A special basis for $C([0,\,1])$

This paper constructs a basis for C([O, 1]) which converges weakly to zero whose elements are nevertheless norm bounded away from zero. Introduction. A basis for a Banach space X is a sequence {x, } CX such that to each yEzX there exists a unique sequence of scalars cn for which the partial sums of E CnXn converge to y in the norm of X. Bases can be distinguished by combinatorial and topological criteria (cf. [4]). Foiaa and Singer asked in [1] if C([0, 1]) possesses a basis of type wco, that is, a basis that converges weakly to zero whose elements have norms bounded away from zero. This paper constructs such a basis using a scheme that may be useful in looking for other peculiar bases in C( [0, 1]). J. R. Holub has recently [2] constructed a basis in co that is weakly closed yet intersects every weak neighborhood of zero defined by a single linear functional. The referee has pointed out that Holub's example and the construction below easily imply the existence of such a basis in C( [0, 1]). A method of construction. A sequence of piecewise linear functions which form a basis in C([0, 1]) can be constructed in the following way. The sequence of functions will be accompanied by a parallel sequence of points in [0, 1]. These points will be called joints. The joints are to mark the intervals on which the functions are actually linear. The functions and joints will be grouped into successive stages, each stage having some finite number of functions and an equal number of joints. The kth function (joint) of the nth stage will be denoted fn,k (an,k). To simplify notation an,k will sometimes be written a(n, k). To form a basis thef.,k must be linearly ordered. Do this by sayingfi,j precedesfn,k if iconstruction notice that the distance between any two adjacent points of Jn,kU { 0, 1 is 2-rn for some nonnegative integer m Received by the editors February 24, 1970 and, in revised form, May 18, 1970. AMS 1969 subject classifications. Primary 4625.