Echelon Spaces of Order »

The notion of Echelon spaces was introduced by Köthe [2] as a means of constructing sequence spaces by taking the intersection of multiples of I1 (see below for definitions). The definition was extended by Dieudonné and Gomes [l] to include lp for \^p< °o. It is the purpose of this note to consider the case, p = co. AH of the terminology defined below is due to Köthe. A sequence of complex numbers whose tth coordinate is Xi will be denoted by x or (xi). A sequence of sequences such that the ¿th coordinate of the &th sequence is xf will be denoted by (xw). If x, y are sequences and a is a complex number, we denote the pointwise sum, pointwise product and pointwise scalar product by x+y, xy, ax respectively. If each yi is different from zero, we define the pointwise quotient, x/y. If A is a set of sequences, then xA = {xy \ y EA }. We define an ordering which is compatible with these arithmetic operations by taking as positive cone, P = {x\ x.-^O for all i and x<^0 for at least one i}. The symbol, X, will denote a set of sequences which is a vector space under the above operations. Given X, we define its a-dual, Xx= {u |xGX=>£<11 \xiUi\ < «J}. The relation, (x, u) — £41i xíUí places X, Xx in duality. For example, if lúpúTM and we let l" = {x I £<-i I *»'| "< °° } Of P = °° i the sum is replaced by the sup), then if \ = lp, \x = l" where l/p+l/q = l. The vector space, X, is said to be perfect if X =XXX. It is easy to see that Xx is always perfect. If A is a set of sequences, we define the normal hull An={x\ ByEA3\xi\ á|y<| for alii}. If A=An, then A is said to be normal. Since X, Xx are in duality, we may speak of simply bounded subsets of X. Now Xx is clearly normal and from this it follows that the normal hull of a simply bounded set is simply bounded (as a subset of Xxx perhaps, if X is not normal). Let (aw) be a sequence of sequences such that af>0 for all i, k and a(i) ^a(*+1). If 1 ̂ p^ », we define the Echelon space of order p corresponding to (ak) to be,