Generalized functions as sequence spaces with ultranorms

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Generalized functions as sequence spaces with ultranorms

Maximilian F. Hasler

To cite this version:

Maximilian F. Hasler. Generalized functions as sequence spaces with ultranorms. Integral Transforms and Special Functions, Taylor & Francis, 2006, 17 (2-3), pp.149-156. �10.1080/10652460500437807�.

�hal-00758570�

arXiv:0705.2739v1 [math.FA] 18 May 2007

ultranorms

Maximilian F. Hasler, Université Antilles-Guyane, D.S.I., F97275 Shoelher

GF '04, Novi Sad, 27September 2004

Abstrat

Wereviewourreentformulation(withA.Delroix,S.Pilipovi¢andV.Val-

morin)ofColombeautypealgebras asHausdorsequenespaeswith ultra-

norms,dened by sequenes of exponential weights. We extendprevious re-

sultsandgiveseveralnewperspetivesrelatedtoehelontypespaes,possible

generalisations,asymptotialgebras,oneptsofassoiation,andappliations

thereof.

Keywords: Generalized funtion; topologial algebra; sequene spae. MSC:

46F30;46A45;46H05.

1 Introdution

Colombeau'sNew Generalized Funtions [1℄ are today themost widely used asso-

iativedierentialalgebrasontainingtheδdistribution.Theirtopologyisstudied sinethelate1990s[3℄, andinvestigationintopologialduals ofsuh spaesis now

emergingasimportanttopi ofresearhin thiseld.

We dene suh algebrasright from the start asspaes with ultranorms [5, 4℄,

whihisnaturalandespeiallyusefulforpratialuseofthetopology,withnoneed

forvaluations.Ouronstrutionallowsforalgebrasontainingultradistributionsand

periodihyperfuntions[6℄.Withoutspeializingtoaonretespae,wededuegen-

eralresultsaboutompleteness,embeddingofdualsandfuntoriality,andgeneralize

knownoneptsofassoiation,revealing aspetsofthe underlyingstruture rather

hiddenin otherapproahes.Ourapproahalsoshowsbettertheloselink withthe

lassialtheoryofsequenespaes.

2 The basi onstrution

Considerasequener= (rn)_n∈(R₊)^{N dereasingtozero.Foraseminorm} p^onan RorCvetorspaeE^{,thisdenesamap}||| · |||_p,r :E^N→R₊ = [0,∞]^,

f = (fn)_n 7−→ |||f|||=|||f|||_p,r = lim sup

n→∞

p(fn)rn

.

Lemma2.1 (a) If 0<lim infp(fn)≤lim supp(fn)<∞^,then|||f|||= 1^.

(b) Forallf, g∈E^N, λ∈C^∗:|||f +g||| ≤max(|||f|||,|||g|||)^and|||λf|||=|||f|||^.

() If E ^{isatopologial algebra,then} |||f·g||| ≤ |||f||| · |||g|||^.

Proof.Aslimrn = 0^,wehavelimk^rn= 1^foranyk >0^{,thus(a).Writing}p(λfn)≤

|λ|p(fn)^andp(fn+gn)≤2 max(p(fn), p(gn))^{,wehave(b),andusing}∃C >0∀x, y∈ E:p(x y)≤C p(x)p(y)^{,weget()in thesameway.}

Denition 2.2 Thergeneralizedsemi-normedspae(E, p)^{isthefatorspae} Gr(E, p) =Fr(E, p)/Kr(E, p)^{, where}

Fr(E, p) =

f ∈E^N| |||f|||_p,r<∞ , Kr(E, p) =

f ∈E^N| |||f|||_p,r= 0 .

Proposition2.3 Themap||| · |||_p,r ^denesapseudometridp,r ^onFr(E, p)^,making

ita topologial ring,if E ^{is atopologial algebra. As} Kr(E, p) ^isthe intersetion of neighborhoodsof zero, Gr(E, p) ^{isthenthe assoiatedHausdortopologial ring, on}

whih dp,r ^{iswell-denedandan} ultrametri.

Proof.Thisisadiret onsequeneoftheDenitionandpreeding Lemma.

Example2.4 For E = C and p = | · |^{, we obtain the ring of} rgeneralized omplex numbers,C_r=Gr(C,| · |)^.Forrn =_log¹_n (n >1)^,thisareColombeau's generalizednumbersCe,sinelim sup|xn|^1/logn<∞ ⇐⇒ ∃γ∈R:|xn|=o(n^γ),

andlim|xn|^1/logn= 0 ⇐⇒ ∀γ∈R:|xn|=o(n^γ)^.

The hoie rn=n^{−1/m (with}m >0^{),leads to}ultraomplexnumbersC^p!

m

.

Proposition2.5 ThespaesGr(E, p)^(resp.Fr(E, p)^{)aretopologial algebrasover}

thegeneralizednumbers(resp.overFr(K,| · |)^{)equippedwith}||| · |||^{topology,butthey}

aren'ttopologial vetorspaesoverthe eldK=RorC.

Proof.This is seenbyobserving that Lemma 2.1-() alsoholds for f ∈C^N, while Lemma 2.1-(b)impliesthat |||λ f||| ^{doesnotgotozerowhen}λ→0^.

Example2.6 To obtain rgeneralized Sobolev algebras GW^s,∞(Ω) = Gr W^s,∞(Ω), ps,∞

, we hoose E = W^s,∞(Ω) ^{with norm} ps,∞ = P

|α|<s

k∂^α· k_L^∞.

This generalizes toanynormedalgebra.

Theorem2.7 ((equivalentsales)) If r = (rn)_n^, s = (sn)_n ^{derease to zero}

suh that s = O(r)^{, then} Fr(E, p) ⊂ Fs(E, p), Ks(E, p) ⊂ Kr(E, p). ^{In par-}

tiular, if lim

n→∞

sn

rn = C ∈ R^∗

+^{, then} |||f|||_p,s = (|||f|||_p,r)^{C, and thus} Fs(E, p) = Fr(E, p), Ks(E, p) =Kr(E, p) ^and Gs(E, p) =Gr(E, p).

Proof. If sn = cnrn ^with lim supcn = C ∈ R^∗

+^{, we have} log|||f|||_p,s = lim sup(snlogp(fn)) = lim sup(cnrnlogp(fn)) ≤ Clim sup(rnlogp(fn)) = Clog|||f|||_p,r^{, whereweassumed}lim logp(fn)≥0^,i.e. |||f||| ≥1^.Otherwise,≤^must

bereplaedby≥^{,leadingtotheinverseinlusionfor}K^.

Remark 2.8((relation to Maddox' sequene spaes)) Our spaes

Kr(C,|·|)^andFr(C,|·|) ^{arethe same as}c0(r) =T

k∈N

x∈C^N|lim|xn|k^1/rn= 0

and ℓ∞(r) = S

k∈N

x∈C^N|sup|xn|k^−1/rn <∞ ^{, introdued in [7 , 8 ℄ and}

studied extensively by Maddox and his students [9 , 10 ℄. To see this, observe that

∃k∈N: sup|xn|k^−1/rn <∞ ⇐⇒ ∃k: lim sup|xn|^rn ≤k ⇐⇒ |||x|||_r <∞, ^and

∀k: lim|xn|k^1/rn= 0 ⇐⇒ ∀ε >0 :|xn|=o(ε^1/rn) ⇐⇒ |||x|||_r= 0.

These spaes belong to the lasses of ehelon resp. o-ehelon spaes. As we al-

ways require limrn = 0^{,both are MontelandShwartz spaes. Whilethe itedwork}

on sequene spaes isrestritedto(C,|·|)^{,our studies onernmore general spaes.}

However, most of the spaes onsideredin the sequel an be writtenas intersetion

and/or union of ehelon and o-ehelon type spaes. This also allows the general-

ization of the present onstrutionto any abstrat topologial module E^{, as will be}

disussedinaforthomingpubliation.

Denition 3.1 Ther^{extension of aloally onvexspae} (E,P)^{isthe fator}

spaeGr(E,P) =Fr(E,P)/Kr(E,P) = T

p∈P

Fr(E, p) T

p∈P

Kr(E, p)^.

Theorem3.2 If (E,P)^{isatopologial algebra, i.e.} ∀p∈ P ∃¯p∈ P ∃C >0∀x, y ∈ E :p(x y)≤Cp(x) ¯¯ p(y), ^thenFr(E,P) ^{isasubalgebraof} E^N, Kr(E,P)^{isan ideal}

of Fr(E,P)^,and(dp,r)_p∈P ^isafamilyof pseudo-distanes onGr(E,P)^{making ita}

Hausdo topologial algebraoverC_r.

Proof. Lemma 2.1-(b) yields for f, g ∈ Fr^, λ ∈ C, and p ∈ P : |||λf+g|||_p ≤ max(|||f|||_p,|||g|||_p)^,thusFr^andKr^areClinearsubspaes.Continuityofmultiplia- tionin(E,P)^{givesasin2.1-(),}∀p∈ P, ∃¯p∈ P:|||f g|||_p≤ |||f|||_p¯· |||g|||_p¯^{. Thus}Fr

is aCsubalgebraof E^{N, and} Kr ^{an idealof}Fr^.The inequalitiesalsoimply onti- nuityofadditionandmultipliation,thus Fr ^{isatopologial}F|·|,r^algebra,andGP

isagaintheassoiatedHausdorspae.

Example3.3 The lassial simplied Colombeau algebra [1 ℄ is obtained for

rn= _log¹_n ^andP =

p^µ_ν :f 7→ sup

|α|≤ν,|x|≤µ

|f^(α)(x)| _µ,ν∈N ^onE=C^∞(Ω)^.

As a last generalization of the base spae, onsider a family of semi-normed

algebras (E_ν^µ, p^µ_ν)_µ,ν∈N ^{with embeddings} ∀µ, ν ∈ N : E_ν^µ+1 ֒→ E^µ_ν^, E_ν^µ ֒→ E_ν+1^µ

resp.E_ν+1^µ ֒→E_ν^{µ. Let}−→

E = proj lim

µ→∞ ind lim

ν→∞ E_ν^µ,resp.←−

E = proj lim

µ→∞ proj lim

ν→∞ E_ν^{µ, and}

assumethat forallµ∈Ntheindutivelimitisregular,i.e. asubsetisbounded i itisabounded subsetof (E_ν^µ)^Nforsomeν∈N.Nowlet

Fr(−→ E) =n

f ∈−→

E^N∀µ∈N,∃ν ∈N:f ∈(E_ν^µ)^N∧ |||f|||_p^µ_{ν, r}<∞o , Fr(←−

E) = f ∈ ←−

E^N | ∀µ, ν ∈ N : |||f|||_p^µ_{ν, r} < ∞ , ^{and the obvious denition for} Kr(←→

E)^{,wherewewrite}←→· ^forboth,−→· ^and←−·^.Thenagain,Gr(←→

E) =Fr(←→

E)/Kr(←→ E)

isatopologialalgebrafortherespetivelimittopology.

Example3.4 In [6 ℄ we showed how to embed Gevery lass ultradistributions in

ColombeaualgebrasG^(p!m,p!m

′)=G_rm′(E^(m)), G^{p!m,p!m

′}=G_rm′(E^{m}),^wherer^m_n = n^{m1 and} E^(m), E^{{m} are the projproj resp. projind limit type spaes of Beurling}

resp.Roumieuultradierentiablefuntions,denedtroughspaesonwhihp^m,µ_ν (f) = sup

|x|≤µ,α∈N^s

ν^|α|

α!^m|f^(α)(x)| ^resp. q_ν^m,µ=p^m,µ_1/ν ^{are nite.}

Denition 3.5 Consider the spaes Oλ ^{of holomorphi funtions on} Ωλ = z∈C| _λ¹ <|z|< λ ^{with nite norm} q^λ =k·k_L^∞_(Ωλ)^{. Analyti funtionson the}

unit irle arethenA(T) = ind lim

λ→1 Oλ^.Let−→

E = ind lim

λ→1 (A1(T), q^λ)^,where A1(T) = ind lim

m→1⁻ ind lim

ν→∞

n

f ∈ A(T)| kf^(α)k_L^∞_(T) =

α→∞O(ν^αα!^m)o .

Then,rgeneralizedhyperfuntions on TaredenedasGH,r(T) =Gr(−→ E),^quo-

tientspaeof Fr(−→ E) = S

λ>0

Fr(A1(T), q^λ)^byKr(−→ E) = S

λ>0

Kr(A1(T), q^λ).

Thesametypeof ultranormanbeusedtoharaterizegeneralizedfuntions f

on the unit irle by means of their Fourieroeients (fbk)_k ∈ C^Z, for whih we dene |||(fbk)_k∈Z|||^±_r = max

|||(fbk)_k∈N|||_|·|,r, |||(fb−k)_k∈N|||_|·|,r .

Fourier oeients of analyti funtions f ∈ A(T)^{, Shwartz} distributions

T ∈ E^′(T) ^and hyperfuntions H ∈ B(T) ^are haraterized by |||(fbk)_k|||^±_(·)−1 < 1^,

Proposition3.6 ((Fourierharaterization)) ThesamespaesFr(−→

E)^,Kr(−→ E)

are obtainedinthe previous denitionif q^{λ isreplaedby}qb^λ:f 7→sup

k∈Z

λ^|k||fbk|^.

Proof. If f ∈ Fr(−→

E)^, |||f|||_qλ,r < ∞^{, there is} C > 0 ^{suh that} q^λ(fn)^rn < C ^for

alln^{. Cauhy's}inequalitiesinΩλ ^thengive|fˆn(k)| ≤q^λ(fn)λ^{−|k|, thus}|fˆn(k)|^rn ≤ C λ^{−|k|rn forall}k∈Z,whene |||f|||_ˆqλ,r <∞^. Conversely,iff ∈−→

E^N,|||f|||_ˆqλ,r <∞^,

wehave qˆ^λ(fn)^rn < C ^{for some} C >0 ^{and all} n^{, i.e.} |fˆn(k)| < C^1/rnλ^{−|k| for all} k∈Z.Consequently,thereisM >0^suhthatq^λ(fn)≤M C^1/rn,thus|||f|||_qλ,r<∞^.

TheproofforK ^{goesthesameway.}

Convolutionwith molliers φn =P

|k|≤1/rnz^{k allows to embed} hyperfuntions

B(T)^intoGH,r(T)^{,preservingtheusualprodutof}A1(T)^[6℄.

Proposition3.7 ((Completeness)) Without assuming ompleteness of

←→ E^, Fr(←−

E) ^{isomplete, and}Fr(−→

E)^issequentiallyomplete.

Proof. If (f^m)_m ^{is a Cauhy sequene in} Fr(←−

E)^{, there are inreasing sequenes} (mµ),(nµ)∈ N^N suh that ∀µ ∈ N,∀k, ℓ ≥mµ^, lim sup

n→∞

p^µ_µ f_n^k−f_n^ℓrn

< ₂¹µ ^and

more preisely ∀k, ℓ ∈ [mµ, mµ+1] ∀n ≥nµ : p^µ_µ f_n^k−f_n^ℓrn

< ₂¹µ. ^Let µ(n) =¯ sup{µ|nµ≤n}^{,andonsider thesequene}f¯= (f_n^mµ(n)¯ )_n. ^Thenwehavef^m→f¯

in Fr(←−

E)^{. Indeed,for}ε^and p^µ_ν^{0 given,take}µ > µ0, ν ^{suh that} ₂¹µ < ¹₂ε^.As p^µ_ν ^is

inreasingin bothindies,wehaveform∈[mµ+s, mµ+s+1]^: p^µ_ν⁰(f_n^m−f¯n)^rn≤p^µ_µ(f_n^m−f_n^mµ+s+1)^rn+Pµ(n)−1¯

µ^′=µ+s+1p^µ_µ^′′(f_n^mµ′ −f_n^mµ′+1)^rn.

Forn > nµ+s^,onehasn≥nµ(n)¯ ^,thus p^µ_ν⁰(f_n^m−f¯n)^rn<Pµ(n)¯ µ^′=µ+s 1

2^µ′ <₂²µ < ε^.

AsFr(←−

E)^{isametrisablespae,thisimplies}ompleteness.

ForCauhynetsinFr(−→

E)^{,weusethatforall}µ^thereisν(µ)^suhthatp^µ_ν(µ)≤p^µ+1_ν(µ+1)

andp^µ_ν(µ)(f_n^m−f_n^p)^rn< εµ,^where(εµ)_µ ^{dereasestozero.With this,weanprove}

thesequentialompletenessofFr(−→

E)^{bythesameargumentsasabove.}

Remark 3.8((disreteness ofinduedtopology)) In [6 ℄ we have shown that

anet (δn)_n∈←→

E^{Nsuh that}∀ψ∈←→

E , limn→∞

R

R^sδn·ψ=ψ(0)^{,annot bebounded}

in

←→

E^{, under very weak} assumptions. From this we dedue that the topology of any algebraontaining δ^and←→

E ^{must indue the disretetopologyon}←→ E^.

4 Families of sales and asymptoti algebras

Wenowgeneralizethegrowthonditions.Considerafamilyr= (r^m)_m^ofsequenes (r^m_n)_n ^{dereasingtozeroas}n→ ∞^{.Supposethateither}

(I) ∀m∈N:r^m=O(r^m+1)^{, or (II)} ∀m∈N:r^m+1 =O(r^m)^.

Theorem4.1 Dene Fr(←→ E) =T

m∈NFr^m(←→

E), Kr(←→ E) = S

m∈NKr^m(←→

E) ^{in ase}

(I), resp. Fr(←→ E) = S

m∈NFr^m(←→

E), Kr(←→ E) = T

m∈NKr^m(←→

E) ^{in ase (II). Then}

again, Gr(←→

E) =Fr(←→

E)/Kr(←→

E)^{isan algebra.}

Proof. UsingKr^m(←→

E)· Fr^m(←→

E)⊂ Kr^m(←→

E) ^{andTheorem 2.7, itis easy to verify}

that inbothases,Fr(←→

E)^{isasubalgebraand}Kr(←→

E)^{is anidealthereof.}

Example4.2 For r_n^m= 1 ^ifn≤m^,0 ^{elsewhere,we get}Egorov-typealgebras.

Denition 4.3 Let a = (am:N→R₊)_m∈Z ^{be an asymptoti sale, i.e.} ∀m ∈ Z,

am+1 =o(am), a−m= 1/am ^and ∃M ∈Z :aM =o(a²_m)^{. The asymptotialgebra}

dened by a and a loally onvex algebra (E,P) ^{is the fator spae} A(a)(E,P) = f ∈E^N| ∃m∈Z ∀p∈ P : p(f) =O(am)

{f ∈E^N| ∀m∈Z ∀p∈ P : p(f) =o(am)}.

Example4.4 (i)am(n) =n^{−m leads toColombeautype} generalizedalgebras.

(ii) am= 1/exp^{m (}m^{-fold iterated}exp ^{funtion)gives} exponentialalgebras[2 ℄.

Theorem4.5 For r_n^m=|logam(n)|^{−1,we have}Gr(E,P) =A₍a)(E,P)^.

Proof. If p(fn) < C am(n) = C e^{1/rm (for} am > 1^{), then} lim sup(p◦f)^rm < ∞

andf ∈ Fr(E,P)^.Conversely,iflim sup(p◦f)^1/|logam¯|< C^thenp◦f ≤(am)^logC, (am, C >1)^{.Usingthethirdpropertyofsales,}∃M :p◦f =o(aM)^.

Nowonsider ∀m¯ : p◦f = o(am¯)^{. Take}m ∈N. Then, for any q ∈ N, there is mˆ

suhthatamˆ =o(amq)^andp◦f =o(amˆ) =o(amq) =o((e^−1/rm)^q) =o((e^−q)^1/rm)^.

Thereforelim sup(p◦f)^rm≤e^−q,andasq^{wasarbitrary,wehave}|||f|||_p,rm = 0^,thus f ∈ Kr(E,P)^.

Finallyassume∀m¯ : lim supp(fn)^rm¯ = 0^,i.e. ∀C >0 : (p◦f)^1/|logam¯|=o(C)^{, thus} p(f) = O(C^|logam¯| =O(am¯|logC|)^{. Now for any} m^{, let} m¯ =m+ 1 ^and C = 1/e^.

Thenp(f) =O(am¯) =o(am)^{,as required.}

Aseondkindofasymptoti algebrasisoftheform

A^(a)(E,P) =

f ∈E^N| ∀σ <0 ∀p∈ P: p(f) =o(aσ) {f ∈E^N| ∃σ >0 ∀p∈ P: p(f) =o(aσ)}^,

where a = (aσ)_σ∈R ^{is a sale (i.e.} ∀σ > ρ, aσ = o(aρ)^{, et.), indexed by a real}

number. As the subalgebra is heregiven as intersetion and the idealas union of

sets, thisaseisnotoveredbythepreviousone.

Proposition4.6 For r^m = _|loga¹

1/m|^{, we have} A^(a)(E,P) = F_r^′(P)/K_r^′(P)^{, with} F_r^′(P) =F_r^′(E,P) =n

f ∈E^N| ∀m∈N∀p∈ P:|||f|||_p,rm≤1o

and

K^′_r(P) =K^′_r(E,P) =n

f ∈E^N| ∃m∈N∀p∈ P:|||f|||_p,rm <1o .

Example4.7 aσ(n) = e^{−n σ gives algebras with infraexponential growth [3 ℄, of}

partiular interestfor embeddings of periodihyperfuntions.

5 Funtorial properties

A mapϕ:←→ E →←→

F ^{obviouslyextends anoniallyto} Gr(ϕ) :Gr(←→

E)→ Gr(←→ F)^iffor

allf ∈ Fr(←→

E)^andk∈ Kr(←→

E)^,wehave (F1) : ϕ(f) = (ϕ(fn))_n∈ Fr(←→

F), ^and (F2) : ϕ(f +k)−ϕ(f)∈ Kr(←→ F).

Denition 5.1 The r^{extension of a map} ϕ :←→ E →←→

F ^{satisfying the above}

onditions (F1),(F2)^{, is dened as the map} Gr(ϕ) : Gr(←→

E) → Gr(←→

F) ^{suh that} [f]7→ϕ(f) +Kr(←→

F)^,where f ^isany representative of[f] =f+Kr(←→ E)^.

Example5.2 Linear mappings ϕ ^{of loally onvex vetor spaes} (E,P) →(F,Q)

are ontinuousi ∀q∈ Q ∃p∈P ∃c >0 ∀x∈E: q(ϕ(x))≤c p(x) ^.

Then, ∀f ∈E^N:|||ϕ(f)|||_q,r≤ |||f|||_p,r^,whene(F1)^and(F2)^{,usinglinearity.}

Weonsideragainasequeneofsales(r^m)^suhthatr^m+1≤r^{m.Letusdenote}

Denition 5.3 In ase r^m+1 ≤ r^{m, an inreasing map} g : R₊ → R₊ is alled

r^{moderate i} ∀m ∈ N ∃M ∈ N : g(F_r^+m) ⊂ F_r⁺M^{, and} r^{ompatible i it is}

ontinuous at0and∀M ∈N∃m∈N:h(K⁺r^m)⊂ K⁺_rM^.

In aser^m+1≥r^{m,the denitionshold with}∀m ∃M ↔ ∀M ∃m^exhanged.

Thesenotionsallowtoharaterizemapsthatextendanoniallyto Gr^:

Denition 5.4 Amapϕ: (E,P)→(F,Q)^isontinuouslyr^{temperatei:(a)}

there isanr^{moderate funtion}g ^suhthat

∀q∈ Q ∃p∈ P ∀f ∈E:q(ϕ(f))≤g(p(f))^,

and(b) thereisanr^{moderate funtion} g ^andanr^{ompatiblefuntion} h

suhthat ∀q∈ Q ∃p∈ P ∀f, k∈E:q(ϕ(f+k)−ϕ(f))≤g(p(f))h(p(k))^.

Theorem5.5 Anyontinuously r^{temperate map} ϕ^{extendsanoniallyto}Gr(ϕ) : Gr(E,P)→ Gr(F,Q), ^{andthis anonial extension isontinuous for the topologies}

induedby (||| · |||_p,rm)_p∈P,m∈N ^resp. (||| · |||_q,rm)_q∈Q,m∈N^.

Proof.Condition(a)ofDenition 5.4implies(F1)^,and(b)gives(F2)^.Weomitthe

straightforwardalulations,abitlengthyinviewofthefourasestobetreated[4℄.

Continuity ofGr(ϕ) ^{isobtained in thesamewayas}(F2)^{, replaing}p(f)∈ F_r^{+m by}

|||f|||_p,m≤K^,andp(k)∈ K⁺_r^{m by}|||k|||_p,m≤ε^.

6 Assoiation in

r

Inseveralsituations,e.g.whensolvingPDE,strongequalityisimpossibletoobtain

ornotneeded,andapproximationexpressedbyassoiation issuient.

Denition 6.1 Generalizednumbers[x],[y]∈C_r areassoiatedix−y ^isanull

sequene, [x] ≈[y] ⇐⇒ x−y ∈ N =

x∈C^N|limx= 0 ^{. For} s ∈R, they are

s^assoiated,[x]≈^s [y]^,ix−y∈N^(s)=

x∈C^N|xn=o(e^−s/rn) ^.

Remark 6.2 (i)The denitioniswell-posedsineKr(C,|·|)⊂N^.

(ii) WehaveN^(s)=e^−s_r N^,where er= (e^rn1 )_n ^{represents apositive unitof} C_r. (iii) All elements of the open unit ball are assoiated to zero, |||x|||_|·|,r < 1 =⇒ x∈N^,andx∈N =⇒ |||x|||_|·|,r ≤1^,but _r¹_n →

n→∞∞^{also veries}|||¹_r|||_|·|,r = 1^.

Denition 6.3 IfJ ^{isanadditivesubsetof}Fr(←→

E)^ontainingKr(←→

E)^,twoelements F, G∈ Gr(←→

E)^are J^assoiated, F≈

J G^,iF−G∈ J/Kr(←→ E).

Proposition6.4 IfJ ^{isabsolutelyonvex, the relation}≈

J

isstableunder multipli-

ation withelements ofthe losedunitball.

Clearly,≈

J

isompatiblewith derivationiJ ^{isstable under}dierentiation.

Denition 6.5 We all F, G ∈ Gr(E,P) ^{strongly assoiated,} F ≃ G^{, i} ∀p ∈ P :dp,r(F, G)<1^{.For any}s∈R,strong s^{assoiation isdenedas}

F ≃^s G ⇐⇒ ∀p∈ P:dp,r(F, G)< e^−s ⇐⇒ F ≈

J^(s)G

whereJ^(s)=

f ∈E^N| ∀p∈ P:|||f|||_p,r< e^−s =:B^(P)_e−s^.

Remark 6.6 If F≃^s G^{for all} s >0^,thenF =G^,beauseT

s>0J^(s)=oGr(E,P)^.

In order to dene weak assoiation, notie that a ontinuous bilinear form

h·,·i:←→

E ×^D→CanoniallyextendstoGr(←→

E)×^D→C_r (f.Example5.2).This allowstodene,foranyonvexsubsetM ^ofFr(C,| · |)^ontaining0C_r^,subspaesJ

oftheformJM = f ∈←→

E^N| ∀ψ∈^D:hf, ψi ∈M ^.