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arXiv:1711.06577v1 [math.FA] 17 Nov 2017

A simpler description of the

κ-topologies on

the spaces

_D

_L

p

,

L

p

,

M

1

Christian Bargetz

∗

_{, Eduard A. Nigsch}

†

_{, Norbert Ortner}

‡

November 2017

Abstract

The κ-topologies on the spaces DLp, Lp 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 semi-norms generating theκ-topologies on the above spaces of functions and measures defined by integrability properties. In addition, we present a sequence-space representation of the spaces DLp 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 DL1p,L

p _{and M}1 .

Keywords: locally convex distribution spaces, topology of uniform conver-gence on compact sets, p-integrable smooth functions, compact sets

MSC2010 Classification: 46F05, 46E10, 46A13, 46E35, 46A50, 46B50

∗_{Institut f¨ur Mathematik, Universit¨at Innsbruck, Technikerstraße 13, 6020 Innsbruck,}

Austria. e-mail: christian.bargetz@uibk.ac.at.

†_{Wolfgang Pauli Institut, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria. e-mail:}

eduard.nigsch@univie.ac.at.

‡_{Institut f¨ur Mathematik, Universit¨at Innsbruck, Technikerstraße 13, 6020 Innsbruck,}

1

Introduction

In the context of the convolution of distributions, L. Schwartz introduced the spaces DLp of C8-functions whose derivatives are contained in Lp and the

spaces DL1q of finite sums of derivatives of L

q_{-functions, 1 ď p, q ď 8. The}

topology of DLp is defined by the sequence of (semi-)norms

DLp ÑR_`, ϕ ÞÑp_mpϕq:“ sup

|α|ďm

kBαϕk_p,

whereas D1

Lq “ pDLpq1 for 1{p`1{q “1 if p ă 8 and D1

L1 “ pB9q1, carry the

strong dual topology.

Equivalently, for 1ďpă 8, by barrelledness of these spaces, the topology of

DLp is also the topologyβpDLp,D_L1qq of uniform convergence on the bounded

sets of D1

Lq. In the case of p “ 1, the duality relation pB9

1_q1 _{“ DL1, see [3,}

Proposition 7, p. 13], provides that DL1 also has the topology of uniform

convergence on bounded sets ofB91_{. Following L. Schwartz in [26],B9}1_denotes

the closure of E1 inB1 “DL18 which is not the dual space of B9 but in some

sense is its analogon for distributions instead of smooth functions. Forp“ 8,

DL8 carries the topology βpDL8,D1

L1q due to pD_L11q1 “DL8 [26, p. 200].

By definition, the topology ofD1

Lq is the topology of uniform convergence on

the bounded sets of DLp and B9 for q ą1 and q“1, respectively.

In [25], the spaces DLp_,c and D_L1q_,c are considered, where the index c

desig-nates the topologiesκpDLp,D_L1qq andκpD_L1q,DLpqof uniform convergence on

compact sets of D1

Lq andDLp, respectively. Let us mention three reasons why

the spaces Lq

c and DL1q_,c are of interest:

(1) Let E and F be distribution spaces. IfE has the ε-property, a kernel distribution Kpx, yq PD1

xpFyq belongs to the spaceExpFyqif it does so

“scalarly”, i.e.,

Kpx, yq PExpFyq ðñ @f PF1 :xKpx, yq, fpyqy PEx.

The spaces Lq

c, 1ăq ď 8, and DL1q_,c, 1ďq ď 8, have the ε-property

[25, Proposition 16, p. 59] whereas for 1 ď p ď 8, Lp _{and D}1 Lp do

not. For Lp, 1 ďp ă 8, this can be seen by checking that Kpx, kq “ pk1_{p

{p1`k2 x2

qq PD1_pRqbpc

0satisfiesxKpx, kq, ay PLpfor alla Pℓ 1

but is not contained in Lp _bp

εc0; in the case of p“ 8 one takes Kpx, kq “ Ypx´kqinstead. For D1

Lq a similar argument withKpx, kq “δpx´kq

(2) The kernelδpx´yqof the identity mapping Ex ÑEy of a distribution

space E belongs to E_c,y1 ε Ex. Thus, e.g., the equation

δpx´yq “

8 ÿ

k“0

HkpxqHkpyq

where the Hk denote the Hermite functions, due to G. Arfken in [2,

p. 529] and P. Hirvonen in [15], is valid in the spacesLq

c,ybpεLpx,DL1q_,c,ybpε

DLp_,x if 1ďpă 8, orM

1

c,ybpεC0,x, see [22], or, classically, inSy1bpSx.

(3) The classical Fourier transform

F: Lp ÑLq, 1ďpď2, 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 PLcpLp_x, Lq_yq –LεpLpc,y, L q

c,xq “L q

y bpεLqc,x

“Lqc,xbpεLqy

if pą1, and by e´ixy _PL8

c,xpC0,yqif p“1. Obvious generalizations are

e´ixy PDLq_,c,xp

8 ď

k“0

pLqqk,yq, 2ăq ă 8, and

e´ixy PDL8_,c,xpO

0 c,yq.

Whereas the topology of DLp, 1 ď p ď 8, can be described either by the

seminorms pm or, equivalently, by

pB: DLp ÑR, p_Bpϕq “sup

SPB

|xϕ, Sy|, B ĎDL1q bounded,

the topology of DLp_,c only is described by

pC: DLp ÑR, p_Cpϕq “sup

SPC

|xϕ, Sy|, C ĎDL1q compact

for 1ăpď 8 and

pC: DL1 ÑR, p_Cpϕq “sup

SPC

|xϕ, Sy|, CĎB91 compact.

An analogue statement holds for Lp

Thus, our task is the description of the topologies κpDLp,D_L1qq, κpDL1,B91q,

κpLp_{, L}q_{q and κpM}1

,C0q (for 1ăpď 8) by seminorms involving functions

and not sets (Propositions 4, 8, 16 and 18). As a byproduct, the compact sets in D1

Lq and L

q _{for 1 ďq ă 8 are characterised in Proposition 3 and in}

Proposition 17, respectively. In addition, we also give characterisations of the compact sets of B91 and M1

.

The notation generally adopted is the one from [24–28] and [17]. However, we deviate from these references by defining the Fourier transform as

Fϕpyq “

ż

Rn

e´iyxϕpxqdx, ϕ PSpRn_{q “ S.}

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 pτhfqpxq :“ fpx´hq. Besides

the spaces Lp_pRn_{q “ L}p_{, DL}

ppRnq “ DLp, 1 ď p ď 8, we use the space

M1

pRn_{q “M}1

of integrable measures which is the strong dual of the space

C0 of continuous functions vanishing at infinity. In measure theory the

mea-sures in M1

usually are called bounded measures whereas J. Horv´ath, in analogy to the integrable distributions, calls them integrable measures. Here we follow J. Horv´ath’s naming convention.

The topologies on Hausdorff locally convex spaces E, F we use are

βpE, Fq – the topology (onE) of uniform convergence onbounded sets ofF,

κpE, Fq – the topology (on E) of uniform convergence on absolutely convex compact subsets of F, see [17, p. 235].

Thus, if 1ăpď 8 and 1_{

p`1{q“1,

M1

c “ pM 1

, κpM1 ,C0qq, Lpc “ pLp, κpLp, Lqqq,

DL1_,c “ pDL1, κpDL1,B91q,

DLp_,c “ pDLp, κpDLp,D_L1qqq.

We use the spaces

Hs,p“F´1pp1` |x|2q´s{2

FLpq 1ďpď 8, sPR_,

see [23, Def 3.6.1, p. 108], [1, 7.63, p. 252] or [36]. The weighted Lp_-spaces

are

Lp_k “ p1` |x|2q´k{2

In addition, we consider the following sequence spaces. By s we denote the space of rapidly decreasing sequences, by s1 its dual, the space of slowly increasing sequences. Moreover we consider the space c0 of null sequences

and its weighted variant

The space s1 is the non-strict inductive limit

s1 “lim_ÝÑ

k

pc0q_´k

with compact embeddings, i.e. an (LS)-space, see [12, p. 132].

The uniquely determined temperate fundamental solution of the iterated metaharmonic operatorp1´∆nqk, where ∆n is then-dimensional Laplacean,

is given by

is defined by analytic continuation with respect to sPC_{. The symbol Pf in}

[26, (II, 3; 20), p. 47] is not necessary because LsP HpCsqbpS1 is an entire

holomorphic function with values in S1pRn

xq[26, p. 47]. In virtue of

Particular cases of this formula are:

L0 “δ, L_´2_k “ p1´∆_nqkδ, p1´∆_nqkL2_k “δ, k PN.

For s ą 0, Ls ą 0 and Ls decreases exponentially at infinity. Moreover, Ls P L

1

for Res ą 0. In contrast to [29, p. 131] and [36], we maintain the original notation Ls for the Bessel kernels and we write Ls˚ instead of Js.

The spaces DL1q can be described as the inductive limit

see, e.g., [26, p. 205]. The space B91 _{of distributions vanishing at infinity has}

top1´∆nqm an application of de Wilde’s closed graph theorem provides the

topological equality since by [3, Proposition 7, p. 65] B91 _{is ultrabornological}

and lim_ÝÑmp1´∆nqmC0 has a completing web since it is a Hausdorff inductive

limit of Banach spaces.

2

“Function”-seminorms in

_D

_Lp_,c

,

1

ă

p

ď 8

In order to describe the topology of DLp_,c p1 ă p ă 8) by

“function”-seminorms it is necessary to characterise the compact sets of the dual space

D1

and endowed with the strong topologyβpD1

Lq,DLpq. The description ofDL8_,c

is already well-known [10].

Due to the definition of the space DLp, 1ďpă 8, as the countable

projec-tive limit Ş8_m“0H 2_m,p

of the Banach spaces H2_m,p

, which are called “poten-tial spaces” in [29, p. 135], we conclude that the strong dual D1

Lq coincides

with the countable inductive limit Ť8_m“0H ´2_m,q

. Note that the topological identity follows from the ultrabornologicity ofpD1

Lq, βpD

1

Lq,DLpqq, which

fol-lows for example by the sequence-space representation D1

Lp – s1 bp ℓ

q _given

independently by D. Vogt in [31] and by M. Valdivia in [30], by means of Grothendiecks Th´eor`eme B [14, p. 17]. The completeness ofD1

Lq implies the

regularity of the inductive limit Ť8_m“0H´

2m,q _{[6, p. 77].}

An alternative proof of the representation of DL1q 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 onLq.

We first show that the (LB)-space D1

Lq, 1 ďq ă 8 is compactly regular [6,

6. Definition (c), p. 100]:

Proposition 1. If 1ďq ă 8 the (LB)-space D1

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 as:

@mPN0 Dk ąm @εą0 @ℓąk DC ą0 : kSk2k,q ďεkSk2m,q `CkSk2ℓ,q @S PH

´2_m,q .

(Note that H´2_m,q

ãÑ H´2_k,q

ãÑ H´2_ℓ,q

.) For (Q) see [32, Prop. 2.3, p. 62]. By definition,

kL2_k˚Sk_q ďεkL2_m˚Sk_q`CkL2_ℓ˚Sk_q @S PH´ 2_m,q

is equivalent to

L2_pk´ℓq˚S

qďε

L2_pm´ℓq˚S

q`CkSkq @S P H

´2_pm´ℓq,q

But this inequality follows from Ehrling’s inequality [36], which states that for 1ďq ă 8 and 0ăs ăt,

@εą0DC ą0 : kJsϕk_q ďεkϕk_q`CkJtϕk_q, ϕPS.

By density of S in H´2_pm´ℓq,q

this implies the validity of (Q).

Remarks 2. (a) By means of M. Valdivia’s and D. Vogt’s sequence space representation DLp – sbp ℓp given in [30, Thm. 1, p. 766], and [31,

(3.2) Theorem, p. 415] and B9 – c0 bp s, we obtain by [14, Chapter

II, Thm. XII, p. 76] that DL1q – ℓ

q _{bp s}1_{, 1 ď q ď 8. Using this}

representation, a further proof of the compact regularity of the (LB)-space D1

Lq is given in Section 5.

(b) Differently, the compact regularity of the (LB)-space D_L11 is proven in

[10, (3.6) Prop., p. 71].

(c) If q “ 2, the space D_L12 is isomorphic to the (LB)-space

Ť8 k“0pL

2

q´k.

The compact regularity of the (LB)-spaces

8 ď

k“0

pLpq´k, 1ďpď 8,

immediately follows from the validity of condition (Q). For p “ 1 the compact regularity of the spaceŤ8_k“0pL

1

q´k was shown in [9, (3.8), Satz

(a), (b), p. 28; (3.9) Bem., (a), p. 29].

Proposition 3. Let 1ďq ă 8. A set CĎD1

Lq is compact if and only if for

some mP N_{0, L2m˚C is compact in L}q_.

Proof. “ð”: The compactness of L2_m˚C in Lq implies its compactness in D1

Lq and, hence, C “L´2m˚ pL2m˚Cq is compact in DL1q.

“ñ”: In virtue of Proposition 1 there is m P N0 such that C is compact in H´2_m,q

. The continuity of the mapping ϕ ÞÑL2m ˚ϕ, H´ 2_m,q

ÑLq _implies

the compactness of L2_m ˚C inLq.

The following proposition generalizes the description of the topology of the space Bc “ pDL8, κpDL8,D1

L1qq by the “function”-seminorms

pg,mpϕq “ sup |α|ďm

kgBα_ϕk

8, g PC0, mPN0

for ϕ PB “DL8 in [10, (3.5) Cor., p. 71].

Proposition 4. Let 1ăpď 8 and 1_{

p`1{q“1. The topology κpDLp,D_L1qq

of DLp_,c is generated by the seminorms

DLp ÑR_`, ϕ ÞÑp_g,mpϕq:“kgp1´∆_nqmϕk_p, g P C0, m PN0,

or equivalently by

ϕÞÑ sup

|α|ďm

kgBαϕk_p, g PC0, mP N0.

Proof. Due to [10, (3.5) Cor., p. 71] it suffices to assume 1 ă p ă 8. We denote the topology on DLp generated by tp_g,m : g P C0, m P N0u by t.

Moreover, B1_,p shall denote the unit ball inLp.

(a) tĎκpDLp,D_L1qq:

If Ug,m :“ tϕ P DLp : p_g,mpϕq ď 1u is a neighborhood of 0 in t we have

Ug,m “ ppUg,mq˝_q˝ _{by the theorem on bipolars. We show that U}˝ g,m is a

compact set in D1

Lq. We have

ϕ PUg,m ðñgpL´2m˚ϕq PB1,p

ðñ sup

ψPB1,q

|xψ, gpL´2_m˚ϕqy| ď1

ðñ sup

ψPB1,q

|xL´2_m˚ pgψq, ϕy| ď1

Hence, U˝

pgB1_,qqsatisfies the three criteria of the M. Fr´echet-M. Riesz-A.

Kolmogorov-H. Weyl Theorem [28, Thm. 6.4.12, p. 140]:

(iii) C is Lq_{-equicontinuous because}

kτhpµ˚ pgϕqq ´µ˚ pgϕqk_q ďkτhµ´µk1kgk₈

(b) κpDLp,D_L1qq Ăt:

If C˝ is a κpDLp,D_L1qq-neighborhood of 0 with C a compact set in D

1

Lq then,

by Proposition 3, there exists m P N0 such that the set L2m˚C is compact

in Lq_{. By means of Lemma 5 below there is a function g P C}

0 such that L2_m ˚C ĎgB1_,q, hence C Ď L_´2_m˚ pgB1_,qq Ď U_g,m˝ . Thus, C˝ Ě Ug,m, i.e., C˝ _{is a neighborhood in t.}

Lemma 5. Let 1 ď q ă 8. If K Ď Lq is compact then there exists g P C0

such that K ĎgB1,q.

Proof. Apply the Cohen-Hewitt factorization theorem [11, (17.1), p. 114] to the bounded subset K of the (left) Banach module Lq _{with respect to the}

Banach algebra C0 having (left) approximate identity te´k

2

|x|2 _:ką0u.

Remark 6. Denoting by τpB,D1

L1q the Mackey-topology on B “ DL8 we

even have for p“ 8, B “DL8:

Bc “ pB, κpB,DL11qq “ pB, tq “ pB, τpB,D_L11qq,

since D_L11 is a Schur space [10, p. 52].

3

The case

p

“

1

Using the sequence-space representationB91 –s1bpc0 given in [3, Theorem 3,

p. 13], the compact regularity of the (LB)-spaceB91 _{can be shown similarly to}

the proof of Proposition12. Moreover, one has the following characterisation of the compact sets of B91.

Proposition 7. A set C Ď B91 is compact if and only if for some m P N0, L2_m ˚C is compact in C0.

Proof. The proof is completely analogous to the one of Proposition 3.

Proposition 8. The topology κpDL1,B91q of DL1_,c is generated by the

semi-norms

DL1 ÑR_`, ϕ ÞÑp_g,mpϕq:“kgp1´∆_nqmϕk

1, g PC0, m PN0,

or equivalently by

ϕ ÞÑ sup

|α|ďm

kgBα_ϕk

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

9

B1. Similarly to the proof of Proposition 4, we have to show that the set

L2_pℓ´mq˚ pgB1_,C0q is a relatively compact subset of B9

1_{, where B}

1_,C0 denotes

the unit ball of C0. We pick ℓ ą m and, by the compact regularity of B91

and the Arzela-Ascoli theorem, we have to show that L2_pℓ´mq ˚ pgB1_,C0q is

bounded as a subset of C0 and equicontinous as a set of functions on the

Alexandroff compactification of Rn_{. Since ℓąm, we have that L}

2_pℓ´mq PL 1

. For every ϕ PB1_,C0, Young’s convolution inequality implies

}L2_pℓ´mq˚ pgϕq}₈ď }L2_pℓ´mq}1}gϕ}₈ ď }L2_pℓ´mq}1}g}₈,

i.e. L2_pℓ´mq˚ pgB1_,C0q ĎC0 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

}τhpL2_pℓ´mq˚ pgϕqq ´L2_pℓ´mq˚ pgϕq}₈ “ }pτhL2_pℓ´mq´L2_pℓ´mqq ˚ pgϕq}₈

ď }pτhL2_pℓ´mq´L2_pℓ´mqq}1}g}₈

for all ϕ in the unit ball of C0. From this inequality we may conclude that L2_pℓ´mq˚ pgB1_,C0qis 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| PC0 and |L2_pℓ´mq| “L2_pℓ´mq. Moreover, Lebesgue’s theorem

on dominated convergence implies that the convolution of a function in C0

and an L1

-function is contained in C0. Finally, by the above reasoning the

inequality

|pL2_pℓ´mq˚ pgϕqqpxq| ď ż

Rn

|gpx´ξq||L2_pℓ´mqpξq|dξ“ pL2_pℓ´mq˚ |g|qpxq

shows that L2_pℓ´mq˚ pgB1_,C0q is equicontinuous at infinity.

The proof that κpDL1,B91q is finer than t is completely analogous to the

corresponding part of Proposition4if we can show that every compact subset of C0 is contained ingB1_,C0 for some g P C0. Let C ĎC0 be a compact set.

By the Arzela-Ascoli theorem, it is equicontinuous at infinity, i.e., for every

k P N _{there is an Rk such that |hpxq| ď 1{k for h P C and all |x| ą Rk.}

This condition implies the existence of the required function g P C0 with the

4

Properties of the spaces

_D

_Lp_,c

In [25, p. 127], L. Schwartz proves that the spacesBc “DL8_,c andB “DL8

have the same bounded sets, and that on these sets the topology κpB,D1 L1q

equals the topology induced by E “ C8. Moreover, κpB,D_L11q is the finest

locally convex topology with this property. By an identical reasoning we obtain:

Proposition 9. Let 1ďpă 8.

(a) The spaces DLp and DLp_,c have the same bounded sets. These sets are

relatively κpDLp,D_L1qq-compact and relatively κpDL1,B91q-compact for

1ăpă 8 and p“1, respectively.

(b) The topology κpDLp,D_L1qq of DLp_,c is the finest locally convex topology

onDLp which induces on bounded sets the topologies ofE or D1 or D_L1p.

In the following proposition we collect some further properties of the spaces

DLp:

Proposition 10. Let 1 ď p ď 8. The spaces DLp_,c are complete,

quasi-normable, semi-Montel and hence semireflexive. DLp_,c is not infrabarrelled

and hence neither barrelled nor bornological. DLp_,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 D1

Lq and B91 are

bornological.

(2) DLp_,c is quasinormable since its dualD_L1q 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 DLp_,c it is a

semi-Montel space.

(4) Infrabarrelledness and the Montel-property would imply that DLp_,c is

Montel which in turn implies the coincidence of the topologies κpDLp,D_L1qq

and βpDLp,D_L1qq. This is a contradiction, since together with the compact

regularity of the inductive limit representation D1 Lq “

Ť

H´2_m,q

this would imply that for some mthe unit ball ofH´2m,q _{is compact which is impossible}

(5) DLp_,c is a Schwartz space [13, Def. 5, p. 117] because it is quasinormable

and semi-Montel.

(6) Using the sequence-space representation DLp_,c –ℓp_c bps, we can conclude

by [14, Ch. II§3 n°2, Prop. 13, p. 76] thatDLp_,c is nuclear if and only ifℓp_c is

nuclear. We first consider the case pą1. The nuclearity of ℓp

c would imply

cy is the space of absolutely summable and of

uncon-ditionally summable sequences in ℓpc, 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 H¨older’s inequality, we observe that

›

q _{which together with the characterisation of}

uncon-ditional convergence in [33, Theorem 1.15] and the condition that compact subsets ofℓq _{are small at infinity implies that the sequence is unconditionally}

convergent. On the other hand taking pk´αq8

k“1 Pc0 with α “1´

qj,k is not an absolutely summable

sequence in ℓp c.

For the casep“1 we use the Grothendieck-Pietsch criterion, see [18, p. 497], and observe that Λpc0_,`q “ℓ

Remark 11. The above proof actually shows that ℓp

c, for 1ď pď 8 is not

nuclear. From this we may also conclude that DL1p_,c, 1 ď p ď 8, is not

nuclear.

Analogous to the table with properties of the spacesDF_{, D}1F _{(defined in [17,}

p. 172,173]) in [5, p. 19], we list properties of DLp and DLp_,c in the following

property DLp DLp_,c

complete ` `

quasinormable ` `

metrizable ` ´

bornological ` ´

barrelled ` ´

(semi-)reflexive `(1ăpă 8) `

semi-Montel ´ `

Schwartz ´ `

nuclear ´ ´

5

Sequence space representations of the spaces

D

Lp_,c

,

D

_L1_q,c

and the compact regularity of

D

_L1q

P. and S. Dierolf conjectured in [10, p. 74] the isomorphy

DL8_,c “Bc –ℓ8

c bp s,

where ℓ8c “ pℓ8, κpℓ8, ℓ 1

qq “ pℓ8, τpℓ8, ℓ1

qq. This conjecture is proven in [7, 1. Theorem, p. 293]. More generally, we obtain:

Proposition 12. Let 1ďp, qď 8.

(a) DLp_,c –ℓp_c bp s;

(b) D1 Lq_,c –ℓ

q cbp s1.

Proof. (1) By means of M. Valdivia’s isomorphy (see Remark 2 (a)) DL1q –

ℓq_{bp s}1 _{it follows, for q ă 8, by [8, 4.1 Theorem, p. 52 and 2.2 Prop., p. 46]:}

DLp_,c –ℓ

p

c bps, 1{p`1{q“1.

If p “ 1, DL1_,c – pB91q1

c – pc0bp s1q1_c –ℓ 1

c bp s by [8, 4.1 Theorem, p. 52 and

2.2 Prop., p. 46] and [3, Prop. 7, p. 13]. The casep“ 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 Fr´echet spaces E, F:

and hence, for E “ℓp and F “s, Ebp F “ℓpbp s which implies

DL1q_,c – pℓ

p_{bp sq}1 c –ℓ

q c bps

1

if 1ăqď 8, 1_{

p`1{q“1. If q“1,

DL11_,c – pB9q1_c – pc0bpsq1_c –ℓ 1 c bp s1.

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 DLp_,c and D_L1q_,c yield a

further proof of the quasinormability of these spaces. Indeed, ℓpc and ℓqc 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 s1 remains quasinormable by [14, Chapter II, Prop. 13.b, p. 76].

The compact regularity ofDL1q as a countable inductive limit of Banach spaces

is proven in Proposition 1. By means of the sequence space representation

DL1q –ℓ

q_{bp s}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

s1bp E “lim_ÝÑ

k

ppc0q_´kbp_εEq “lim

ÝÑ

k

pc0bp_εEq_´k “lim

ÝÑ

k

pc0pEqq_´k

are valid.

(b) The inductive limit lim_ÝÑkpc0pEqq_´k is compactly regular.

Proof. (1) The assertion follows from [16, Theorem 4.1, p. 55].

(2) The compact regularity of lim_ÝÑkpc0pEqq_´kis a consequence of [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 as:

@m PN_{0 Dk ąm @εą0@l ąk DC ą0 @x“ pxjqj P pc0pEqq´m:}

sup

j

j´kkxjkďεsup j

j´mkxjk`Csup j

j´lkxjk.

For the sequences pxjqj P pc0pEqq_´m the sequence pkx_jkq_j is contained in s1 _“lim

ÝÑkpc0q´k. Thus, (Q) is fulfilled because s

Corollary 15. The spaces B91 _{– lim}

ÝÑkpc0pc0qq´k and D 1

Lq – lim ÝÑkpc0pℓ

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

_c1

.

Motivated by the description of the topology κpℓp_{, ℓ}q_{q of the sequence space} ℓp

c by the seminorms

ℓp ÑR_{, x“ pxjqjPN ÞÑpgpxq “} ˜₈

ÿ

j“1

gpjq |xpjq|p

¸1_{p

“kg¨xk_p,

g P c0, see I. Mack’s master thesis [21] for the case p “ 8, we also

investi-gate the description of the topologies κpLp_{, L}q_{q and κpM}1

,C0q by means of

“function”-seminorms.

Denoting the space of bounded and continuous functions by B0

(see [24, p. 99]) we recall that B0

c “ pB 0

, κpB0 ,M1

qqhas the topology of R. C. Buck.

Proposition 16. Let 1ăpď 8. The topology κpLp_{, L}q_{q of L}p

c is generated

by the family of seminorms

Lp ÑR_{`, f ÞÑpg,hpfq “kh˚ pgfqk}

p, g P C0, hPL 1

.

Proof. (a) tĎκpLp_{, L}q_{qwheretdenotes the topology generated by the}

semi-norms pg,h above and let Ug,h :“ tf P Lp : pg,hpfq ď 1u be a neighborhood

of 0 in t. We show that

(i) Ug,h “ pgpˇh˚B1_,qqq˝,

(ii) gpˇh˚B1,qq is compact inLq.

ad (i): If S P gpˇh˚B1_,qq Ď Lq there exists ψ P B1_,q with S “ gpˇh˚ψq. For f PLp _{with p}

g,hpfq ď 1 we obtain

|xf, Sy| “ˇˇxf, gphˇ˚ψqyˇˇ“ |xh˚ pgfq, ψy|

and

sup

SPgpˇ_h˚B

1,qq

so (i) follows.

ad (ii): The 3 criteria of the M. Fr´echet-M. Riesz-A. Kolmogorov-H. Weyl Theorem are fulfilled:

-modulus of continuity is continuous.

(b) tĚκpLp, Lqq:

for f P Lp_{. By applying the factorization theorem [11, (17.1). p. 114] twice,}

there exist g P C0, h P L 1

such that C Ď gpˇh ˚B1_,qq. More precisely,

take L1

as the Banach algebra with respect to convolution and C0 as the

Banach algebra with respect to pointwise multiplication, Lp _{as the Banach}

module. As approximate units we use e´k2

An exactly analogous reasoning as for Proposition 16yields

Proposition 18. The topology κpM1

,C0q of the space M 1

c is generated by

the seminorms

M1

ÑR_{`, µÞÑpg,hpµq:“kh˚ pgµqk1.}

Proof. For this, note that kµk1 “ supt|xϕ, µy| :ϕ P C0,kϕk₈ ď1uand that

the multiplication

C0ˆM

1

ÑM1

, pg, µq ÞÑg¨µ

and the convolution

L1

ˆM1

ÑL1

,ph, νq ÞÑ h˚ν

are continuous [28, Thm 6.4.20, p. 150]).

Proposition 19. Let C ĎM1

. Then,

C compactðñ Dg P C0 DhPL 1

: C Ďgph˚B1,1q.

Acknowledgments. E. A. Nigsch was supported by the Austrian Science Fund (FWF) grant P26859.

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