Unconditional and shrinking bases in locally convex spaces
L. J. Weill · Pacific Journal of Mathematics · 1969
Let E be a locally convex space with an unconditional Schauder basis {x^ and let {fk} be the sequence of coefficient functionals biorthogonal to {Xk}.Owing to works of R. C. James and S. Karlin it is known that if E is a Banach space then each of the three conditions which follow is necessary and sufficient for {f k } to be a basis for £7* in the strong (norm) topology.(1) E has no subspace topologically isomorphic to the space I 1 .(2) E* is separable in the strong topology.(3) E* is weakly (w(E* 9 E**)) sequentially complete.The primary purpose of this paper is to show that in certain spaces which are more general than Frechet spaces and hence than Banach spaces, each of the above three conditions is necessary and sufficient for (0) {fk} is a strong basis for E*.PROPOSITION 1.1.Let E be a linear topological space 2 .If {%k>fk} is & Schauder basis or a weak Schauder basis for E, then {fk>κ(%k)} is a weak* Schauder basis for E*.It follows that if {f k } is also a strong basis then it is a strong Schauder basis with {π(x k )} as the coefficient functionals.The σ-partial sum for {f k } is then the adjoint of S σ ; i.e., Sί(f) = Σ πfe)(/)Λ -£/(%)/*.TheAs a kind of converse we have the following.PROPOSITION 1.2.Let E be a locally convex space 2 .If {f k F k } is a weak* Schauder basis for E*, then {x k ,f k } is a weak Schauder basis for E, where for each k, x k is that unique element of E such that π(x k ) = F k .The next two propositions require the Barrel Theorem for their proofs.PROPOSITION 1.3.Let E be a barrelled space 2 .If {x k ,f k } is a Schauder basis for E then {f k } is a strong basis for the closed linear span, [77], of {/,}.Actually, in Proposition 1.3, "Schauder basis" may be replaced by "weak Schauder basis" as the following reveals.PROPOSITION 1.4.In a barrelled space a weak Schauder basis is a Schauder basis.A Schauder basis {x k ,f k } is called a shrinking basis if {f k } is a UNCONDITIONAL AND SHRINKING BASES IN LOCALLY CONVEX SPACES 469 basis for E* (in the strong topology).A basis {x k } is boundedly complete if for each sequence {t k } of scalars such that {Σ t k x k }n=i is oo k=ί bounded, the series Σ **#* is convergent.The next proposition due to J. Dieudonne [3, Prop.6] strengthens Proposition 1.1 in the case where E is barrelled.PROPOSITION 1.5.If {x k ,f k } is a Schauder basis in a barrelled space 2 E, then {/J is a weak* boundedly complete basis for E*.Hence, if {%k,fk} is a shrinking basis then {f k } is a boundedly complete basis for E*.The next proposition was proved for Banach spaces by R. C. James [8] and generalized by J. R. Retherford [13].PROPOSITION 1.6.Let {x k ,f k } be a Schauder basis for a barrelled space 2 E. Then E is reflexive if and only if {x k } is both shrinking and boundedly complete.