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1 A simpler description of the κ-topologies on the spaces D L p, L p, M 1 arxiv: v2 [math.fa] 27 Nov 2018 Christian Bargetz, Eduard A. Nigsch, Norbert Ortner November 2017 Abstract The κ-topologies on the spaces D L p, L p and M 1 are defined by a neighbourhood basis consisting of polars of absolutely convex and compact subsets of their (pre-)dual spaces. In many cases it is more convenient to work with a description of the topology by means of a family of semi-norms defined by multiplication and/or convolution with functions and by classical norms. We give such families of seminorms generating the κ-topologies on the above spaces of functions and measures defined by integrability properties. In addition, we present a sequence-space representation of the spaces D L p equipped with the κ-topology, which complements a result of J. Bonet and M. Maestre. As a byproduct, we give a characterisation of the compact subsets of the spaces D 1 L p, Lp and M 1. Keywords: locally convex distribution spaces, topology of uniform convergence on compact sets, p-integrable smooth functions, compact sets MSC2010 Classification: 46F05, 46E10, 46A13, 46E35, 46A50, 46B50 Institut für Mathematik, Universität Innsbruck, Technikerstraße 13, 6020 Innsbruck, Austria. christian.bargetz@uibk.ac.at. Wolfgang Pauli Institut, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria. eduard.nigsch@univie.ac.at. Institut für Mathematik, Universität Innsbruck, Technikerstraße 13, 6020 Innsbruck, Austria. mathematik1@uibk.ac.at. 1

2 1 Introduction In the context of the convolution of distributions, L. Schwartz introduced the spaces D L p of C 8 -functions whose derivatives are contained in L p and the spaces D 1 L q of finite sums of derivatives of Lq -functions, 1 ď p,q ď 8. The topology of D L p is defined by the sequence of (semi-)norms D L p Ñ R`, ϕ ÞÑ p m pϕq : sup B α ϕ p, α ďm whereas D 1 L q pd L pq1 for 1 {p ` 1{q 1 if p ă 8 and D 1 L 1 p 9 Bq 1, carry the strong dual topology. Equivalently, for 1 ď p ă 8, by barrelledness of these spaces, the topology of D L p is also the topology βpd L p, DL 1 qq of uniform convergence on the bounded sets of DL 1 q. In the case of p 1, the duality relation pb9 1 q 1 D L 1, see [3, Proposition 7, p. 13], provides that D L 1 also has the topology of uniform convergence on bounded sets of B9 1. FollowingL.Schwartz in[26], B9 1 denotes the closure of E 1 in B 1 DL 1 which is not the dual space of B9 but in some 8 sense is its analogon for distributions instead of smooth functions. For p 8, D L 8 carries the topology βpd L 8, D 1 L q due to pd 1 1 L q 1 D 1 L 8 [26, p. 200]. By definition, the topology of DL 1 q is the topology of uniform convergence on the bounded sets of D L p and B 9 for q ą 1 and q 1, respectively. In [25], the spaces D L p,c and DL 1 q,c are considered, where the index c designates the topologies κpd L p, DL 1 qq and κpd L 1 q, D Lpq of uniform convergence on compact sets of DL 1 and D q Lp, respectively. Let us mention three reasons why the spaces L q c and D L 1 q,c are of interest: (1) Let E and F be distribution spaces. If E has the ε-property, a kernel distribution Kpx,yq P D 1 x pf yq belongs to the space E x pf y q if it does so scalarly, i.e., Kpx,yq P E x pf y q P F 1 : xkpx,yq,fpyqy P E x. The spaces L q c, 1 ă q ď 8, and DL 1 q,c, 1 ď q ď 8, have the ε-property [25, Proposition 16, p. 59] whereas for 1 ď p ď 8, L p and DL 1 do p not. For L p, 1 ď p ă 8, this can be seen by checking that Kpx,kq pk 1{p {p1`k 2 x 2 qq P D 1 prq pbc 0 satisfies xkpx,kq,ay P L p foralla P l 1 but is not contained in L p pb ε c 0 ; in the case of p 8 one takes Kpx,kq Y px kq instead. For DL 1 q a similar argument with Kpx,kq δpx kq can be used. 2

3 (2) The kernel δpx yq of the identity mapping E x Ñ E y of a distribution space E belongs to E 1 c,y εe x. Thus, e.g., the equation δpx yq 8ÿ H k pxqh k pyq k 0 where the H k denote the Hermite functions, due to G. Arfken in [2, p.529]andp.hirvonenin[15], isvalidinthespacesl q p c,yb ε L p x, D L 1 p q,c,yb ε D L p,x if 1 ď p ă 8, or Mc,y 1 b p ε C 0,x, see [22], or, classically, in Sy 1 bs p x. (3) The classical Fourier transform F: L p Ñ L q, 1 ď p ď 2, 1 {p ` 1{q 1, is well-defined and continuous by the Hausdorff-Young theorem. By means of the kernel e ixy of F we can express the Hausdorff-Young theorem by e ixy P L c pl p x,lq y q L εpl p c,y,lq c,x q Lq y p b ε L q c,x L q c,x p b ε L q y if p ą 1, and by e ixy P L 8 c,xpc 0,y q if p 1. Obvious generalizations are 8ď e ixy P D L q,c,xp pl q q k,y q, 2 ă q ă 8, and k 0 e ixy P D L 8,c,xpO 0 c,y q. Whereas the topology of D L p, 1 ď p ď 8, can be described either by the seminorms p m or, equivalently, by p B : D L p Ñ R, p B pϕq sup xϕ,sy, B Ď DL 1 bounded, q SPB the topology of D L p,c only is described by p C : D L p Ñ R, p C pϕq sup xϕ,sy, C Ď DL 1 compact q SPC for 1 ă p ď 8 and p C : D L 1 Ñ R, p C pϕq sup xϕ,sy, C Ď B 9 1 compact. SPC An analogue statement holds for L p c, 1 ă p ď 8 and M1 c. 3

4 Thus, our task is the description of the topologies κpd L p, DL 1 qq, κpd L 1, B9 1 q, κpl p,l q q and κpm 1, C 0 q (for 1 ă p ď 8) by seminorms involving functions and not sets (Propositions 4, 8, 16 and 18). As a byproduct, the compact sets in DL 1 and q Lq for 1 ď q ă 8 are characterised in Proposition 3 and in Proposition 17, respectively. In addition, we also give characterisations of the compact sets of B9 1 and M 1. The notation generally adopted is the one from [24 28] and [17]. However, we deviate from these references by defining the Fourier transform as ż Fϕpyq e iyx ϕpxqdx, ϕ P S pr n q S. R n We follow [26, p. 36] in denoting by Y the Heaviside-function. The translate of a function f by a vector h is denoted by pτ h fqpxq : fpx hq. Besides the spaces L p pr n q L p, D L ppr n q D L p, 1 ď p ď 8, we use the space M 1 pr n q M 1 of integrable measures which is the strong dual of the space C 0 of continuous functions vanishing at infinity. In measure theory the measures in M 1 usually are called bounded measures whereas J. Horváth, in analogy to the integrable distributions, calls them integrable measures. Here we follow J. Horváth s naming convention. The topologies on Hausdorff locally convex spaces E,F we use are βpe,f q the topology (on E) of uniform convergence on bounded sets of F, κpe,f q the topology (on E) of uniform convergence on absolutely convex compact subsets of F, see [17, p. 235]. Thus, if 1 ă p ď 8 and 1 {p ` 1{q 1, We use the spaces Mc 1 pm1,κpm 1, C 0 qq, L p c pl p,κpl p,l q qq, D L 1,c pd L 1,κpD L 1, B9 1 q, D L p,c pd L p,κpd L p, D 1 L qqq. H s,p F 1 pp1 ` x 2 q s{2 FL p q 1 ď p ď 8,s P R, see [23, Def 3.6.1, p. 108], [1, 7.63, p. 252] or [36]. The weighted L p -spaces are L p k p1 ` x 2 q k{2 L p, k P Z. 4

5 In addition, we consider the following sequence spaces. By s we denote the space of rapidly decreasing sequences, by s 1 its dual, the space of slowly increasing sequences. Moreover we consider the space c 0 of null sequences and its weighted variant pc 0 q k tx P C N : lim jñ8 j k xpjq 0u. The space s 1 is the non-strict inductive limit s 1 lim ÝÑk pc 0 q k with compact embeddings, i.e. an (LS)-space, see [12, p. 132]. The uniquely determined temperate fundamental solution of the iterated metaharmonic operator p1 n q k, where n is the n-dimensional Laplacean, is given by L 2k wherein the distribution 1 2 k`n{2 1 π n{2 Γpkq x k n{2 K n{2 k p x q F 1 pp1 ` x 2 q k q 1 L s x s n 2 s`n 2 1 π n{2 Γps{2q 2 Kn s 2 p x q is defined by analytic continuation with respect to s P C. The symbol Pf in [26, (II, 3; 20), p. 47] is not necessary because L s P H pc s q pb S 1 is an entire holomorphic function with values in S 1 pr n xq [26, p. 47]. In virtue of FL s p1 ` x 2 q s{2 P O M,x we, in fact, have L s P H pc s q pb O 1 C and L s L t L s`t for s,t P C [26, (VI, 8; 5), p. 204]. Particular cases of this formula are: L 0 δ, L 2k p1 n q k δ, p1 n q k L 2k δ, k P N. For s ą 0, L s ą 0 and L s decreases exponentially at infinity. Moreover, L s P L 1 for Res ą 0. In contrast to [29, p. 131] and [36], we maintain the original notation L s for the Bessel kernels and we write L s instead of J s. The spaces DL 1 q can be described as the inductive limit 8ď p1 n q m L q lim H 2m,q ÝÑm, m 0 5

6 see, e.g., [26, p. 205]. The space 9 B 1 of distributions vanishing at infinity has the similar representation 9 B 1 8ď p1 n q m C 0 lim ÝÑm p1 n q m C 0, m 0 by [26, p. 205]. If we equip p1 n q m C 0 with the final topology with respect to p1 n q m an application of de Wilde s closed graph theorem provides the topological equality since by [3, Proposition 7, p. 65] B9 1 is ultrabornological and ÝÑm lim p1 n q m C 0 has a completing web since it is a Hausdorff inductive limit of Banach spaces. 2 Function -seminorms in D L p,c, 1 ă p ď 8 In order to describe the topology of D L p,c p1 ă p ă 8) by function - seminorms it is necessary to characterise the compact sets of the dual space DL 1 q (1 ă q ă 8), defined in [26, p. 200] as D 1 L q pd L pq1, 1 p ` 1 q 1, and endowed with the strong topology βpd 1 L q, D L pq. The description of D L 8,c is already well-known [10]. Due to the definition of the space D L p, 1 ď p ă 8, as the countable projective limit Ş 8 m 0 H2m,p of the Banach spaces H 2m,p, which are called potential spaces in [29, p. 135], we conclude that the strong dual DL 1 coincides q with the countable inductive limit Ť 8 m 0 H 2m,q. Note that the topological identity follows from the ultrabornologicity of pdl 1 q,βpd L 1 q, D Lpqq, which follows for example by the sequence-space representation DL 1 p s1 pb l q given independently by D. Vogt in [31] and by M. Valdivia in [30], by means of Grothendiecks Théorème B [14, p. 17]. The completeness of DL 1 q implies the regularity of the inductive limit Ť 8 m 0 H 2m,q [6, p. 77]. An alternative proof of the representation of DL 1 q as the inductive limit of the potential spaces above can be given using [4, Theorem 5] and the fact that 1 n is a densely defined and invertible, closed operator on L q. We first show that the (LB)-space DL 1 q, 1 ď q ă 8 is compactly regular [6, 6. Definition (c), p. 100]: Proposition 1. If 1 ď q ă 8 the (LB)-space DL 1 q is compactly regular. 6

7 Proof. Compactly regular (LF)-spaces are characterised by condition (Q) ([34, Thm. 2.7, p. 252], [35, Thm. 6.4, p. 112]) which in our case reads P N 0 Dk ą ą ą k DC ą 0 : S 2k,q ď ε S 2m,q `C S P H 2m,q. (Note that H 2m,q ãñ H 2k,q ãñ H 2l,q.) For (Q) see [32, Prop. 2.3, p. 62]. By definition, L 2k S q ď ε L 2m S q `C L 2l S q is equivalent to L2pk lq S q ď ε L2pm lq S q `C S P H P H 2pm lq,q But this inequality follows from Ehrling s inequality [36], which states that for 1 ď q ă 8 and 0 ă s ă ą 0 DC ą 0 : J s ϕ q ď ε ϕ q `C J t ϕ q,ϕ P S. By density of S in H 2pm lq,q this implies the validity of (Q). Remarks 2. (a) By means of M. Valdivia s and D. Vogt s sequence space representation D L p s pb l p given in [30, Thm. 1, p. 766], and [31, (3.2) Theorem, p. 415] and B 9 c 0 pb s, we obtain by [14, Chapter II, Thm. XII, p. 76] that DL 1 q lq pb s 1, 1 ď q ď 8. Using this representation, a further proof of the compact regularity of the (LB)- space DL 1 q is given in Section 5. (b) Differently, the compact regularity of the (LB)-space D 1 L 1 is proven in [10, (3.6) Prop., p. 71]. (c) If q 2, the space D 1 L 2 is isomorphic to the (LB)-space Ť 8 k 0 pl2 q k. The compact regularity of the (LB)-spaces 8ď pl p q k, 1 ď p ď 8, k 0 immediately follows from the validity of condition (Q). For p 1 the compact regularity of the space Ť 8 k 0 pl1 q k was shown in [9, (3.8), Satz (a), (b), p. 28; (3.9) Bem., (a), p. 29]. The next proposition characterises compact sets in D 1 L q. 7

8 Proposition 3. Let 1 ď q ă 8. A set C Ď DL 1 q is compact if and only if for some m P N 0, L 2m C is compact in L q. Proof. ð : The compactness of L 2m C in L q implies its compactness in D 1 L q and, hence, C L 2m pl 2m Cq is compact in D 1 L q. ñ : In virtue of Proposition 1 there is m P N 0 such that C is compact in H 2m,q. The continuity of the mapping ϕ ÞÑ L 2m ϕ, H 2m,q Ñ L q implies the compactness of L 2m C in L q. The following proposition generalizes the description of the topology of the space B c pd L 8,κpD L 8, D 1 L 1 qq by the function -seminorms p g,m pϕq sup gb α ϕ 8, g P C 0, m P N 0 α ďm for ϕ P B D L 8 in [10, (3.5) Cor., p. 71]. Proposition 4. Let 1 ă p ď 8 and 1 {p ` 1{q 1. The topology κpd L p, D 1 L qq of D L p,c is generated by the seminorms D L p Ñ R`, ϕ ÞÑ p g,m pϕq : gp1 n q m ϕ p, g P C 0,m P N 0, or equivalently by ϕ ÞÑ sup gb α ϕ p, g P C 0,m P N 0. α ďm Proof. Due to [10, (3.5) Cor., p. 71] it suffices to assume 1 ă p ă 8. We denote the topology on D L p generated by tp g,m : g P C 0,m P N 0 u by t. Moreover, B 1,p shall denote the unit ball in L p. (a) t Ď κpd L p, D 1 L qq: If U g,m : tϕ P D L p : p g,m pϕq ď 1u is a neighborhood of 0 in t we have U g,m ppu g,m q q by the theorem on bipolars. We show that Ug,m is a compact set in DL 1 q. We have ϕ P U g,m ðñ gpl 2m ϕq P B 1,p ðñ sup ψpb 1,q xψ,gpl 2m ϕqy ď 1 ðñ sup ψpb 1,q xl 2m pgψq,ϕy ď 1 ðñ ϕ P pl 2m pgb 1,q qq. 8

9 Hence, Ug,m L 2m pgb 1,q q Ď DL 1 q. By Proposition 3, Ug,m is compact in DL 1 if there exists l P N q 0 such that L 2l U g,m L 2pl mq pgb 1,q q is compact in L q. Choosing any l ą m, it suffices to show that C : L 2pl mq pgb 1,q q satisfies the three criteria of the M. Fréchet-M. Riesz-A. Kolmogorov- H. Weyl Theorem [28, Thm , p. 140]: (i) Because l ą m, µ : L 2pl mq P L 1 and hence, for ϕ P B 1,q, i.e., C is bounded in L q. µ pgϕq q ď µ 1 g 8, (ii) The set C has to be small at infinity: for ϕ P B 1,q, }Y p. Rqpµ pgϕqq} q ď ˆ ď Y p. Rq py p R pgϕq q 2. qµq ˆ ` Y p. Rq py p. R pgϕq q 2 qµq ˆż ż q ď ˇ µpξqpgϕqpx ξqdξ ˇ dx x ěr ξ ďr{2 1{q ` py p. R{2qµq pgϕq q ż ˆż ď µpξqpgϕqpx ξq dx 1{q q dξ ξ ďr{2 x ěr ` Y p. R{2qµ 1 g 8 ż ˆż ď µpξq dξ pgϕqpzq q dz ξ ďr{2 z ěr{2 ` Y p. R{2qµ 1 g 8 ď µ 1 Y p. R{2qg 8 ` Y p. R{2qµ 1 g 8. 1{q Hence, lim RÑ8 Y p. Rqpµ pgϕqq q 0 uniformly for ϕ P B 1,q. (iii) C is L q -equicontinuous because τ h pµ pgϕqq µ pgϕq q ď τ h µ µ 1 g 8 tends to 0 if h Ñ 0, uniformly for ϕ P B 1,q. 9

10 (b) κpd L p, DL 1 qq Ă t: If C is a κpd L p, DL 1 qq-neighborhood of 0 with C a compact set in D L 1 then, q by Proposition 3, there exists m P N 0 such that the set L 2m C is compact in L q. By means of Lemma 5 below there is a function g P C 0 such that L 2m C Ď gb 1,q, hence C Ď L 2m pgb 1,q q Ď Ug,m. Thus, C Ě U g,m, i.e., C is a neighborhood in t. Lemma 5. Let 1 ď q ă 8. If K Ď L q is compact then there exists g P C 0 such that K Ď gb 1,q. Proof. Apply the Cohen-Hewitt factorization theorem [11, (17.1), p. 114] to the bounded subset K of the (left) Banach module L q with respect to the Banach algebra C 0 having (left) approximate identity te k2 x 2 : k ą 0u. Remark 6. Denoting by τpb, D 1 L q the Mackey-topology on B D 1 L 8 even have for p 8, B D L 8: we B c pb,κpb, D 1 L 1qq pb,tq pb,τpb, D 1 L 1qq, since D 1 L 1 is a Schur space [10, p. 52]. 3 The case p 1 Using the sequence-space representation B 9 1 s 1 pbc 0 given in [3, Theorem 3, p. 13], the compact regularity of the (LB)-space B 9 1 can be shown similarly to the proof of Proposition 12. Moreover, one has the following characterisation of the compact sets of B9 1. Proposition 7. A set C Ď 9 B 1 is compact if and only if for some m P N 0, L 2m C is compact in C 0. Proof. The proof is completely analogous to the one of Proposition 3. Proposition 8. The topology κpd L 1, 9 B 1 q of D L 1,c is generated by the seminorms D L 1 Ñ R`, ϕ ÞÑ p g,m pϕq : gp1 n q m ϕ 1, g P C 0,m P N 0, or equivalently by ϕ ÞÑ sup gb α ϕ 1, g P C 0,m P N 0. α ďm 10

11 Proof. We first show that the topology t generated by the above seminorms is finer than the topology of uniform convergence on compact subsets of B9 1. Similarly to the proof of Proposition 4, we have to show that the set L 2pl mq pgb 1,C0 q is a relatively compact subset of B9 1, where B 1,C0 denotes the unit ball of C 0. We pick l ą m and, by the compact regularity of B9 1 and the Arzela-Ascoli theorem, we have to show that L 2pl mq pgb 1,C0 q is bounded as a subset of C 0 and equicontinous as a set of functions on the Alexandroff compactification of R n. Since l ą m, we have that L 2pl mq P L 1. For every ϕ P B 1,C0, Young s convolution inequality implies }L 2pl mq pgϕq} 8 ď }L 2pl mq } 1 }gϕ} 8 ď }L 2pl mq } 1 }g} 8, i.e. L 2pl mq pgb 1,C0 q Ď C 0 is a bounded set. Since a translation of a convolution product can be computed by applying the translation to one of the factors, we can again use Young s convolution inequality to obtain }τ h pl 2pl mq pgϕqq L 2pl mq pgϕq} 8 }pτ h L 2pl mq L 2pl mq q pgϕq} 8 ď }pτ h L 2pl mq L 2pl mq q} 1 }g} 8 for all ϕ in the unit ball of C 0. From this inequality we may conclude that L 2pl mq pgb 1,C0 q is equicontinuous at all points in R n. Therefore we are left to show that it is also equicontinous at infinity. In order to do this, first observe that g P C 0 and L 2pl mq L 2pl mq. Moreover, Lebesgue s theorem on dominated convergence implies that the convolution of a function in C 0 and an L 1 -function is contained in C 0. Finally, by the above reasoning the inequality ż pl 2pl mq pgϕqqpxq ď gpx ξq L 2pl mq pξq dξ pl 2pl mq g qpxq R n shows that L 2pl mq pgb 1,C0 q is equicontinuous at infinity. The proof that κpd L 1, 9 B 1 q is finer than t is completely analogous to the corresponding part of Proposition 4 if we can show that every compact subset of C 0 is contained in gb 1,C0 for some g P C 0. Let C Ď C 0 be a compact set. By the Arzela-Ascoli theorem, it is equicontinuous at infinity, i.e., for every k P N there is an R k such that hpxq ď 1{k for h P C and all x ą R k. This condition implies the existence of the required function g P C 0 with the above property. 11

12 4 Properties of the spaces D L p,c In [25, p. 127], L. Schwartz proves that the spaces B c D L 8,c and B D L 8 have the same bounded sets, and that on these sets the topology κpb, D 1 L 1 q equals the topology induced by E C 8. Moreover, κpb, D 1 L 1 q is the finest locally convex topology with this property. By an identical reasoning we obtain: Proposition 9. Let 1 ď p ă 8. (a) The spaces D L p and D L p,c have the same bounded sets. These sets are relatively κpd L p, DL 1 qq-compact and relatively κpd L 1, B9 1 q-compact for 1 ă p ă 8 and p 1, respectively. (b) The topology κpd L p, D 1 L qq of D L p,c is the finest locally convex topology on D L p which induces on bounded sets the topologies of E or D 1 or D 1 L p. In the following proposition we collect some further properties of the spaces D L p: Proposition 10. Let 1 ď p ď 8. The spaces D L p,c are complete, quasinormable, semi-montel and hence semireflexive. D L p,c is not infrabarrelled and hence neither barrelled nor bornological. D L p,c is a Schwartz space but not a nuclear space. Proof. (1) The completeness follows either from [19, (1), p. 385], or from [17, Ex. 7(b), p. 243]. It must be taken into account that DL 1 and B 9 1 are q bornological. (2) D L p,c is quasinormable since its dual DL 1 q is boundedly retractive (see the argument in [10, p. 73] and use [13, Def. 4, p. 106]) which, by [6, 7. Prop., p. 101] is equivalent with its compact regularity (Proposition 1). (3) Since bounded and relative compact sets coincide in D L p,c it is a semi- Montel space. (4) Infrabarrelledness and the Montel-property would imply that D L p,c is Montel which in turn implies the coincidence of the topologies κpd L p, DL 1 qq and βpd L p, DL 1 qq. This is a contradiction, since together with the compact regularity of the inductive limit representation DL 1 Ť H 2m,q this would q imply that for every m there is m 1 such that the unit ball of H 2m,q is contained and relatively compact in H 2m1,q. In case m 1 ď m, the continuous 12

13 inclusion H 2m1,q Ď H 2m,q would give that the unit ball of H 2m,q is compact, i.e., that H 2m,q is finite dimensional; in case m 1 ě m, by tranposition the inclusion H 2m,q Ñ H 2m1,q and a fortiori the inclusion H 2m,q Ñ L q would be compact, which cannot be the case [1, Example 6.11, p. 173]. (5) D L p,c is a Schwartz space [13, Def. 5, p. 117] because it is quasinormable and semi-montel. (6) Using the sequence-space representation D L p,c l p c p bs, we can conclude by [14, Ch. II 3 n 2, Prop. 13, p. 76] that D L p,c is nuclear if and only if l p c is nuclear. We first consider the case p ą 1. The nuclearity of l p c would imply l 1 tl p c u l1 pb π l p c l1 pb ε l p c l1 xl p c y where l 1 tl p cu and l 1 xl p cy is the space of absolutely summable and of unconditionally summable sequences in l p c, respectively, see [18, pp. 341, 359]. We now proceed by giving an example of an element of the space at the very right which is not contained in the space at the very left. Fix ε ą 0 small enough. Using Hölder s inequality, we observe that 8ÿ δ jk k p1`εq{pf 8ÿ 1 k ˇ j 1 k p1`εq{pf kˇ ď C}f} q k 1 for every f pf k q 8 k 1 P lq which together with the characterisation of unconditional convergence in [33, Theorem 1.15] and the condition that compact subsets of l q are small at infinity implies that the sequence is unconditionally convergent. On the other hand taking pk α q 8 k 1 P c 0 with α 1 1`ε yields p 8ÿ j 1 δ jk k p1`εq{p g k q k } p 8ÿ j 1 k 1 j p1`εq{p α 8ÿ j 1 1 j 8 fromwhichwemayconcludethat pδ jk k p1`εq{p q j,k isnotanabsolutelysummable sequence in l p c. For the case p 1 we use the Grothendieck-Pietsch criterion, see [18, p. 497], and observe that Λpc 0,`q l 1 c. Choosing α p1{kq 8 k 1 provides the necessary sequence with pα k {β k q 8 k 1 R l1 for every β P c 0,` with β ě α. Remark 11. The above proof actually shows that l p c, for 1 ď p ď 8 is not nuclear. From this we may also conclude that DL 1 p,c, 1 ď p ď 8, is not nuclear. Analogous to the table with properties of the spaces D F, D 1F (defined in [17, p. 172,173]) in [5, p. 19], we list properties of D L p and D L p,c in the following table (1 ď p ď 8): 13

14 property D L p D L p,c complete ` ` quasinormable ` ` metrizable ` bornological ` barrelled ` (semi-)reflexive ` (1 ă p ă 8) ` semi-montel ` Schwartz ` nuclear 5 Sequence space representations of the spaces D L p,c, DL 1 q,c and the compact regularity of D 1 L q P. and S. Dierolf conjectured in [10, p. 74] the isomorphy D L 8,c B c l 8 c pbs, where l 8 c pl 8,κpl 8,l 1 qq pl 8,τpl 8,l 1 qq. This conjecture is proven in [7, 1. Theorem, p. 293]. More generally, we obtain: Proposition 12. Let 1 ď p,q ď 8. (a) D L p,c l p c p bs; (b) D 1 L q,c lq c p bs 1. Proof. (1) By means of M. Valdivia s isomorphy (see Remark 2 (a)) D 1 L q l q pbs 1 it follows, for q ă 8, by [8, 4.1 Theorem, p. 52 and 2.2 Prop., p. 46]: D L p,c l p c p bs, 1 {p ` 1{q 1. If p 1, D L 1,c p 9 B 1 q 1 c pc 0 pbs 1 q 1 c l 1 c p b s by [8, 4.1 Theorem, p. 52 and 2.2 Prop., p. 46] and [3, Prop. 7, p. 13]. The case p 8 is a special case of [7, 1. Theorem, p. 293]. (2) The second Theorem on duality of H. Buchwalter [20, (5), p. 302] yields for two Fréchet spaces E,F: pe pb ε F q 1 c E1 c p b π F 1 c, 14

15 and hence, for E l p and F s, E pbf l p pbs which implies if 1 ă q ď 8, 1 {p ` 1{q 1. If q 1, D 1 L q,c pl p pbsq 1 c l q c p bs 1 D 1 L 1,c p 9 Bq 1 c pc 0 p bsq 1 c l1 c p bs 1. An alternative proof for (2) can also by given using [8, 4.1 Theorem, p. 52 and 2.2 Prop., p. 46] again. Remark 13. The sequence space representations of D L p,c and D 1 L q,c yield a further proof of the quasinormability of these spaces. Indeed, l p c and lq c are quasinormable since their duals are Banach spaces and thus, they fulfill the strict Mackey convergence condition [13, p. 106]. The claim follows from the fact that the completed tensor product with s or s 1 remains quasinormable by [14, Chapter II, Prop. 13.b, p. 76]. Thecompactregularityof D 1 L q asacountableinductivelimitofbanachspaces is proven in Proposition 1. By means of the sequence space representation D 1 L q lq pbs 1 which has been presented in Remark 2 (a) we can give a second proof: Proposition 14. Let E be a Banach space. (a) The inductive limit representations are valid. s 1 pbe lim ÝÑk ppc 0 q k pb ε Eq lim ÝÑk pc 0 pb ε Eq k lim ÝÑk pc 0 peqq k (b) The inductive limit lim ÝÑk pc 0 peqq k is compactly regular. Proof. (1) The assertion follows from [16, Theorem 4.1, p. 55]. (2) The compact regularity of lim ÝÑk pc 0 peqq k isaconsequenceof[35,thm.6.4, p. 112] (or [34, Thm. 2.7, p. 252]) and the validity of condition (Q) (see [32, Prop. 2.3, p. 62]). The condition (Q) reads P N 0 Dk ą ą ą k DC ą px j q j P pc 0 peqq m : sup j j k x j ď εsup j j m x j `Csup j l x j. j For the sequences px j q j P pc 0 peqq m the sequence p x j q j is contained in s 1 lim ÝÑk pc 0 q k. Thus, (Q) is fulfilled because s 1 is an (LS)-space. 15

16 Corollary 15. The spaces B9 1 lim ÝÑk pc 0 pc 0 qq k and DL 1 lim pc q ÝÑk 0pl q qq k, for 1 ď q ď 8, are compactly regular countable inductive limits of Banach spaces. 6 Function -seminorms in L p c and M 1 c. Motivated by the description of the topology κpl p,l q q of the sequence space l p c by the seminorms 8ÿ p 1{p l p Ñ R, x px j q jpn ÞÑ p g pxq gpjq xpjq g x p, g P c 0, see I. Mack s master thesis [21] for the case p 8, we also investigate the description of the topologies κpl p,l q q and κpm 1, C 0 q by means of function -seminorms. Denoting the space of bounded and continuous functions by B 0 (see [24, p. 99]) we recall that B 0 c pb 0,κpB 0, M 1 qq has the topology of R. C. Buck. j 1 Proposition 16. Let 1 ă p ď 8. The topology κpl p,l q q of L p c by the family of seminorms is generated L p Ñ R`,f ÞÑ p g,h pfq h pgfq p, g P C 0,h P L 1. Proof. (a) t Ď κpl p,l q q where t denotes the topology generated by the seminorms p g,h above and let U g,h : tf P L p : p g,h pfq ď 1u be a neighborhood of 0 in t. We show that (i) U g,h pgpȟ B 1,qqq, (ii) gpȟ B 1,qq is compact in L q. ad (i): If S P gpȟ B 1,qq Ď L q there exists ψ P B 1,q with S gpȟ ψq. For f P L p with p g,h pfq ď 1 we obtain xf,sy ˇˇxf,gp ȟ ψqyˇˇ xh pgfq,ψy and sup xf,sy h pgfq p p g,h pfq, SPgpȟ B 1,qq 16

17 so (i) follows. ad (ii): The 3 criteria of the M. Fréchet-M. Riesz-A. Kolmogorov-H. Weyl Theorem are fulfilled: (a) The set gpȟ B 1,qq is bounded in L q. (b) The set gpȟ B 1,qq is small at ą 0 DR ą 0 such that gpxq Y p x Rq ă ε for all x P R n and hence, Y p. Rqgpȟ B 1,qq q ď ε h 1. (c) The set gpȟ B 1,qq is L q -equicontinuous: }τ s pgpȟ fqq gpȟ fq} q ď ď pτs g gqτ s pȟ fq q ` gpτs pȟ fq pȟ fqs q ď τ s g g 8 h 1 ` g 8 τs ȟ ȟ 1. If s tends to 0, τ s g g 8 Ñ 0 because g is uniformly continuous, and τ s ȟ ȟ 1 Ñ 0 because the L 1 -modulus of continuity is continuous. (b) t Ě κpl p,l q q: For a compact set C Ď L q we have to show that there exist g P C 0, h P L 1 such that p g,h pfq gph fq p ě sup xf, Sy spc for f P L p. By applying the factorization theorem [11, (17.1). p. 114] twice, there exist g P C 0, h P L 1 such that C Ď gpȟ B 1,qq. More precisely, take L 1 as the Banach algebra with respect to convolution and C 0 as the Banach algebra with respect to pointwise multiplication, L p as the Banach module. As approximate units we use e k2 x 2, k ą 0, in the case of C 0 and p4πtq n{2 e x 2 {4t, t ą 0, in the convolution algebra L 1. Then, sup xf, Sy ď sup ˇˇxf,gp ȟ ψqyˇˇ sup xh pgfq,ψy h pgfq p, SPC ψpb 1,q ψpb 1,q which finishes the proof. Proposition 17. Let 1 ď q ă 8 and C Ď L q. Then, C compact ðñ Dg P C 0 Dh P L 1 : C Ď gph B 1,q q. 17

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