Orthogonally Complemented Subspaces in Banach Spaces

Abstract

In this paper, we extend the Moreau (Riesz) decomposition theorem from Hilbert spaces to Banach spaces. Criteria for a closed subspace to be (strongly) orthogonally complemented in a Banach space are given. We prove that every closed subspace of a Banach space X with dim X ≥ 3 (dim X ≤ 2) is strongly orthognally complemented if and only if the Banach space X is isometric to a Hilbert space (resp. strictly convex), which is complementary to the well-known result saying that every closed subspace of a Banach space X is topologically complemented if and only if the Banach space X is isomorphic to a Hilbert space.

Keywords:

• Banach space
• Duality mapping
• Hilbert space
• Orthogonally complemented subspace
• Strict convexity

AMS Subject Classification:

• 46B20

ACKNOWLEDGMENTS

The research was supported in part by the National Science Foundation (19971023) and the Science Foundation Grant of Heilongjiang Province.