Complemented Subspaces of Sums and Products of Banach Spaces (*). G. ~r - V. B. ~r Summary. - We prove that, when X is one o/ the Banaeh spaces 1~ (l<p< c~) or co, then every in]inite-dimensional complemented subspace o] X Iv (resp. X(N)) is isomorphic to one o] the ]ollowing spaces: co, X, co• X N (resp. q~, X, ~ 0 X, X(N)). Therefore, X N and X(N) are primary. We also give some consequences and related results. O. - Introduction. In the structure theory of Frdchet spaces, it is a general problem to determine what complemented subspaces of a given Frdchet space look like. So far~ the at- tention has been focused on the classical nuclear spaces of power series type and on some Dragilev spaces, for which a number of results are available (see [3], [4], [11], [13] and [14]). In particular, we mention that the situation is completely clear in the case of stable i nuclear power series spaces of finite type: thanks to results of B. S. MITJAGI~ [15], [16] (see also D. VoG~ [11]), complemented subspaces of such spaces are of the same type. However, all the above spaces have a continuous norm. For Fr~chet spaces without a continuous norm there are no results except the well-known fact that ~o always appears as a complemented subspace. It is the purpose of this paper to initiate the study of complemented subspaces in Frdchet spaces without a continuous norm by looking at some very special spaces of this kind. The spaces considered are of the form X ~v, where X is one of the familiar Banach spaces l ~ (l<p< ~) or co, and for these we find that their complemented subspaces are indeed of the form which one would expect, enabling us to conclude that the spaces considered are primary. It is then natural to investigate what hap- pens in the dual situation, i.e. for direct sums X (N), and we find that, in this case too, complemented subspaces are of the expected form, so that also these spaces are primary. This result follows immediately by duality from the former one in the reflexive case X= 1~ (1 ~p ~ c~), while for X----co, P or 1* the proofs are inde- pendent and rather delicate. (*) Entrain in Redazione il 25 febbraio 1987. The second author acknowledges partial support from the Italian Ministero della Pub- blica Istruzione. Indirizzo degli AA. : G. M~TAsu~: Facolt~ di Scienze M.F.N., Universits della Basili- cata, Via N. Sauro 85 (Rione Francioso), 85100 Potenz~, Italia; V.B. MOSCA~ELLI: Dipar- timento di Matematica, Universi~k, C.P. 193, 73100 Lecce, Italia. 176 G. ~r - V. B. ~OSCATELLI: Complemented subspaees o] sums, etc. Finally, we conclude with an Appendix containing results of an independent interest and remarks on injeetive spaces. We use standard terminology (see, e.g, [5]) and, in particular, we denote by ~o (resp. by ~) the product (resp. direct sum) of countably many copies of the scalar field (~-R or C). Also, for two locally convex spaces E and ~, we write E ~ F and E ~/~ to mean respectively that E is topologically isomorphic to /z or to a complemented subspaee of F. We use also the by-now accepted term quo~ection (coined in [1]) for the projective limit of a projective sequence (E~, R~) of Banach sp~ces E. and surjective linking m~ps R~: ~.+~-+E~ and for this we introduce the notation quoj~ (E~, R~). l~on-trivial (or twisted) quojections, i.e. quojections that are not isomorphic to products of Banach spaces, were first constructed in [7] by the second author. A distinctive feature of all quojections, which will be useful to us and that can be desumed from Proposition 3 of [lJ, is that their quotients are again quojections and hence ]~anach if they have a continuous norm. :~inally~ we recall that a locally convex space E is primary if, whenever E=~G, then either F~ or G~E. 1. - Complemented subspaees of (c0)zr and (l~) N (1 ~p ~ c~). Let E = proj, (E~ R.) be the projective limit of a projective sequence of locally convex spaces ~. and linking maps R.: E~+~-+ E.. By definition "~ : { (xn)~ l~ E~t: Xn: "~Xn-~l for all n} is a (dosed) subspace of ~[ ~. It is clear that, for every strictly increasing sequence (n(k)) oN, we also have that E = projk (E~k),/~(~)), where the maps /~(~): E~(~+~)-+ 2~.(7~) are obtained from the R~'s by composition. l~ow, let P,~:/~-+E~ be the map defined by Pax ~-x~. Then R~P~+~-~ P~ and the following lemm~ holds: ~E~A 1.1. - _Let ~ be a closed subspaee o] E. I] ~.-~ _P~,(E) and 1~ is the restriction o] R~ to F~+~, then fF = proj~ (F~,/~.). P~OOF. - Set G = proj. (/~,,,/~.). If x ~/~, then x = (x.) with x~= R~x~+l~tz~ for all ~. Hence x~ =/~.x~+l and x ~ (7. Conversely, if y ~ G, then y = (y~) with y~E_~, and y.:~.y.+l:t~.y.+~. For each n there is an x~eF such that _P~x~: y~. If k~ n, put R~: R~ ... R._IR~_I; then Pkx~: R.7oP~x': R.7r y~. This shows that limn x ~-- y and hence y e ~, since /~ is closed. Thus F ~ G alge- braicully and, as easily seen~ also topologically. G. IV~TAYUNE - V. :B. ~r Complemented subspaces o/ sums, ere. 177 TItEORElVs 1.2. -- Let X be one o/the Bancwh space l~ (l<p< oo) or Co. I] E< X x and dim E ~ 00, then E ,'~ o~, X, o~ x X or X N. PaooF. - We begin by representing X x as quoj~ (X '~, S,,), where S~: X~+I-> X ~ is the canonical projection (xl, ..., x+~, x~+~) --> (xl, ..., x~). Because E c X x, by Lem- ma 1.1 for each n there exists E~ c X ~ such that E ---- projn (/~, ~). Note that the maps ~: E,+~ :->/~, restrictions of the S,;s, are surjective, but the subspaces E~ need not be closed. However, since E is complete we may also write E----proj: (F~, S,,), where we have denoted by F~ the closure of/~ in X ~ and again by S, the restric- tion of S~ to F~+~, in general not surjective but with dense image in F~. Recalling that X ~ X ~, we see that each F. is isomorphic to a closed subspaee of X. Now we use the fact that E is a quojection to write E = quoj~ (E~, R:). If dim (ker R:) < co for all but a finite number of n, one easily obtains the first three cases. Therefore, by passing to subsequences if necessary, we assume s now on that (1) dim (ker R~) : dim (ker S~) ----co for all n, and also that we have the situation indicated in the following commutative diagram )- E~+2 :~"+~) E~+~ ~ > E. > ... (2) ... --+. F~+~ ~ F= ~ ..., where the maps R~ and L~ are surjecti+-e, while the S~ and Q~ need not be so. Introducing the canonical maps (see just before Lemma 1.1) (3) P~: E -~E~ and P~: E -~Fn, one immediately has that (4) P.---- L~P~ and P~----Q~P~+~. By assumption there is a continuous projection T: XX-+E and we naturally introduce the maps (5) T~= P~T: XN-* E~ and T~= P.T: XN-> F~ , for which we have 178 G..-YfE~A~'U~E - V. B. i~[0SCATELZI: Complemented subspaees o/ sums, etc. I~ote that each 2"~ is surjective because so are P and P~ the Ia~ter in view of the fact that E ---- quoj. (E~, R,). Moreover, since E~ is Banaeh and Y~ is a quotient map, for every n there exists k(n)~ N such that T~, hence also P~_~, factors canonically through X ~(~~ i.e. the diagrams X ~ ~ > ~ X N -----+~ j~ Xk(n) Xk(n+l) commut% where we have denoted again by T~ (reap. P~) the restriction of T. to X ~(~) (resp. of P~ to X~(~+~)). For the rest of the proof we consider three cases separately. (a) X= ~" (~<p < ~). Assuming, of course, dim E~= c~ we see that the map/'1: X ~(~) -->E~ is not compact, because it is surjeetive. Hence, by (6) the map T~: X~(~)->~F~ is not compact. Because F~ c X by construction and X ~(~) ~_ X, by [6] (Proposition 2.a.2 and the remark following Proposition 2.e.3) there exist sub@aces M~ c X ~(~) and ]/1 c/~ such that //1 ~_ X, H1 is complemented in X and the map P~: M~-~//1 is an isomorphism onto. Since S~P~+~= T,,, it is dear that also all the maps are isomorphisms onto. l~ow note that Sl(x, y)= x for all (% y)~ F2 and hence, if N-~ ker $I, then 2V = F~(~ ({0}• Because S~: T~(M1) -+T~(M1) = H~ is an isomorphism onto, there exists a continuous linear re@A: M~--> {0} • with T~(M~) = {(x, Ax): x~H~}. It follows that, if i%: X-->H~ is a continuous projection, then the map r: X • defined by r(x, y) -~ (p~x, Aplx) is a projection onto T~(M~) with kerr = {@, y): plx = 0} = (kerp~) • Now, recalling the second diagram in (7), we consider the map P~: X ~(81 --> iv2. Since N ---- ker S~ r i~ and dim N ----- ~z, the m@P~:Pj~(~)-+2vc{o}• is not compact and again, by the argument above which also @phes to sub@aces of l ~, we can find M~ c X k(8) and H~ r N such that H~X, H~ is complemented in {0}• and the m@T~: M2->H~ is an isomorphism onto.