$\DeclareMathOperator\Lip{Lip}$Let $X$ be a real Banach space. The dual $X^*$ is a closed subspace of $\Lip_0(X)$. ($\Lip_0(X)$ denotes the space of real-valued Lipschitz functions $f:X\to\mathbb{R}$ satisfying $f(0)=0$ with Lipschitz constant as norm).
Is there anything known about the complementability of $X^*$ inside $\Lip_0(X)$? If $X$ is finite dimensional, then $X^*$ is trivially complemented in $\Lip_0(X)$. Are there any examples of infinite dimensional spaces $X$ whose duals are or are not complemented in $\Lip_0(X)$?
Best Answer
Corollary 5.4 in my book (Lipschitz Algebras, 2nd edition): Let $V$ be a Banach space. Then there is a norm 1 linear projection from ${\rm Lip}_0(V)$ onto $V^*$. If $V$ is separable then there is a weak* continuous norm 1 linear projection.
This follows from Theorem 2 of "On nonlinear projections in Banach spaces" by Lindenstrauss (Michigan Math J 11 (1964), 263-287).