The subject of the present diploma thesis is known as ”The Complemented Subspace Problem”. This states that, if in some infinite dimensional Banach space every closed subspace of which is complemented on it, then it is isomorphic to a Hilbert space. This problem was answered in the positive by Lindenstrauss and Tzafriri in the paper [ J.Lindenstrauss and L.Tzafriri, On the complemented subspaces problem, Israel J. Math. (1971), 263-269 ].The structure of the thesis has as follows.
The first chapter includes some (preliminary) results such as definitions and concepts, which are necessary for the development of our topic. The second chapter is devoted to a special case of the Complemented Subspace Problem, where we assume that the Banach space X (which is in question) has an unconditional basis. The proof of this special case is quite intriguing, as it applies a lot of important results. Some of them are Zippin’s Theorem (for the homogeneity of the basis of the lp and c0) and the theorem of Lindenstauss and Tzafriri which characterizes the lp and c0 spaces through complemented subspaces which are produced from block basic sequences of an unconditional basis.
In the end, the third chapter contains the proof of the general case of the Complemented Subspaces Problem. Actually, they are given two proofs for the problem, the first one is the original proof of Lindenstrauss and Tzafriri. The point here is that both of proofs are based on Dvoretzky’s theorem which we cite in the first chapter. The difference between the two proofs lies to the extent which Dvoretzky’s theorem is involved in the proof.