Approximation property in Banach Spaces.pdf (original) (raw)
The notion of a basis in a Banach space was introduced by J. Schauder in 1927. If a Banach space has a basis then it is also separable. The problem whether every separable Banach space has a Schauder basis appeared for the first time in 1931 in Banach’s book "Theory of Linear Operations". It was clear to Banach, Mazur and Schauder that this question was related to the "approximation problem". This is mentioned in Banach’s book as a remark to the chapter on compact operators. If a Banach space X has a Schauder basis it also has the approximation property, since the natural projections of X onto its finite dimensional subspaces form a bounded sequence of finite rank operators converging pointwise on X to the identity operator. The approximation problem is equivalent to whether a Banach space has the "approximation property". A Banach space X has the approximation property if the identity operator on X is the limit for the topology of uniform convergence on compact subsets of X of a sequence of finite rank operators. The approximation property in its various forms was thoroughly analyzed by A. Grothendieck in his thesis "Produits tensoriels topologiques et espaces nucléaires" published in 1955. But it was not until 1972, that P. Enflo solved both questions in the negative. He found a subspace of co, which he showed does not have the approximation property and consequently does not have a basis. Almost immediately, T. Figiel and A. M. Davie greatly simplified his proof. The later using a probabilistic lemma constructed a separable closed subspace of l1 without the approximation property. Moreover, they also showed that both co and every lp space with 1 p < 1 , (p 6= 2) has subspaces without the approximation property. In this work we present some of the equivalent properties to the approximation property due to A. Grothendieck, and we make a detailed exposition of the proof by A. M. Davie.
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