach Spaces ◽  
Error Bounds ◽  
Norm Inequalities ◽  
Differentiable Functions ◽  
Fréchet Differentiable ◽  
General Norm

In this paper we establish some error bounds in approximating the integral by general trapezoid type rules for Fréchet differentiable functions with values in Banach spaces.

The wigner property for CL-spaces and finite-dimensional polyhedral banach spaces

2021 ◽  
pp. 1-17
Author(s):  
Dongni Tan ◽  
Xujian Huang
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Phase Function ◽  
Linear Isometry ◽  
Basic Properties ◽  
Finite Dimensional ◽  
Image Position

Abstract We say that a map $f$ from a Banach space $X$ to another Banach space $Y$ is a phase-isometry if the equality \[ \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \] holds for all $x,\,y\in X$ . A Banach space $X$ is said to have the Wigner property if for any Banach space $Y$ and every surjective phase-isometry $f : X\rightarrow Y$ , there exists a phase function $\varepsilon : X \rightarrow \{-1,\,1\}$ such that $\varepsilon \cdot f$ is a linear isometry. We present some basic properties of phase-isometries between two real Banach spaces. These enable us to show that all finite-dimensional polyhedral Banach spaces and CL-spaces possess the Wigner property.

scholarly journals Geometric Invariants of Surjective Isometries between Unit Spheres

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2346
Author(s):  
Almudena Campos-Jiménez ◽  
Francisco Javier García-Pacheco
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Unit Sphere ◽  
Geometric Property ◽  
The Other ◽  
Geometric Invariants ◽  
Finite Dimensional ◽  
Surjective Isometry ◽  
Finite Dimensional Case

In this paper we provide new geometric invariants of surjective isometries between unit spheres of Banach spaces. Let X,Y be Banach spaces and let T:SX→SY be a surjective isometry. The most relevant geometric invariants under surjective isometries such as T are known to be the starlike sets, the maximal faces of the unit ball, and the antipodal points (in the finite-dimensional case). Here, new geometric invariants are found, such as almost flat sets, flat sets, starlike compatible sets, and starlike generated sets. Also, in this work, it is proved that if F is a maximal face of the unit ball containing inner points, then T(−F)=−T(F). We also show that if [x,y] is a non-trivial segment contained in the unit sphere such that T([x,y]) is convex, then T is affine on [x,y]. As a consequence, T is affine on every segment that is a maximal face. On the other hand, we introduce a new geometric property called property P, which states that every face of the unit ball is the intersection of all maximal faces containing it. This property has turned out to be, in a implicit way, a very useful tool to show that many Banach spaces enjoy the Mazur-Ulam property. Following this line, in this manuscript it is proved that every reflexive or separable Banach space with dimension greater than or equal to 2 can be equivalently renormed to fail property P.

Extensions Of Subdifferential Calculus Rules in Banach Spaces

1996 ◽  
Vol 48 (4) ◽  
pp. 834-848 ◽  
Author(s):  
A. Jourani ◽  
L. Thibault
Keyword(s):  
Banach Spaces ◽  
Clarke Subdifferential ◽  
Subdifferential Calculus ◽  
Finite Dimensional ◽  
Calculus Rules ◽  
Approximate Subdifferential ◽  
The Clarke Subdifferential

AbstractThis paper is devoted to extending formulas for the geometric approximate subdifferential and the Clarke subdifferential of extended-real-valued functions on Banach spaces. The results are strong enough to include completely the finite dimensional setting.

On bases, finite dimensional decompositions and weaker structures in Banach spaces

1971 ◽  
Vol 9 (4) ◽  
pp. 488-506 ◽  
Author(s):  
W. B. Johnson ◽  
H. P. Rosenthal ◽  
M. Zippin
Keyword(s):  
Banach Spaces ◽  
Finite Dimensional
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