On the Zero-Set of Real Polynomials in Non-Separable Banach Spaces

Abstract

We show constructively that every homogeneous polynomial that is weakly continuous on the bounded subsets of a real Banach space whose dual is not weak${}^{\ast }$-separable admits a closed linear subspace whose dual is not weak${}^{\ast }$-separable either where the polynomial vanishes. We also prove that the same can be said for vectorvalued polynomials. Finally, we study the validity of this result for continuous 2-homogeneous polynomials.