Is Birkhoff-James orthogonality an orthogonality in the sense of Ratz?


**Orthogonality in the sense of Ratz:**


Suppose $X$ is a real vector space with $\dim X\geq2$ and $\perp$ is a binary relation on $X$ with the following properties:


1. *Totality* of $\perp$ for zero: $x\perp 0$ and $0\perp x$ for all $x$;
2. *Independence*: If $x,y\in X-\{0\}$ and $x\perp y$, then $x$ and $y$ are linearly independent.
3. *Homogeneity*: If $x,y\in X$ and $x\perp y$, then $\alpha x\perp\beta y$ for all $\alpha,\beta\in\mathbb{R}$.
4. *Thalesian Property*: If $P$ is a 2-dimensional subspace of $X$, $x\in P$ and $\lambda$ is a nonnegative real number, then $\exists$ $y\_0\in P$ such that $x\perp y\_0$ and $x+y\_0\perp \lambda x-y\_0$.


**Birkhoff-James orthogonality:**


$x$ is said to be Birkhoff-James orthogonal to $y$ if $\|x\|\leq\|x+\lambda y\|$ for all $\lambda\in\mathbb{R}$.


I could figure out that properties 1, 2 and 3 of Ratz's conditions for orthogonality are satisfied by Birkhoff-James orthogonality. The last condition bothers me.


**Question.** Is Birkhoff-James orthogonality an orthogonality in the sense of Ratz, or not?


Thanks in advance!



Birkhoff-James orthogonality and Ratz's orthogonality
Yes, indeed, Birkhoff-James orthogonality is an orthogonality in the sense of Rätz. A proof appears on p.36 of



> 
> J. Rätz, On orthogonally additive mappings, Aequations Math. 28
>  (1985), 35-49.
> 
> 
> 


The argument there is a refinement of the proof given on p.188 of 



> 
> K. Sundaresan, Orthogonality and nonlinear functionals on Banach
>  spaces. Proc. Amer. Math. Soc. 34 (1972), 187-190.
> 
> 
> 


Since it is a non trivial proof, I just refer the interested reader to the above papers.


