Orthogonality, Projections and Equivalent Bases

Abstract

If there is a basis for a closed linear subspace Y of Banach space X, a very general condition allows to define a projection of X on Y On the other hand, projections are very useful tools for existence proofs of bases. Indeed, they can be applied in the proof of the existence theorem of Nikol’Skiĭ given in the first paragraph. The theorem is quite essential in the theory and in the application of the theory of bases and, accordingly, we have to make reference to it in many of the subsequent theorems and corollaries. The second paragraph refers to the very nice fact that the concept of orthogonality can be extended from Hilbert spaces to the normed linear spaces. It gives the relation between total orthogonal systems, simple N ₁-spaces and monotone bases. The last section is concerned with equivalent bases, block bases and the theorem that every infinite dimensional Banach space contains an infinite dimensional subspace with a basis.