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⊥.  2005 Elsevier Inc. All rights reserved.