Summary:
The basis problem, the question if every separable Banach
space has a Schauder basis, which was formulated by Mazur in the
Scottish book in 1936 (problem 153, date 6 November 1936), was one
of the main problems in functional analysis until 1972, when the
Swedish mathematician Per Enflo gave a negative answer. Mazur had
also formulated the approximation problem. The first
mathematician who dealt systematically with this problem, which asks
if every separable Banach space has the approximation property, i.e.
if in each separable Banach space the identity operator can be
approximated uniformly on compact subsets by finite rank operators,
was Grothendieck in 1955. Both problems were eventually
solved by Enflo in 1972, who constructed a separable Banach space
failing to have the approximation property and hence also failing to
have a basis. Later, many other counterexamples were
given. Davie and Szankowski constructed closed subspaces of l_p, p#2 and
c_0 which fail to have a basis. We describe these constructions as well
as other constructions of Banach spaces which provide a negative
answer to variants of the approximation problem.
Keywords:
Schauder basis, Approximation property, Bounded approximation property, Enflo
