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 $ \|x+\lambda y\|_{X} \geq \|x\|_{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 $\varphi$ on $X$ such that $\|\varphi\| = 1$, $\varphi(x,x)=\|x\|^2$ and $\varphi(x,y)=0$. A similar result holds for $C^*$-algebras. A key point in our approach to orthogonality is the representations of bounded bilinear maps via projective tensor product spaces.