Köthe echelon spaces à la Dieudonné

Indag. Mathern., N.S., 5 (l), 51-60 KGthe echelon spaces CI la Dieudonnk March 28, 1994 by Miguel Florencio, Pedro J. Paljl and Carmen Shez Departamento de Matemcitica Aplicada II. E.S. Ingenieros Industriales, Av. Reina Mercedes s/n, 41012-Sevilla, Spain Communicated by Prof. J. Korevaar at the meeting of December 21,1992 ABSTRACT Let (gn) be a sequence of locally integrable functions defined on a Radon measure space. The echelon space associated to (g,,) was defined by J. Dieudonne as the Kothe-dual of (g”). i.e. the space n of all locally integrable functions f such that all the integrals s If g,l are finite. Denote by Ax the Kiithe-dual of A. We prove that A@(,$ Ax)) IS a Frechet space with dual Ax. This result gives its correct sense to a wrong affirmation of J. Dieudonne and validates those instances where it has been used. As a tool to prove this result, we study the problem of when the strong dual of a perfect space coincides with its Kiithe-dual and give some necessary and sufficient conditions. 1. INTRODUCTION The Kothe-Toeplitz theory of perfect sequence spaces [7, $301 has been one of the most influential in the study of the structure of locally convex spaces. This theory, and in particular the duality between echelon and co-echelon sequence spaces (see e.g. [l] and [6]), has provided the specialists with plenty of hints, examples and counterexamples. Different extensions of this theory are obtained by replacing the C in the definition of G. K&he and 0. Toeplitz with a suitable S. One of the first moves in this direction was made by J. Dieudonne. It is in the last section of his seminal paper [3, $161 that J. Dieudonne considers echelon spaces of functions. His definition is as follows: let X be a locally compact and a-compact Hausdorff topological space with a Radon measure p. The echelon space associated to a sequence (g,) of locally integrable functions is defined as the Kothe-dual of the sequence: 51 n := f : f is locally integrable and &n(f) = s If.g,I dp < +oo, n = 1,2,. . . x After giving this definition, he affirms that A endowed with the topology defined by the seminorms ( pg,) is a Frechet space and that its topological dual coincides with its Kothe-dual Ax. This is not correct, as was pointed out by J.A. Lopez Molina [8, Ex. (p. 187)] with the following example: consider the unit interval with its Lebesgue measure and the echelon space A associated to the function g(x) = exp( - l/x). Then h = x[~,~I is a function in ilx because the functions in A are, by definition, locally integrable. However, h is not continuous for the seminormp, because the sequence (k xp~,k]) from n satisfies lip pg(k. XIO,llk]) = lip s k. exp(-l/x) dx 5 lip exp(-k) = 0, 0 butforallk= 1,2,...wehave jkqqo,,,k,(x).h(x)dx= ykdx= 1. 0 0 This mistake led J.A. Lopez Molina to consider an alternative definition of echelon space: require the functions in n to be only measurable instead of locally integrable. The theory of echelon spaces defined in this way has been developed and generalized by J.A. Lopez Molina [8] and [9], J.C. Diaz [2] and K. Reiher [12]. Our purpose in this paper is to prove that J. Dieudonne’s affirmation is essentially correct in the sense that an echelon space A is a Frechet space when endowed with the strong topology p(A, Ax) and its topological dual equals its Kothe-dual Ax. This result will, in turn, validate those instances in which J. Dieudonne’s affirmation has been used precisely in its correct sense, as in [ll, Cor. 11. In $2 we recall J. Dieudonne’s definition of perfect spaces and give some necessary and sufficient conditions for the topological dual of a perfect space to coincide with its Kothe-dual. These conditions, which are of independent interest, generalize and unify some previously known results and will be used in $3, where the result announced in the preceding paragraph appears. We refer the reader to W. Rudin’s book [13] for the results concerning measure theory and integration, and to G. Kothe’s monograph [7] for the theory of locally convex spaces. 2. WHEN DOES THE TOPOLOGICAL DUAL COINCIDE WITH THE KGTHE-DUAL? Although some of our results in this paper can be given in a more abstract measure-theoretic frame, we shall stick to J. Dieudonne’s original formulation [3, #lo-161. In what follows, X stands for a locally compact, Hausdorff topological space that is a-compact, so that we can write X = U, X,, where every X, is 52 compact and X, c int(X,+i) for all m E N. Let p be a positive Radon measure on X and G be the space of all (equivalence classes of) locally integrable functions from X into the field [ib of real or complex numbers. For a subset A c R we denote by A” the set of all g E $2 such that f. g is integrable for each f E A; Ax is called the K&he-dual of A. A linear subspace A of 0 is said to be a perfect space (or Kiithe space) if (Ax)” = A. In particular, Ax is always perfect. For instance, L1 (,u) and L”(p) are perfect spaces, and each one is the Kothe-dual of the other. The space fl is also perfect and Qx is the space @ of all (equivalence classes of) measurable essentially bounded functions with compact support. When A contains @, the spaces A and Ax are put into duality by means of the canonical bilinear form (f> g) = L f(x) g(x) 44x) (f E 4 g E Ax 1. Let B stand for the unit ball of L”(p). A subset H of R is called normal if f t E H for all f E H and t E B or, equivalently, if f E H and g is a measurable function such that /g(x) 1 5 1 f(x) 1 ,+a.e. on X, then g is also in H. The normal hull of a set H c 0 is defined by {t f : f E H, t E B}. One important fact is that the normal hull of a weakly bounded set is also bounded [3, Prop. 61. This means that the strong topology /?(A, A”) is generated by the seminorms where H runs through the absolutely convex, normal and o(A’, A)-bounded subsets of Ax. If A is a perfect space, then A(p(A, Ax)) is complete [3, Th. 51. Denote by A’ the topological dual of A(p(A, Ax)). The problem of when A’ equals Ax or, equivalently, when the strong topology p(A, Ax) coincides with the Mackey topology ,u(A, A”), has been addressed by several authors (we shall give precise references in the notes following our Theorem l), and different necessary or sufficient conditions have been considered. In our first result we unify and extend previously known results. Theorem 1. Let A be a perfect space and denote by A’ the topological dual of A endowed with the strong topology /?(A, A”). C onsider the following conditions.. (i) A’ = Ax or, equivalently, p(A, Ax) = P(A, Ax). (ii) For every function f E A and every sequence (A,) of measurable sets such that lim, p(A,) = 0, the sequences (f XA,) and (f -f . xx,) converge to zero for the strong topology P(A, Ax). (iii) If (fn) is a decreasing sequence in A that converges to zero p-a.e., then ( fn) converges to zero for the strong topology ,@A, Ax). (iv) A(p(A, Ax)) is separable. (v) The space Cc(X ) of continuousfunctions with compact support is dense in A for the strong topology p(A, A”). 53 Then we have (iv) + (i) @ (ii) @ (iii) + (v). Besides, if the measure space (X, p) is separable, then (iv) is equivalent to (i) - (iii). Moreover, cyX is a metrizable space, then all the conditions above are equivalent. Proof. We start by proving the equivalence of(i) - (iii). (i) + (ii): Take f E A. T o see that (f -f XX,,) converges strongly to zero, fix a strong seminorm pH, where H is an absolutely convex, normal and a(A’, A)- bounded subset of Ax. Condition (i) tells us that H is relatively compact for the weak topology a(Ax , _4). Consider the mapping T:gEAx-T(g)=f.gEL’(p). Since A is normal, the function f. h is in A for every h E L”( p), and we have (T(g),h)(,l(,),,,(,)) = i f.g.hd~ = (gJ.h)(,x,,). Therefore, T is a(A’, A) - a(L’( p), L”(p)) continuous and consequently T(H) is relatively compact for the weak topology a(L’( p), I.,“( p)). Then, by [3, Th. 41, there is a compact set K c X such that sup J” IT(g)]dp:gEH J” jf.gldp:gEH J’\K X\K TakingmENsuchthatKcX,wehavepH(f-f.xxm)<lforn>m. On the other hand, if p(A,) + 0 but (f . XA.) does not tend to zero for the Mackey topology, then we can find a normal set H c Ax, absolutely convex and a(Ax , A)-compact, and an increasing sequence of indices (nk) such that but sup Aj 1f.gldl.L: gEH >l forallk=1,2,... “k Consider the decreasing sequence (Fk) of measurable sets defined by Fk := Ujzk A,. Then P(h) 5 xj>k P(&), h ence limk p(Fk) = 0. For each k E N define the set H/c:= gEH: j- If.gldp> 1 Fk > These sets (Hk) are non-empty, o(A" , A)-closed and they have the finite intersection property because for kl < k2 we have J 2 IfWd~ If.hldp I for every h E Ax so that HkZ c Hk,. Since H is o(A’, A)-compact, there is a function g E H such that g E Hk for all k or, equivalently, J If. gl dp 2 1 for all k E N. Fk 54 But f. g E L’(p). Therefore, there is some 6 > 0 such that if p(A) < 6 then J, If . gl dp < 1, and this is in contradiction with the inequality above because limk p(Fk) = 0. (ii) + (iii): Let (fn) b e a d ecreasing sequence in A that converges to zero p-a.e. and take a normal and a(nx, A)-bounded set H. Applying (ii) to f = fi E A we can find an index N E N such that Now, let cy =PH(xx,,,). We may assume that cy # 0. (Otherwise, every g E H would be zero /l-a.e. in X, and the proof of this implication would be finished.) Consider the measurable sets defined for each II E N by: A, := {x E -=,: fn(x) > 1/(4d’)}. Since p(XN) < +oc and (fn) converges to zero p-a.e., we have that lim, CL(&) = 0. By condition (ii), there is m E N such that ~~(fi . XA,) < $. Now, for n 2 m we have that Ifn(x)I < 1/(4a) on XN\&.Therefore PH(h XX,) < PH(h . XA,) +PH(fn ’ %\A,) 5 PH(fi XA,,,) + (l/(4a)) ‘PHh.\A,) < ; + (1/(4a)). cy = 1. Hence, if n > m we have pH(fn) 5 pH(fn -fn XX,) +pH(fn XX,) < ; + ; = 1. (iii) =+ (i): W e a ways have Ax c A’, On the other hand, take 4 E A’. Fix an 1 index N E N. For each measurable set A c XN, define G(A) = I. Condition (iii) yields that G is a a-additive measure. Indeed, if (A,) is a sequence of disjoint measurable sets in X, with union A, apply (iii) to the functions defined by t, := XA - 2 X.4, E @ c A, foreachm= 1,2,... n=l Then (tm) converges to zero for the strong topology. Hence I = C,“=i ~(xA,) so that G(A) = C,“=, G(A,). Let us see now that G is absolutely continuous with respect to p: if we have a measurable set A C XN with p(A) = 0, then XA = 0 p- a.e., thus G(A) = I = 0. Apply the Radon-Nikodym Theorem to deduce the existence, on each XN, of a function gN E L’(p, Xv) such that G(A) = JgN dp for A c xv. A It is clear that if N 2 M, then gN = gM on X, so that the function g = limN gN is well-defined and locally integrable. Moreover, for each compact K and each measurable set A c K, we have G(A) = s g dl.L. A 55 We prove now that g E Ax and 4(f) = (f, g). Suppose, without loss of generality, that g > 0. We proceed in several steps. (1) Since 4 is linear, we have that 4(f) = (g,f) for f a simple function with compact support. (2) For f E @ (the space of measurable and essentially bounded functions with compact support), let (&) b e a net of simple functions with compact support such that lim, Il(f -fu) XA llm = 0 w h ere A is the support off. If H is a normal and a(Ax, A)-bounded set, then we have limPH(f -fo?) = lim sup a J If -fol] . IhI dp: h E H > ~~11~ .PH(XA) = 0. N This proves that (fol) converges to f for the strong topology p(A, Ax). Using this, (1) above and that f = lim, fa in Lm(p, A), we obtain (3) Now take a positive function f E A. For each 12 E N, let fn be the function in @ defined by: h(x):= {y iff(x)<nandxEX, > otherwise. Then (f - fn) is a decreasing sequence that converges to zero p-a.e. By condition (iii), fn converges to f for the topology @(A, A”). Using this and the Monotone Convergence Theorem in L’ (p) we have (4) Finally, for arbitrary f E A the equality 4(f) = (f, g) follows by linearity. (iv) + (i): W e a wa s 1 y h ave Ax c A’. Now, take 4 E A’. Then there is some absolutely convex, closed and o(Ax, A)-bounded subset H of Ax such that 4 E H”“, the bipolar of H in A’. Since A(p(A, A”)) is separable, H”” is metrizable [7, §21.3.(4)] and compact for the topology o(A’, A). Therefore, H is sequentially dense in H”“. But, on the other hand, H is o(Ax, A)-sequentially complete because it is closed in A’(c~(il’, A)) and this space is sequentially complete [3, Prop. 121. Hence, c5 E H”” = H c Ax. Consequently, A’ c Ax. (i) + (v): For every non-zero g E Ax there is a function h continuous and having compact support such that (h, g) # 0 [3, p. 981. Then, by the Hahn-Banach separation theorem, Cc(X) is dense in A for any topology such that the dual of A isAx. (i) - (iii) + (iv) when the measure space (X, p) is separable: For every nonzero g E A” there is a simple function h having compact support such that (h,g) # 0. Condition (i) and the Hahn-Banach separation theorem ensure that the space SC(X) of simple functions having compact support is dense in 56 n(,B(A, A”)). Therefore, we have to prove that S,(X) is separable for the strong topology /?(A, A”). Since the measure space (X, p) is separable, there is a countable family C of measurable sets (we may, and do, assume that this family contains all the sets of the form Y f~ X, for Y E C and n E N) such that for every measurable set A there is a sequence ( Yj) from C with limi p(AnYj) = 0, where D stands for the symmetric difference operator. If, in particular, A is contained in X,,, then both XA and xX,\A are in SC(X) c A. Since limj p(An Y,) = 0, we can apply condition (ii) to xA andxx,\A = X& - XA (recall that A C X,) to obtain, for the strong topology P(4 AX )9 O=limxAx~ar,=limxA(xA+x~-2xAx~)=li~(xA-xAxY;) .I I and 0 = lim(xx;, -xA)xAar, =lijn(x~ -XA)(XA +,YY,-~xAxY,) i = Ii,? k,~xr, - x~xY,). It follows that limj (XA - x~~,~~,)) = 0 for the strong topology p(il, Ax). This ensures that the countable set D of all simple functions of the form C cryxy, where each Y in the sum is from C and each QY is rational, is dense in S,(X) for the strong topology /?(A, /lx). (v) =+ (iv) when X IS metrizable: According to condition (v), we have to prove that Cc(X) endowed with the restriction of the topology @(il, A”) is separable. Now, if X is metrizable, then each of the spaces C(Xn)( 1) 11,) is separable. For every n E FU, let D, be a countable dense set in C(Xn)(\j II,). Now, for f E C(Xn) take g E D, such that If(x) -g(x)\ I 1 for all x E X,. Let H be a normal and a(n”, A)-bounded subset of Ax. Since S and g are supported in X,, we have ~~(f‘-g) = SUP .f If-s/. Ihldp: h E H xn This shows that D = (J, D, is a countable set dense in Cc(X) for the strong topology p(n, A);). q Notes. J. Dieudonni: [3, $13 (p. 107)J claimed that (iv) =+ (i). Y. K6mura [5, Th. 1.31 showed the equivalence of(i) and (iv) for X = [w”. G.G. Lorentz 110, Th. 31 proved that (iii) + (i) for the case when X is a finite interval in the real line and A is a normed space. R. Welland [18, Th. 21 proved the equivalence of(i), (iii) and (iv) under the hypothesis that the measure space (X, ,u) is separable; here we have shown that this hypothesis is not really necessary to prove (i) ej (iii). Condition (ii) may be easier to use that (iii) and the proof of (ii) + (iii) above follows the 57 ideas given by A.C. Zaanen in [19,§72]. Finally, condition (v) has been considered by G. Silverman for translation invariant perfect spaces [17, Th. 2 and Th. 31. 3. THE STRONG DUAL OF AN ECHELON SPACE Let (g,) be an increasing sequence of non-negative locally integrable functions. The Kothe-dual of (gn) is called the echelon space associated to this sequence and we shall denote it by A(g,). Thus A(a) = f E Q: ~g,(f) := j- If(x)1 .8,(x) dp(x) < +cc x foralln=1,2,... When (g,) reduces to a simple function g, the space A(g) is commonly denoted by Li and has many interesting properties (see [3, #IO-121, [4] and [15]). Note that we can write A(g,) = n, Li.. As we said in the introduction, the main purpose of this paper is to give the following result. Theorem 2. Let A = A(g,,) be the echelon space associated to an increasing sequence (g,) of non-negative locally integrable functions. Then A is a perfect space with Kiithe-dual Ax = U, (Lin)‘. Moreover, A(p(A, Ax)) is a Frtchet space with dual Ax and the strong topology ,B(A, A”) is generated by thefamily of seminorms pg,: f ~A+p,~(f) :=EIf(x)l.g.(x)d~(x), n= 1,2,... qm: f6 A-+ qm(f) :=L If(x)Idp(x), m= I,&... Proof. First, we prove the theorem when (gn) reduces to a single function g E 6). According to [ 15, Prop. 11, the strong topology ,B( Li, (Li) ’ ) is given by the family of seminorms { pg and qm, m = 1,2, . .} where pg( f) = Jx ) f) g dp. The space Lk is perfect because it is the Kothe-dual of {g} and therefore, it is complete when endowed with the strong topology [3, Thm. 51. Hence Lh(p(Li, (Lh)“)) is a Frechet space. To prove that the topological dual of this space is (Li)‘, we shall apply condition (ii) of Theorem 1 above. Let f E Lj. Obviously, lim, qm(f -f . xx”) = 0 for every m E N, and since f. g E L’(n), we also have lim pg( f -f XX.) = lip s 1 f gl dp = 0. n x\xl Now let (An) be a sequence of measurable sets with lim, p(A,) = 0. Since f XX, is integrable for each m, we have li,m qm(f . x.4,) = lip j If I xx, dp = 0. A. 58 On the other hand, using again thatf g E L’(p), we have l$p,(f. XA,) = 1,” l IfI . gdp = 0. A” This finishes the proof for LL. We turn now to the general case. Let T be the topology defined by both sets of seminorms { ps, : II = 1,2, . . .} and {qm : m = 1,2, .}. Since n c Ljn for every n E N we have that (Lin)’ c Ax and therefore U, (LL”)” c Ax. Denote by r,, the corresponding strong topology ,B(Li., (Li”)’ ) on Lj.. Since g, < g,+l, the inclusion $+, (7,+1) + Lin (~~1 IS continuous, so that A(T) is the reduced (because Cc(X) is dense in each of these spaces by (i) + (v) in Theorem 1) countable projective limit of a family of Frechet spaces. Therefore, .4(~) is a Frechet space. Denote by A’ the dual of n(r). Note that T = @(A, A’). On the other hand, A’ c u (L;n(Tn))’ = u (L;“y c AX. ” n The proof will be finished if we show that il” c A’. Take h E Ax and assume, without loss of generality, that h 2 0. Call T the linear form induced by h on il, T(f) = (f, h). Fork = 1,2,. . define the functions h/‘(X) = i h(x), if h(x) < k and x E xk 0, otherwise, then (hk) is an increasing sequence that converges pointwise to h. We can apply the Monotone Convergence Theorem to deduce that T(f) = Sf.hd,u=lip ~f~hkd~=li~(f,hk). x x Now, observe that every hk E @ and that the seminorm I(., hk) 1 is dominated by k qk. Therefore, T is the pointwise limit of the sequence ((., hk)) of r-continuous linear forms. By the Banach-Steinhaus Theorem, T is also r-continuous, i.e. hEA’. q One can see now that the trouble in J. Dieudonne’s affirmation was that he missed the family of seminorms {qnl : m E N}. ACKNOWLEDGMENT This research has been supported by La Consejeria de Education y Ciencia de la Junta de Andalucia. REFERENCES 1. Bierstedt, K.D., R. M&e and W. H. Summers ~ Kiithe sets and Kiithe sequence spaces. Functional Analysis, Holomorphy and Approximation Theory, North-Holland Mathematical Studies 71, ElsevieriNorth-Holland, Amsterdam, Oxford and New York, 27791 (1982). 2. Diaz, J.C. -Strictly regular echelon Kiithe spaces. Arch. Math. (Basel) 46,360-367 (1986). 3. Dieudonne, J. - Sur les espaces de Kathe. J. Analyse Math. 1,81-115 (1951). 59