product of vector subspaces of homogenous polynomial of a fixed degree

Question

Denote $P_k$ to be the $K$ vector space of homogeneous polynomials in $K[x_1,x_2,\dots,x_n]$ with total degree $k$. Is it easy to find the minimum dimension $d$ such that any vector subspace $U\le P_k$ of dimension $d$ yields $U\cdot P_k = P_{2k}$, where here $U\cdot P_k$ is the vector space spanned by the product i.e. $$U\cdot P_k:= \langle fg : f\in U, g\in P_k \rangle$$