Group-Like Structure Underlying the Unit Ball in Real Inner Product Spaces

1990 ◽  
Vol 18 (3-4) ◽  
pp. 355-364 ◽  
Author(s):  
Abraham A. Ungar
Keyword(s):  
Unit Ball ◽  
Inner Product ◽  
Product Spaces ◽  
Inner Product Spaces ◽  
Real Inner Product Spaces

On Hua's inequality in real inner product spaces

2002 ◽  
Vol 33 (3) ◽  
pp. 265-268 ◽  
Author(s):  
J. Pecaric
Keyword(s):  
Inner Product ◽  
Product Spaces ◽  
Inner Product Spaces ◽  
Real Inner Product Spaces

Generalization of results in [1] concerning the Hua's inequality in real inner product spaces are given.

Four-point characterizations of real inner product spaces

2002 ◽  
Vol 75 (1) ◽  
pp. 97-105
Author(s):  
Raymond Freese ◽  
Edward Z. Andalafte
Keyword(s):  
Inner Product ◽  
Product Spaces ◽  
Inner Product Spaces ◽  
Real Inner Product Spaces

Angles in Normed Linear Spaces and a Characterization of Real Inner Product Spaces

1986 ◽  
Vol 129 (1) ◽  
pp. 197-204 ◽  
Author(s):  
Charles R. Diminnie ◽  
Edward Z. Andalafte ◽  
Raymond W. Freese
Keyword(s):  
Inner Product ◽  
Product Spaces ◽  
Linear Spaces ◽  
Inner Product Spaces ◽  
Normed Linear Spaces ◽  
Real Inner Product Spaces

scholarly journals ON HUA'S INEQUALITY IN REAL INNER PRODUCT SPACES

1997 ◽  
Vol 27 (3) ◽  
pp. 227-232
Author(s):  
S. S. DRAGOMIR ◽  
G.-S. YANG
Keyword(s):  
Inner Product ◽  
Product Spaces ◽  
Inner Product Spaces ◽  
Real Inner Product Spaces

A generalization of the celebrated Hua's inequality in real inner product spaces and some connected results are given.

scholarly journals On The Continuous Dependence Of Solutions To Orthogonal Additivity Problem On Given Functions

2015 ◽  
Vol 29 (1) ◽  
pp. 19-23
Author(s):  
Karol Baron
Keyword(s):  
Continuous Dependence ◽  
Inner Product ◽  
Product Spaces ◽  
Additivity Problem ◽  
Inner Product Spaces ◽  
Real Inner Product Spaces ◽  
The Given ◽  
Continuous Dependence Of Solutions ◽  
Orthogonal Additivity

AbstractWe show that the solution to the orthogonal additivity problem in real inner product spaces depends continuously on the given function and provide an application of this fact.

scholarly journals Symmetries of Projective Spaces and Spheres

2018 ◽  
Vol 2020 (7) ◽  
pp. 2205-2240 ◽  
Author(s):  
György Pál Gehér
Keyword(s):  
Product Space ◽  
Transition Probability ◽  
Inner Product Space ◽  
Inner Product ◽  
Quantum Mechanical ◽  
Product Spaces ◽  
Inner Product Spaces ◽  
Wigner’S Theorem ◽  
Real Inner Product Spaces ◽  
Wigner's Theorem

Abstract Let H be either a complex inner product space of dimension at least two or a real inner product space of dimension at least three, and let us fix an $\alpha \in (0,\tfrac{\pi }{2} )$. The purpose of this paper is to characterise all bijective transformations on the projective space P(H) which preserve the quantum angle $\alpha$ (or Fubini–Study distance $\alpha$) between lines in both directions. (Let us emphasise that we do not assume anything about the preservation of other quantum angles). For real inner product spaces and when $H=\mathbb{C}^2$ we do this for every $\alpha$, and when H is a complex inner product space of dimension at least three we describe the structure of such transformations for $\alpha \leq \tfrac{\pi }{4}$. Our result immediately gives an Uhlhorn-type generalisation of Wigner’s theorem on quantum mechanical symmetry transformations, that is considered to be a cornerstone of the mathematical foundations of quantum mechanics. Namely, under the above assumptions, every bijective map on the set of pure states of a quantum mechanical system that preserves the transition probability $\cos ^2\alpha$ in both directions is a Wigner symmetry (thus automatically preserves all transition probabilities), except for the case when $H=\mathbb{C}^2$ and $\alpha = \tfrac{\pi }{4}$ where an additional possibility occurs. (Note that the classical theorem of Uhlhorn is the solution for the $\alpha = \tfrac{\pi }{2}$ case). Usually in the literature, results which are connected to Wigner’s theorem are discussed under the assumption of completeness of H; however, here we shall remove this unnecessary hypothesis in our investigation. Our main tool is a characterisation of bijective maps on unit spheres of real inner product spaces which preserve one spherical angle in both directions.

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