Projecting Lipschitz functions onto spaces of polynomials

Abstract

The Banach space P( 2 X) of 2-homogeneous polynomials on the Banach space X can be naturally embedded in the Banach space Lip 0 (B X ) of real-valued Lipschitz functions on B X that vanish at 0. We investigate whether P( 2 X) is a complemented subspace of Lip 0 (B X ). Especially important for us is when X is the separable Hilbert space, in which case the question is related to an open question posed by Gilles Godefroy asking whether Lip 0 (ℓ 2 ) has the approximation property. Moreover, our results can be interpreted as failure of a polynomial counterpart to a classical result by Joram Lindenstrauss, asserting that P( 1 X) = X * is complemented in Lip 0 (B X ).