key: cord-0041963-3dt8hdqh authors: Bierstedt, Klaus D.; Bonet, José title: Weighted (LF)‐spaces of Continuous Functions date: 2006-11-11 journal: nan DOI: 10.1002/mana.19941650104 sha: d1cbcfbebfdd4227ca98bbeddd9d000307538dff doc_id: 41963 cord_uid: 3dt8hdqh [Chemical structure: see text]. denoted by v = v((Y)). Clearly, the original idea "behind" this method was prompted by EHRENPREIS' notion of "analytically uniform spaces" (cf. [2] ), but it came as a surprise to discover (in [lo] ) that a rather simple, canonical approach already worked in quite general situations. In the case of (LB)-spaces V C ( X ) and V o C ( X ) (and, especially, for Kothe echelon spaces of order c c and 0), the role of the "projective hulls" C v ( X ) and C E ( X ) is very well understood by now, and characterizations of the algebraic and topological equalities V C ( X ) = C Q X ) resp. % C ( X ) = C & ( X ) are known (e.g., see [I] , [3] , [5] , [6] , [71, [lo] , [l 11, [22] ). While several interesting problems with holomorphic functions have remained open, [ 101 already contained a projective description theorem for weighted (LB)-spaces of holomorphic functions which was quite satisfactory from the point of view of the main applications. In fact, it was deduced from the corresponding result for spaces of continuous functions-proved with a partition of unity argument-by an application of Baernstein's open mapping lemma. In this paper the (much more complicated) case of the weighted (LF)-spaces V C ( X ) and ?;C(X) of continuous functions on a locally compact space X and of their projective hulls C v ( X ) and C & ( X ) is treated, for the first time. We hope that the present results will prove helpful in future studies of weighted (LF)-spaces of holomorphic functions (and in their applications). Here we extend several projective description techniques from the (LB)-to the (LF)case. But we also make use of ideas from PALAMODOV'S homological theory of acyclic and weakly acyclic (LF)-spaces and from RETAKH'S characterizations (cf. within functional analysis (i.e., replacing homological tools) and derived versions of RETAKH'S conditions which can be evaluated in concrete cases (and thus are more suitable for applications). In the important paper [23], VOGT characterized the regularity, completeness and (weak) acyclicity of Kothe (LF)-sequence spaces EP, 1 ,< p < rx) or p = 0. (The only remaining case; viz., the characterization of the weakly acyclic spaces of type Em, was then settled by the authors in [8] .) VOGT already mentioned in [23] that part of his results could be extended to spaces of continuous functions. Here we include such extensions and, to some extent, use VOGT'S methods, results (in particular, Lemma 2.3 is taken from his work) and, above all, his conditions (wQ) and (Q). On the other hand, we do have a different point of view: We concentrate on projective descriptions and exploit the associated system V of weights and the projective hulls C v ( X ) and CVo(X) as far as possible. ~ In Section 2, we quote the unpublished lecture notes 1201 of VOGT at several places. Such references are always complemented by others to the published work of VOGT, but we have preferred to keep the references to [20] since these notes contain full proofs. The present study is also intended to provide better insight into the structure of (LF)-spaces, and it is relevant in connection with some questions in the abstract theory of such spaces: It is an open problem (due to GROTHENDIECK) whether regular (LF)-spaces must be complete, and it is also not known if sequentially retractive (LF)-spaces are always acyclic. In our context here, however, both questions have affirmative answers. The article is divided in five sections; in the first one we establish the notation, recall a result from [lo] and prove two technical lemmas (for later use). In Section 2 the al-gebraic identity Y C ( X ) = C V ( X ) is characterized in terms of VOGT'S condition (wQ) for the sequence Y (Proposition 2.4). We show that (wQ) is also equivalent to both the regularity and the completeness of Y C ( X ) (Theorem 2.7). In Section 3 we turn to the corresponding algebraic equality Y o C ( X ) = C E ( X ) (or, equivalently, to the completeness of YoC(X)). It holds if and only if Y satisfies VOGT'S condition (Q), and there are several other equivalences (Theorem 3.3). A formulation of (Q) which resembles the regularly decreasing condition of [lo] is included (Lemma 3.2), and we also describe when Y,C(X) and Y C ( X ) satisfy the conditions (M) resp. (M,) of RETAKH (Theorems 3.3, 3.4, 3.6) . The topological identity Y C ( X ) = C v ( X ) is discussed in Section 4. If X is o-compact or discrete and Y satisfies (wQ), we characterize this identity in the spirit of [l], [5] and [22] (Theorem 4.7). The first part of Section 5 is devoted to a natural condition (2) which arises in some applications. Finally, examples of sequences V' for which (wQ) or (Q) hold resp. do no hold are given. Our notation is standard; we refer the reader to 1151. For a locally compact space X we denote by C ( X ) and C J X ) the space of all continuous functions on X and the space of the continuous functions on X with compact support, respectively. In the sequel, X will always denote a locally compact (Hausdorff) space. For every n E N, V, = (un,k)keN is a sequence of (strictly) positive continuous functions ("weights") on X . We denote by Y the sequence (V,)neN, and we assume that the following two conditions are satisfied: (1) v , , k ( X ) < On,& + ( x ) for all n, k E N and x E X , (2) ~, , k ( X ) > u , + l , k ( X ) for all n, kEN and X E X . For each n EN, we define the following weighted Frdchet spaces of continuous functions VkCN: I I f Iln,k:=Sup u vx X\K} * According to our assumptions CV,(X) (resp., C ( V,,), ( X ) ) is continuously included in CV,, ( X ) (resp., C(V,+ ( X ) ) for every n EN. We assume without loss of generality that (Un,k)kdV, u,,k:={fECK(X); Sup u , , k ( x ) l f ( x ) l < I} for kEN, xex forms a basis of 0-neighbourhoods in the space CV,(X) for each nEN. The weighted (LF)-spaces of continuous functions are defined by YC(X):=ind CV,(X) and Y,C(X):=ind C(V,),(X). n n In order to describe these inductive limits algebraically and topologically, we introduce the (maximal) system of weights associated with V V=V(Y):= {D : X -P R; ij 2 0, D is upper semicontinuous (USC) and Vn E N 3ctn > 0, k(n) E N : ij < ct,un,k(n) on X > . According to our assumption on the basis of 0-neighbourhoods of CV,(X), a nonnegative usc function is belongs to if and only if there is an increasing sequence (k(n))n6N c N with ij Q gi v,,k(n). The projective hulls of the weighted inductive limits are defined as follows: Both spaces are endowed with the locally convex topology defined by the system of seminorms {p;; is~v). If i j~E we define U,:= { f~ C v ( X ) ; padf) Q l}. Clearly V C ( X ) c C v ( X ) and % C ( X ) c C K ( X ) with continuous inclusions, and C K ( X ) and are complete locally convex spaces. It is easy to see that, if we consider an increasing sequence of weighted Frechet spaces of continuous functions CW,(X) c CW,, ( X ) with continuous inclusions, it is possible to select an equivalent system of weights V, in each step such that the sequence V = (Vn)neN satisfies conditions (1) and (2) and V C ( X ) = ind CW,(X) algebraically and topologically. In this sense our two conditions do not mean any restriction when dealing with weighted (LF)-spaces of continuous functions. Our aim is to consider the following projective description problem for weighted (LF)-spaces of continuous functions: When do the equalities hold algebraically and/or topologically?-We will also discuss under which conditions the (LF)-spaces V C ( X ) and V o C ( X ) are complete. Let us recall the following result, which is a particular case of [lo, 1.11 . It is obtained by a partition of unity argument. This already provides the desired topological description in the case (ii) of the projective description problem. Our discussions here can be seen as extensions of the (LB)-case, which was extensively treated (cf. [lo] , [ll] , [6] ). In fact, if (u,) ,,~ is a decreasing sequence of weights on X , the weights (vn,k)k& that give rise to the corresponding (LB)-spaces V C ( X ) = ind Cv,(X) and -Y,C(X) = ind C(v,), ( X ) are, for instance, fl n U,,k(X) = 2kUn(X), n, k E N , X E x . In the (LF)-case, some of our results are extensions of VOGT'S results on the (LF)-We now present two technical lemmas. The first (easy) one will be needed in the Proof.(a)Assumethat thereiszEXwithu(z)> a > S max(v(z), w(z))forsomea>O. Since X is locally compact and u, v, w E C ( X ) , we can find g E C,(X), 0 6 g < 1, with g(z) = 1 and Clearly this g contradicts the hypothesis. g E C,(X), 0 < g 6 1, with g(z) = 1 and The function h:= a-'g E C,(X) yields a contradiction. supp g c (x E X ; a < u(x) and S max (v(x), w(x)) < a> . 6-'w(z) ) and choose and max(v(x), 6-'w(x)) < a } . The second lemma (exploiting the "Ernst-Schnettler trick") will be used in the proofs Lemma 3. Assume that there is an increasing sequence (k(n))ncN in N such that, i f f~ C r ( X ) and sup v,,Jx)If(x)I < 03 for some n E N and some k >, k(n), then f~ CV,, ( X ) . Let (l(n) )nEN be a sequence in N and let F E~ be given. We set L(n):=max(k(n), l(n + 1)) for every nEN (and put l/co = 0, 1/0 = 00 in (2) below). Then, for each m E N , we have: Then f l E C ( X ) and Ifl [ < I f l , whence fi E C r ( X ) . Clearly lfll < -on X and also L(1) 3 k(1). By our assumption, this yieldsf, E CV2(X). On the other hand u2,L(1) < ul,L(l) and L(1) 3 1(2), therefore u2,1(2) < u2,L(1) < u l , L ( l ) . Consequently fl E U2,1(z). (2) The proof is analogous. Observe that for the inductive argument above it is not 0 An inductive limit E = ind En is called regular if every bounded subset of E is contained and bounded in some step En. It is well-known that every complete (LF)-space is regular, but whether the converse holds seems to be an open problem (mentioned by GROTHENDIECK), even for (LB)-spaces. In this section we will characterize the algebraic identity V C ( X ) = C v ( X ) in terms of a condition on the sequence called (wQ), which was introduced by VOGT (see [23] ). We will see that this condition is also equivalent to the regularity (and even to the completeness) of the (LF)-space V C ( X ) . n Definition 1 ([23] ). We say that the sequence Y = ((Un,k)kcN),,& satisfies condition ( w e ) (or is of type (we)) if From now on, we will only use n, m, p, v, M , N for natural numbers in the first index (steps) and k, 1, IC, 1, K , L for natural numbers in the second index (weights in a fixed step). (1) Condition (wQ) can be formulated in the following way: V n 3m 3 n and there exists a 0-neighbourhood u n , k in CV,(X) such that, for any p 2 m, un,k intersected with an appropriate 0-neighbourhood in CV,(X) is absorbed by each 0-neighbourhood in CV,,,(X). (2) It is easy to see that condition (wQ) is always satisfied in the (LB)-case, i.e., if U,,k = 2kun, n, k E N, and (u,,),,~~ is a decreasing sequence of weights on X . (3) The following, a priori stronger, condition is equivalent to (wQ): (wQ*) 3(k(v)),,, Vn 3m 2 n Vp 2 m, 13L, S > 0: Q S max ( min uv,k(v), up,L). This will be helpful in Lemma 3 below and in Section 3. To see that (wQ) implies (wQ*) we fix n E N and find m(n), k(n) as in (wQ) increasing with n. Given 1, p one finds L(n, 1, p), S(n, 1, p) > 0 also increasing with n satisfying the inequality in (wQ). We have already found the sequence (k(v))v,w Given n, we choose m = m(n), and for p m and 1 let us also select L = L(n, 1, p), S = S(n, 1, p) > 0 increasing with n. In order to complete the proof of (wQ*), suppose U~(~) ,~( X ) > Su,,,(x) for some X E X . Since S and L are increasing with n, we get, for 1 < v < n, Thus f~~(~) ,~( x ) < S(n, 1, p) uv,k(v)(x) for 1 Q v ,< n, and the proof is complete. Now we need some preliminary results on associated sequence spaces and their projective and inductive limits. We consider X as an index set, and we put an,k(x):=u,,k(~)-~, n, EN, X E X . Similarly as in [21, Section 41, we set, for n, kEN, and, for n E N , Y,:= u x,k. Since the Banach space x , k is continuously injected in x,k+, (by condition (1) of V ) , we may define Y,:=ind x , k . Clearly, in the notation of (an&) , and it is a regular (LB)-space. By condition (2) of < x , , is continuously included in Y,, and we may define Y:= Y,, endowed with the projective limit topology. In fact, Y = proj Y, is a reduced projective limit. We also put, for n, k E N , With the canonical identifications, the algebraic equality Y' = U Y ; = U Y,* = Y* holds. The following extension of [ll, 2.51 (which can be proved in the same way) will be needed later on. We also need the following lemma, due to VOGT, on the topological properties of the projective limit of (LB)-spaces I ! We include some comments on the proof referring ( 1 ) Y satisfies condition (we), ( 2 ) Y is barrelled, ) is contained and bounded in some Y:, Proof. (4) implies (1) by a direct application of [23, 4.7 and definition before 2.31. By [20, Scheme after 5.111 (see also [21, 3.4] ), in order to check (1)*(2)*(3)=(4), it is enough to prove that condition (wQ) implies Proj' ?i Y = 0 for the projective spectrum CV = ( YJneN. By [20, 4.41 (cf. [21, 2.21) this is equivalent to To show that this holds, we use (wQ*) (see Remark (3) after Definition 1) to find (k(~)),,,. Given n we select m according to (wQ*), and given p 2 m and cp E Y, we find 1 with (PE Ym,l and then choose L and S and finally a finite subset F of X such that ( 1 ) Y has condition (wQ), ( Proof. (1)=> (2) . Fix f e C F ( X ) . For every finite subset J of X , the linear map f J : Y +K, fJ(cp):= cf(x)cp(x), belongs to Y', and we can apply Lemma 2 to conclude that the set A: (2)=(3). Let B be any closed absolutely convex bounded subset of C v ( X ) . Since C v ( X ) is complete, C v ( X ) , is a Banach space, and, by (2) , the canonical injection C v ( X ) , c, V C ( X ) is well-defined and has closed graph. By Grothendieck's factorization theorem, the injection factors continuously through some CV,(X). In particular, B is bounded in CV,(X). (1). By [23, 4.71 (cf. 2.31, from (4) it follows that Now, applying Lemma 1.2 (a), we conclude (wQ). In order to prove that every (LF)-space of type VC(X) is even complete whenever it is regular, we need some preparation. The proof presented here differs from the approach of [23, Section 51, it is closer to our approach in [5] . We start with the following abstract result (compare with [6] Proof. From Grothendieck's factorization theorem, it follows that s and t have the same bounded sets and that (E, t) is a regular (LF)-space. In particular, (E, t ) is the ultrabornological space associated with (E, s). If sb denotes the barrelled topology associated with s, then t is finer than sb, which, in turn, is finer than s. (E, sb) is complete by [15, 4.4.201. Now it is enough to prove t = sb. Let U be an absolutely convex 0-neighbourhood in (E, t). For each n EN, there is l(n)EN such that Un,l(n) c 2-"U. We apply (*) to select (L(n))nEN and put Cm:= 1 Un,L(n) for each m E N . Then (Cm)mPN is an increasing sequence of closed absolutely convex subsets of the barrelled space (E, sb) whose union u Cm is absorbing. By a result of DE WILDE (e.g., see For the rest of this section we assume that V is of type (wQ). By passing to a subsequence of (T/n)neN, we may assume m = n + 1 in (wQ), i.e. (WQ) In this section we investigate the algebraic (and hence algebraic and topological, see Theorem 1.1) equality -y^oC(X) = C G ( X ) . It turns out that if these two spaces coincide, then V , C ( X ) is not only a complete (hence regular) (LF)-space, but it'also satisfies some stronger regularity conditions which are relevant in other contexts. We recall the necessary definitions. Let (E, t) = ind (En, t,) be an (LF)-space. The inductive limit (E, t) is called sequentially retractive if every convergent sequence in (E, t ) is contained and convergent in some step (En, t,). It is called boundedly retractive if for any bounded subset B of (E, t) there is n E N such that B is contained and bounded in (En, t,) and that the topologies t and t, coincide on B. The space (E, t) is said to satisfy the condition (M) (resp., (M,)) of Retakh if there is an increasing sequence (UJnEN of 0-neighbourhoods U , in (En, t,) such that Vn 3m 2 n Vp 2 m : t, and tm induce the same topology on U , (resp., Vn 3m 2 n Vp >, rn : a(E,, E / ) and a@, , EL) induce the same topology on U,,). (M) (resp., (M,)) are called acyclic (resp., weakly acyclic) in um,l(X) G max(Eun,k(X), s v p , L ( x ) ) * We first give some reformulations of condition (Q) which will be helpful in the proof of our main result. Some of them are closer to the original condition equivalent to the identity %C(X) = C G ( X ) in the (LB)-case (the so-called "regularly decreasing" condition of [lo] ). In particular, our lemma shows that condition (Q) in the (LB)-case is equivalent to the regularly decreasing condition. ( 5 ) (we) plus the following "countably regularly decreasing" condition (cRD) : (we, and In all these conditions we may take n < m < p, k c 1 c L, 0 < E < 1, S > 1. (5)*(1). Given n we apply condition (wQ) to find m, > n, k , such that V1, p 3L, S > 0 : U ,~J < S max(vn,kl, Now we apply (cRD) to find m 2 m,, k 2 k , . Given I > k, p > m, E > 0 we apply (cRD) to find L' (larger than the L corresponding to 1, p in (wQ)) and 1 2 6 > 0 such that Since (5)-=(6) is trivial, the proof is complete. there is an increasing sequence ( UJneN of absolutely convex 0-neighbourhoods U,, in CV,(X) such that Vn 3m > n : the spaces C v ( X ) and CV,(X) induce the same topology on U,, Proof. (1)-(5). By Lemma 2, we may assume that property (4) there is satisfied. For each n E N we define w,:= min vv,k(v). Clearly l s v s n u n : = { f~C V n ( X ) ; x e x is a 0-neighbourhood in CV,(X) and U , t U n + l for each n E N . Now, for fixed n e N , we select m > n as in Lemma 2.(4). To see that C r ( X ) and CV,(X) induce the same topology on U,, given 1 and E > 0, we select i j~v as in Lemma 2.(4), hence < max(Ew,, 5) on X , and this implies at once The equivalence of (10) and (11) follows from Theorem 1.1, since CG(X) is the completion of VoC(X), and (10) clearly implies (12). It remains to prove the (difficult) implication (12)-(1). This will be done in several steps. Assume that V&(X) is regular. (a) We first show that V satisfies (wQ); this will be obtained as a consequence of Proposition 2.4. Fix feCV(X). For each compact subset K of X we select q K E C,(X), 0 < qK < 1, such that q K ( x ) = 1 for every x E K , and we form which is a subset of C,(X) and bounded in Cv(X), hence bounded in Ty,C(X) by Theorem 1.1. By (12) there is nEN such that B is bounded in C(V,),(X). This easily implies f~ CV,(X), and we have shown VC(X) = CV(X) algebraically, B:= ( f y K ; K compact c X} , (b) For n, kEN we write w,,k:=(fEC(l/n)o(X); If1 < l / u n , k on X}, and we claim V n 3m 2 n, k : the closure of w n , k in VoC(X) is contained in C(V,), (X) . where U j is a compact neighbourhood of x j for each j E N, Ujfl U j , = 8 if j # j', such that Indeed, for j = 1, we put L = 1 + 1, 6 = 1 in ( A ) to find x, E X with U , ,~( X~) > &f?,,k(XI) and up,l+ l(xl) < Un,k(x1), and we find a compact neighbourhood U , of xI with up,l + (x) c u,,Jx) for all x E U , , Assume that 6, > . .. > 6,; x,, .. ., x,EX; U,, .. ., Us are already selected. Put VoC(x), We find, for each k€N, q k E w, , with f k -(Pk € But ((Pk)keN tends to 0 in and up.l +s + 1 (xs + 1) < 6s + 1 un,k(xs + 1). compact neighbourhood Us + of x, + which does up,l + s + 1 ( x ) < 6 s + l u n , k ( x ) for us+ 1 * This completes the induction. Let us observe that u Uj and u Uj (for any EN) are closed in X . Indeed, if e.g. Y E u Uj\u Uj, every neighbourhood of y contains points of Uj for j larger than any given j,. Since 6,+0 as j-, 00, this implies 0 < 1 < which is impossible. +O, hence u,,,(y) = 0, At this point we select q .~ C,(X), 0 < q j < 1, with q j ( x j ) = 1, supp ' p j c U j for every j EN, and we define g:= 2 qj. Then g E C ( X ) , 0 < g < 1, and g vanishes on X\u Uj. Finally, putting h:=g/un.k E C(X), we get: Indeed, given s E N, 6 > 0 we find r E N with r > s, r > 6-' . If x does not belong to Given p and 1, we apply (wQ*) to find L and S > 0. Since A is inner regular, there is a compact subset K of X with IAl(X\K) < E/S. We put Then X , is a measurable subset of X , and we can define y : C(Q, ( X ) -+ K by putting The following abstract result, needed in the proof of Theorem 6 below, was proved in [8]. For each nEN let F , be a closed subspace of En such that F n c F n + l for every n E N . Then F:=ind F, is also regular. Proof. If the sequence V satisfies condition (Q), then V C ( X ) has even property (M), hence a fortiori (Mo), by Theorem 3. Conversely, if V C ( X ) satisfies property (Mo), then V o C ( X ) also does. Consequently, V satisfies condition (wQ) by Theorem 4. Therefore, by Proposition 2.4, V C ( X ) is a regular (LF)-space, and we can apply Proposition 5 to conclude that V o C ( X ) is also regular, whence V must have condition (Q) by another application of Theorem 3. 0 Remark. In contrast to weighted (LF)-spaces of holomorphic functions, the weighted (LF)-spaces of continuous functions are strict only in trivial cases. In fact, if V C ( X ) (or VoC(X)) is strict, then it is already a Frechet space. This is a consequence of the following observation: If the inclusion CV,(X) 4 then V k 31 k : v,,k < V,J, hence Om$ 6 v,,k < v,,,~, which yields cK(x) = cv,(x). The algebraic coincidence V C ( X ) = C v ( X ) was characterized in section 2. Our purpose here is to give conditions to ensure that these spaces also coincide topologically. For (LB)-spaces V C ( X ) , the problem of a topological equality V C ( X ) = C v ( X ) was intensively treated in [5] , [l] , [6] ; some of our results in this section are extensions of results in the (LB)-case. In the (LB)-case the topological identity was relevant in connection with the characterization of the distinguished Kothe echelon spaces of order 1. We refer to [3] (see also [22]). We recall that a locally convex space E is called &barrelled (resp., KO-quasibarrelled) if every countable intersection U = n Urn of closed absolutely convex O-neighbourhoods Urn in E which is absorbing (resp., which absorbs every bounded set) is a 0neighbourhood in E. Our first aim is to show that if the sequence V satisfies condition (wQ), then C v ( X ) is KO-barrelled. (2), The proof of the following lemma is obvious. Proof. Since C v ( X ) is complete, we only have to show that it is KO-quasibarrelled. We will apply Lemma 1 for (E, t) = ?fC(X), (E, s) = C v ( X ) ( Without loss of generality we may assume that (wQ) is satisfied. We fix (tin)neN c v and (l(n)),eN c N increasing such that l(n) 2 k(n) for every n E N . We check that ( 1 ) V C ( X ) = C v ( X ) holds topologically, ( 2 ) Proof. The equivalence of (1) and (2) follows from Proposition 2.4 since V C ( X ) and C v ( X ) have the same bounded sets if (wQ) is satisfied. Clearly (1) implies (3). To see that (3) ) will now be provided, under some restrictions on X ; viz., we assume that X is a-compact (and locally compact) or discrete. In this case, according to [lo, p. 1121, & V x E F n c ( x ) : F a on X. For each n E N we denote by the system of all non-negative upper semicontinuous functions on X which are dominated by functions of the form i & C(k/v,,k, where ctk > 0 for each EN; in fact, K f l C ( X ) = CV,(X)+. Since X is a-compact or discrete, 16 EFn c(x) : tii G \i, Theorem 7. Assume that X is a-compact or discrete and that Y" satisfies (wQ). m e n ( 1 ) V C ( X ) = C r ( X ) holds topologically, In the next section we will give examples of sequences Y such that Y o C ( X ) = Y C ( X ) , but V does not satisfy condition (wQ). In these examples, Y C ( X ) is a proper topological subspace of C v ( X ) by Theorem 1.1 and Proposition 2.4. We start this section by discussing a condition which is satisfied in many interesting examples. We say that Y= ( ( U n , k ) In the (LB)-setting (C) coincides with the condition (S) (for every n there exists m > n: vm/vn vanishes at infinity on X ) which was discussed in several papers (e.g., see [lo] ). If a sequence Y on a locally compact space X satisfies (C), then X must clearly be a-compact. Proof. It is obvious that condition (C) implies ( ( 1 ) ( Q ) , ( 2 ) Vn 3m, k V p , I, E > 03L, K c X compact VXEX\K: Vrn,,(x) < E max(vn,k(x), vp,L(x)) 3 (3) Vn 3m, kVp, 13L, K c X compact Vx EX\K: urn,l(x) G max (un,k(x) , ufl,L(x)) 9 ( 4 ) ( w e ) . Proof. (1)=>(4) and (2)*(3) are trivial. (2)-(1) and (3)*(4) follow easily by taking appropriate suprema and infima on K. It remains to show that (4)- (2) . To do this we assume (Z) in the form To check (2), we fix n and select m > n and k as in (wQ). Given m, we choose m' 2 m as in (Z). Now, for p, 1, 0 < E G 1, we take 1' > 1 from (C); i.e., such that Z I , , , , ,~/ U~,~, vanishes at infinity on X, and we apply (wQ) to p and 1' to get L, S > 0 with urn,l, ,< S max(u,,,, uPvL). Vm 3m' 2 mVl31' 2 1 : vrn,.l/urn.l~ vanishes at infinity on X . is compact in X, and for x E X\K we have & um',l(x) < um,l'(x) G max(un,k(x)7 ufl,L(x)) 9 which shows (2) In order to construct concrete examples we will from now on consider the following u:X+R, O 0 or + co, and let (rn)neN, (pk)k.N be strictly increasing sequences of positive numbers with r, -+ r, pk --t p. For each n, k E N we put u,,k(x):=u(x)I" w(x)Qk vx E x and V:= ((' n,k)kcN)ncN* To see a more explicit example, let N E N and let g : C N -+ R, be a continuous function such that a + b log(1 + lzl) G g(z) for all z E CN, where a and b are non-negative constants (e.g., N = 1 and g(z) = log(1 + 1~1') or g(z) = (zl", 0 < a < 1). Let h : C N + R b a non-negative positively homogeneous continuous function (for example, N = 1 and h(z) = IIm .I). For arbitrary n, k E N and z E C N we define un,k(z):=exp( -nh(z) + kg(z)); i.e., in this case we have u(z) = exp( -h(z)), w(z) = exp(g(z)), r, = n and pk = k for n, k E N. and for a given k we let 1 = k + 1; then so that um,k/u,,f vanishes at infinity. for p 3 max(r, -rlr prp l ) we have Conversely, for n = 1 we find m 2 1, and for k = 1 we get 1 2 1 as in (C). Accordingly, thus u/w must vanish at infinity on X . 0 Example 5. Assume that p < co and that v/w vanishes at infinity on X . If V = ((u,,k)k&N satisfies condition (Q), then there is a closed subset A of X containing B:= {x E X ; w(x) = l} such that (i) inf (-log u(x)/log w(x)) > 0 and (ii) V6 > 0 3K = K ( 6 ) c X compact VXEX\(AUK): -log u(x)/log w(x) < 6. Proof. Since V satisfies (C) by Example 4, we may suppose that the condition in xeA\B Lemma 2 (3) is fulfilled. Then for n = 1 we find m 3 1, k such that (*I Vl, pLL, K C X compact VXEX\K: U ( X ) '~-~I < W ( X )~~-Q [ or U ( X ) '~-~' < w(x)@~-PI. At this point we choose ~(O)EN such that -(P -Pk) < PI -Pk for every 12 l(0). If we apply (*) to I3 I(0) and p = m + 1, we find LEN and a compact subset K(I) of X such that for x EX\(AUK (I)) u(X)rm-ri > w ( x ) l ( @ k -~) > w(x)Qk-e~, 1 whence U ( X ) '~-~~+ < W ( X ) " -~~ < W ( X )~-~* . On bornological C v ( X ) spaces Analytically Uniform Spaces and their Applications to Convolution Equations Stefan Heinrich's density condition for Frechet spaces and the characterization of the distinguished Kothe echelon spaces Dual density conditions in (DF)-spaces I, Results in Math Dual density conditions in (DF)-spaces 11 Advances in the Theory of Frkchet Spaces Completeness of the (LB)-space V C ( X ) A question of D. Vogt on (LF)-spaces Induktive Limites gewichteter Raume stetiger und holomorpher Funktionen A projective description of weighted inductive limits Kothe sets and Kothe sequence spaces Ultradifferentiable functions and Fourier analysis Regularity conditions on (LF)-spaces Homological methods in the theory of locally convex spaces Barrelled Locally Convex Spaces, North-Holland Math On the dual of a subspace of a countable inductive limit Functional Analysis Dual spaces of weighted spaces Topics in Locally Convex Spaces Lectures on projective spectra of (DF)-spaces Topics on projective spectra of (LB)-spaces Distinguished Kothe spaces Regularity properties of LF-spaces Acknowledgement: This research is part of a project supported by an "Accion Integrada Hispano-Alemana" between the authors and their institutions. We gratefully acknowledge the support of DAAD and MEC. Since pI tends to p as 1 tends to 00, this implies (ii). 0Example. We will say that a is an accumulation point at infinity off : X -+ R if there is a sequence (x,),,~ c X of pairwise distinct points without accumulation point in X for which lim f ( x , ) = a.We can choose v and w on X = N, R or C, for instance, such thatlogo + log w tends to 00 at infinity (so o/w vanishes at infinity) and such thathas a sequence of accumulation points at infinity which are different, but tend to 0.By Example 5 the corresponding sequence Y = ((vn,k)k&,)neN with p c 00 does not satisfy condition (Q). In view of Proposition 1, Lemma 2 and Example 4, this also yields the promised example of a sequence Y such that Y o C ( X ) = V C ( X ) , but Y does not satisfy (wQ). Compare with [21].Remark. Let W = (wJnEN be a decreasing sequence of weights on a locally compact space X which is not regularly decreasing in the sense of [lo] (see concrete examples in [lo, 111) . For 0n.k = 2kvn, n, k E N , f = ((vn,k)kEN)nsN satisfies condition (wQ), but not (Q).