Projecting Lipschitz functions onto spaces of polynomials

Abstract

The Banach space $\mathcal{P}({}^{2}X)$ of $2$-homogeneous polynomials on the Banach space $X$ can be naturally embedded in the Banach space ${\mathrm{Lip}}_0(B_X)$ of real-valued Lipschitz functions on $B_X$ that vanish at $0$. We investigate whether $\mathcal{P}({}^{2}X)$ is a complemented subspace of ${\mathrm{Lip}}_0(B_X)$. This line of research can be considered as a polynomial counterpart to a classical result by Joram Lindenstrauss, asserting that $\mathcal{P}({}^{1}X)=X^{\ast }$ is complemented in ${\mathrm{Lip}}_0(B_X)$ for every Banach space $X$. Our main result asserts that $\mathcal{P}({}^{2}X)$ is not complemented in ${\mathrm{Lip}}_0(B_X)$ for every Banach space $X$ with non-trivial type.

References in corpus (1)

• On the geometry of Banach spaces of the form ${\mathrm{Lip}}_0(C(K))$

Cited by in corpus (1)

• Minimal projections onto spaces of polynomials on real euclidean spheres