Geodesic
Quasiorthogonal sets and conditions for a~Banach space to be a~Hilbert space
Sbornik. Mathematics, Tome 188 (1997) no. 8, pp. 1171-1182

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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. 
@article{SM_1997_188_8_a3,
     author = {P. A. Borodin},
     title = {Quasiorthogonal sets and conditions for {a~Banach} space to be {a~Hilbert} space},
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P. A. Borodin. Quasiorthogonal sets and conditions for a~Banach space to be a~Hilbert space. Sbornik. Mathematics, Tome 188 (1997) no. 8, pp. 1171-1182. http://geodesic.mathdoc.fr/item/SM_1997_188_8_a3/