Position of $L(X,Y)$ in $Li{p}_0(X,Y)$

Abstract

We prove that $L(X,Y)$ is complemented in $Li{p}_0(X,Y)$ (via a norm-one projection) provided that $Y$ is a dual space. Next, we introduce a vector-valued Lipschitz-free space $F_Y(X)$, a linear contraction ${\beta }_X^Y:F_Y(X)\to Y$ and prove that the quotient space $Li{p}_0(X,Y)/L(X,Y)$ is isometrically isomorphic to $L(\ker ({\beta }_X^Y),Y)$ whenever $Y$ is injective. We also consider a $Y$-valued duality pairing between $Li{p}_0(X,Y)$ and $F_Y(X)$ and obtain a necessary and sufficient condition for a Lipschitz map to be linear. As an application, we describe $\ker ({\beta }_{\mathbb{R}}^{\mathbb{R}})$ as the pre-dual of the quotient space $L^{\infty }(\mathbb{R})/\tilde{\mathbb{R}}$ where $\tilde{\mathbb{R}}$ is the set of all constant maps on $\mathbb{R}$.