James orthogonality and orthogonal decompositions of Banach spaces

We establish decompositions of a uniformly convex and uniformly smooth Banach space B and dual space B∗ in the form B=M⊎J∗M⊥ and B∗=M⊥⊎JM, where M is an arbitrary subspace in B, M⊥ is its annihilator (subspace) in B∗, J:B→B∗ and J∗:B∗→B are normalized duality mappings. The sign ⊎ denotes the James orthogonal summation (in fact, it is the direct sums of the corresponding subspaces and manifolds). In a Hilbert space H, these representations coincide with the classical decomposition in a shape of direct sum of the subspace M and its orthogonal complement M⊥: H=M⊕M⊥.