L2-summand vectors in Banach spaces

The aim of this paper is to study the set L-X(2) of all L-2-summand vectors of a real Banach space X. We provide a characterization of L-2-summand vectors in smooth real Banach spaces and a general decomposition theorem which shows that every real Banach space can be decomposed as an L-2-sum of a Hilbert space and a Banach space without nontrivial L-2-summand vectors. As a consequence, we generalize some results and we obtain intrinsic characterizations of real Hilbert spaces.