$\DeclareMathOperator\Lip{Lip}$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 $\Lip\_0(X)$?



Dual Banach space $X^*$ complemented in $\mathrm{Lip}_0(X)$?
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).


