solved negatively by Timothy Gowers and Bernard Maurey in 1992. Schauder bases and duality A basis {en}n≥0 of a Banach space X is boundedly complete if for every sequence {an}n≥0 of scalars such that the partial sums are bounded in X, the sequence {Vn} converges in X. The unit vector basis for ℓp, , is boundedly complete. However, the unit vector basis is not boundedly complete in c0. Indeed, if an = 1 for every n, then for every n, but the sequence {Vn} is not convergent in c0, since ||Vn+1 − Vn|| = 1 for every n. A space X with a boundedly complete basis {en}n≥0 is isomorphic to a dual space, namely, the space X is isomorphic to the dual of the closed linear span in the dual of the biorthogonal functionals associated to the basis {en}. A basis {en}n≥0 of X is shrinking if for every bounded linear functional f on X, the sequence of non-negative numbers tends to 0 when , where Fn is the linear span of the basis vectors em for m ≥ n. The unit vector basis for ℓp, 1 < p < ∞, or for c0, is shrinking. It is not shrinking in ℓ1: if f is the bounded linear functional on ℓ1 given by then for every n. A basis of X is shrinking if and only if the biorthogonal functionals form a basis of the dual . Robert C. James characterized reflexivity in Banach spaces with basis: the space X with a Schauder basis is reflexive if and only if the basis is both shrinking and boundedly complete. James also proved that a space with an unconditional basis is non-reflexive if and only if it contains a subspace isomorphic to c0 or ℓ1. Related concepts A Hamel basis is a subset B of a vector space V such that every element v ∈ V can uniquely be written as with αb ∈ F, with the extra condition that the set is finite. This property makes the Hamel basis unwieldy for infinite-dimensional Banach spaces; as a Hamel basis for an infinite-dimensional Banach space has to be uncountable. (Every finite-dimensional subspace of an infinite-dimensional Banach space X has empty interior, and is no-where dense in X. It then follows from the Baire category theorem that a countable union of these finite-dimensional subspaces cannot serve as a basis.) See also Markushevich basis Generalized