Schauder bases and locally complemented subspaces of Banach spaces

Abstract

The thesis is about Schauder basis in infinite-dimensional Banach spaces and locally complemented subspaces. It starts with the notion of bases and it proves that it is equivalent with that of Schauder basis. It follows with some general theory about bases, and gives the notion of basic sequences and equivalence of bases. It proves that every Banach space has a basic sequence. Next it gives some general theory about unconditional basis. To give an other version of the definition of complemented subspaces, we present adjoint operators and projections. We prove that c0 is not complemented in l∞. The Principle of Local Reflexivity (PLR) is proved and it states that a Bnach space is locally 1-complemented in its didual space. We present Hahn-Banach extension operators and prove that its existence is equivalent with being locally 1-complemented. In the end, the definition for a basic sequence to be (locally) complemented is given and it proves that if a basic sequence is locally complemented, then its biorthogonal functionals can be extended to a basic sequence in the dual space.