J. Schauder introduced the notion of basis in a Banach space 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". If a Banach space has a Schauder basis it
also has the approximation property. A Banach space X has the
approximation property if for every Banach space Y the finite rank
operators are dense in the closed subspace of al…
Read moreJ. Schauder introduced the notion of basis in a Banach space 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". If a Banach space has a Schauder basis it
also has the approximation property. A Banach space X has the
approximation property if for every Banach space Y the finite rank
operators are dense in the closed subspace of all compact
operators from Y to X. Both problems were solved in the
negative in 1972 by Per Enflo. His proof was almost immediately
simplified by A. M. Davie, who using a probabilistic lemma
constructed a separable closed subspace of l_inifnity without
the approximation property. In this thesis 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.
