$\mathsf{L}^{\mathtt{2}}$-summand vectors in Banach spaces

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