Pythagorean orthogonality and additive mappings (Q675212)

In this note \(X\) is a real normed vector space with either isosceles orthogonality or Pythagorean orthogonality as defined by R. C. James. It is known that any odd, isosceles orthogonally additive mapping from \(X\) into an additive Abelian group \(Y\) is unconditionally additive whenever \(\dim X\geq 3\). Here the same result is shown for Pythagorean orthogonality. Namely, on a real normed vector space \(X\) with \(\dim X\geq 3\), the odd Phythagorean orthogonally additive mappings are unconditionally additive. The proof uses the corresponding result for isosceles orthogonality and a detailed analysis of the geometry of normed spaces.