Quasiorthogonal sets and conditions for a Banach space to be a Hilbert space

For a subspace $Y$ of a Banach space $X$ the quasiorthogonal set $Q(Y,X)$ is the set of all $n\in X$ such that $0$ is one of the best approximation elements of $n$ in $Y$. The properties of the sets $Q(Y,X)$ are studied; several criteria in terms of these sets for $X$ to be a Hilbert space are established; in particular, generalizations of the well-known theorems of Rudin–Smith–Singer and Kakutani are proved.