Orthogonality in Banach Spaces via projective tensor product

Abstract

Let $X$ be a complex Banach space and $x,y\in X$. By definition, we say that $x$ is Birkhoff-James orthogonal to $y$ if $\Vert x+\lambda y{\Vert }_X\ge \Vert x{\Vert }_X$ for all $\lambda \in \mathbb{C}$. We prove that $x$ is Birkhoff-James orthogonal to $y$ if and only if there exists a semi-inner product $\phi$ on $X$ such that $\Vert \phi \Vert =1$, $\phi (x,x)=\Vert x{\Vert }^2$ and $\phi (x,y)=0$. A similar result holds for $C^{\ast }$-algebras. A key point in our approach to orthogonality is the representations of bounded bilinear maps via projective tensor product spaces.

8 pages