Rigidity of the topological dual of spaces of formal series with respect to product topologies

FPSAC 2011, Reykjavik, Iceland DMTCS proc. (subm.) , by the authors, 1–12 Rigidity of the topological dual of spaces of f or mal ser ies with respect to product topologies Laurent Poinsot LIPN - UMR 7030, CNRS - Universit ´ e P aris 13, F-93430 V illetaneuse , F rance Abstract. Even in spaces of formal power series is required a topology in o rder to le gitimate so me operations, in particular to comp ute infinite summations. Many topologies can be exploited for different purposes. Combinatorists and algebraists ma y think to u sual order topo logies, or the produ ct topology induce d by a discrete coef ficient field, or some in verse limit topolog ies. Analysists will tak e into accou nt the v alued field structure of real o r comple x num bers. As the main result of this paper we pro ve that the topological dual spaces of formal po wer series, relati ve to the class of product topologies wi th respect to Hausdo rff field topolo gies on the coef ficient field, are all the same, namely the space of polyno mials. As a conseq uence, this kind of rigidity forces linear maps, co ntinuous for any (and then for all) of those topologies, to be defined by very pa rticular infinite matrices similar to row-finite matrices. R ´ esum ´ e. La validit ´ e de certaines op ´ erations, notamment les calculs de sommes de s ´ eries, repose sur l’emploi d’une topologie su r l’espace des s ´ eries forme lles. En Combinato ire ou en Alg ` ebre il est d ’usage de consid ´ erer les topolog ies donn ´ ees par une v aluation, ou les topo logies prod uits relativ ement ` a un corps discret, ou encore des topologies limites projecti ves. D ` es lors que l’Analyse entre en sc ` ene il de vient in ´ evitable de consid ´ erer les v aleurs absolues des corps des nombres r ´ eels et complexes. Il s’av ` ere en fait que les duals topologiqu es des es paces de s ´ eries, par rapport aux topologies produits relativ es aux cop ies d’un m ˆ eme corps topologique s ´ epar ´ e, sont tous isomorphes ` a l ’esp ace des polyn ˆ omes. La preuve de ce r ´ esultat de rigidit ´ e constitue l’apport principal de notre article. L ’ind ´ ependance du dual topologique relativ ement au choix parmi ces topo logies astreint les op ´ erateurs lin ´ eaires, continus pour un e (et alors pour toutes les) topologie(s) de notre collection, ` a apparaˆ ıtre sous la forme de matri ces infinies bien particuli ` eres qui n’ont qu’un nombre fini d’entr ´ ees non nulles sur chaque “ ligne ”. K eywords: T opological fields, topological duals, product topology , infinite matrices 1 Introdu ction Manipulatio n of f ormal power series requ ires some topologica l notions in order to legitimate some c om- putations. For instance, the usual substitution of a power series in one v a riable without constant term into an other , or the existence of the star operation , related to M ¨ obius inversion, ar e usually treated using ei- ther an orde r function (a v aluation) or , eq ui valently (wh ile more imprecise), s aying t hat o nly finitely many terms co ntrib ute to the calculation in each degree. I n both ca ses is used ( e xplicitly or not) a topology given by a filtratio n (an d mor e precisely a kind of Krull topology): the order of some partial sums must increase indefinitely for the sum to b e d efined (and the operation to be legal). Qu ite n aturally other topologies may subm. to DMTCS c  by the authors Discrete Mathematic s a nd Theoretica l Computer Science (DMTCS), Nancy , Fra nce 2 Laur ent P oinsot be used: for instance if X is an infinite s et, then the com pletion of the algebra R h X i of po lynomials in nonco mmutati ve variables (where R is a comm utati ve ring with a unit) with respect to the usual filtration (related to the leng th of a word in th e free monoid X ∗ ) is the set of all series with only finitely many non-ze ro terms f or each given length. The sum of the alphabet does n ot e ven exist in this c ompletion. I n order to take such series into acco unt we must used the pr oduct topology (with a discrete R ) . If som e analytical in vestigations must be perfo rmed (such as convergence ray , differential equation, or fun ctional analysis), the discrete topo logy of K ∈ { R , C } is no more sufficient; the v alued field structu re of K tu rns to be unav o idable. Other topo logies may be used for particular needs. Giv en a to pology , compatible in a ce rtain sense with algebraic operation s, on a space o f formal p o wer series with coefficients in a t opolog ical field, it can be u seful to consider continuo us endomorp hisms since they com mute to infinite sums. Quite amazingly for a very large class of possible topologies (namely produ ct to pologies with respect to separ ated topolog ies on th e base field) it app ears t hat these continu ous linear m aps may be seen as infin ite matrices of a particular k ind (each “ row ” is finitely supported) a nd that, independ ently of the topo logy cho sen in the class. In other terms, a linear map can be represented as some “ ro w-finite ” matrix if, and only if, it is continu ous for all the topologies in th e giv en class (see Section 5). Thus, in ord er to prove an endom orphism to be continuous for som e top ology , it suffices to prove this prop erty fo r t he more convenient topolo gy in the class. Similarly , if an en domorphism is given in the form o f an in finite matrix ( with finitely sup ported “ r o ws ”), then it is kno wn to be con tinuous for ev ery topology in our large collection, and theref ore is able to g et throu gh infinite sums. At this stage we notice that the class of topologies is that o f pro duct topologies r elati ve to every Hausdorff topolog ies on the co ef ficien t field (an in finite field ha s the car dinal number of the power set of its p o we r set o f distinct field topologies, see Remark 1). The explana tion of this phen omenon relies on the fact that the topo logical dual o f a given space of formal series is th e same f or all the to pologies in the big class: the space o f polyno mials. This re s ult, presented in Sectio n 3 and proved in Section 4, is our m ain theo rem (T heorem 5), and we p resent some of its direct consequ ences (in p articular it e x plains the “ rigidity ” of the space of continuous en domorphisms with respect to a change of topolog y in Section 5). In this paper we focus on the linear structure of formal s eries so that we consider any set X rath er than a free monoid X ∗ , and function s rath er t han formal series. 2 Some notations Let R be a commutative ring (i) with a unit 1 R . If M and N are tw o R -modu les, th en H om R ( M , N ) is the set of all R -linear m aps f rom M to N . In particular, the algebraic d ual M ∗ of M is the R - module Hom R ( M , R ) . If X is a set, then R ( X ) is the R -module of all finitely supp orted maps from X to R , th at is, f ∈ R ( X ) if, and o nly if, the support supp ( f ) = { x ∈ X : f ( x ) 6 = 0 } of f is a fin ite set. For e very x 0 ∈ X , we define the c haracteristic function (or Dir a c mass ) δ x 0 : X → R x 7→  1 R if x = x 0 , 0 R otherwise . (1) (i) In the remain der , the te rm “ ring ” with no except ion refers to “ commutati ve ri ng with a unit ”. Rigidity of the topological dual of spaces of formal series with r espect to pr oduct topologies 3 Let us introdu ce the u sual evaluation map ev : R ( X ) ⊗ R R X → R p ⊗ f 7→ X x ∈ X p ( x ) f ( x ) . (2) As usually it is treated as a R - bilinear ma p h· , ·i , called dual pairing (or canonical bilinear form , see ? or ? ), i.e. , ev ( p ⊗ f ) = h p, f i . In particular, for every x ∈ X , ev ( δ x ⊗ f ) = h δ x , f i = f ( x ) , and then π x : f 7→ h δ x , f i is the pr ojection of R X onto Rδ x ∼ = R . The dual pairing has the obvious properties of non-degeneracy: 1. f or every p ∈ R ( X ) \ { 0 } , there is some f ∈ R X such that h p, f i 6 = 0 , 2. f or every f ∈ R X \ { 0 } , there is some p ∈ R ( X ) such that h p, f i 6 = 0 . Let E be a set, an d F be a collection of map s φ : E → E φ , where each E φ is a top ological s pace. The initial topology induced by F on E is the coarsest topology that makes continuous each φ . This topolog y is Hau sdorff if, a nd only if, F sepa r a tes the elements of E ( i.e. , fo r e very x 6 = y in E , ther e exists some φ ∈ F such that φ ( x ) 6 = φ ( y ) ), and E φ is separated fo r each φ ∈ F . Now , if E = F X , whe re X is a set, and F is a topolo gical sp ace, then the pr odu ct top ology (or top ology o f simple con vergence or function topology ) on F X is the initial to pology ind uced by the cano nical projection s π x : f 7→ f ( x ) , x ∈ X , from E onto F . It is c haracterized by the fo llo wing pr operty: For ev ery topolog ical space ( Y , τ Y ) , the map f : Y → F X is continuo us if, an d only if, every map π x ◦ f : Y → F is continuou s (see ? ). Let R be a r ing. It is said to be a top ological ring when it is equip ped with a topolo gy τ (n ot necessarily Hausdorff) for whic h ring operatio ns, x 7→ − x , ( x, y ) 7→ x + y , ( x, y ) 7→ xy , are continu ous when is co nsidered on R × R the produ ct to pology defined by τ o n each factor s . A field K which is also a topolog ical ring is said to be a topo lo gical field when the map x 7→ x − 1 is contin uous on K ∗ = K \ { 0 } (equipp ed with the subspace top ology). If R (resp. K ) is a to pological rin g (re s p. topological field), any R -module M (resp. K -vector space) is said to be a topological R - module (resp. a topological K -vec tor space ) if it is e quipped with a topo logy (Hausdorff or not) that makes contin uous the mod ule maps x 7→ − x , ( x, y ) 7→ x + y , ( λ, x ) 7→ λx . Notice that if R (resp. K ) is a to pological ring (resp. topo logical field), and X is any set, th en R X (resp. K X ) is a topolo gical R -module (resp. top ological K -vector space) – in the obvious way – when endowed with the prod uct top ology . Let R be a topo logical rin g, and M be a topo logical R -module. Then M ′ denotes the topologica l dual of M , that is, the sub-mo dule of M ∗ of continuo us linear forms (ii) . Summability . Ma n y intermed iary results o f this p apers require th e notion , and some properties, of a summable family in a topo logical ring (wh ich is actually hidden but fu ndamental fo r the treatment of formal power series). W e reca ll them without any proof; we freely used them in the seque l, an d refer to ? for furthe r info rmation concern ing this conc ept but also topological rings, fields and modules. (ii) Quite obvi ously , M ∗ is the topologi cal dual of M , when R and M both ha ve th e discrete topology . 4 Laur ent P oinsot Definition 1 Let G be a Hausdorff Abelian gr ou p (that is, an Abelian gr oup – in additive notatio n – with a separated topology su c h that the gr o up operations, add ition and in version, a r e continuo us), and ( x i ) i ∈ I be a family of elements of G . An element s ∈ G is the sum of the summable family ( x i ) i ∈ I if, and only if , for each neighborhoo d V of s ther e e xists a finite subset J ⊆ I such that X j ∈ J x j ∈ V . The sum s of a summable family ( x i ) i ∈ I of elements of G is usually denoted by X i ∈ I x i . Proposition 1 If ( x i ) i ∈ I is a summable family of elements of a Hausdorff A belian gr ou p G having a s um s , then for any permutation σ o f I , s is also the sum of the summable family ( x σ ( i ) ) i ∈ I . Proposition 2 If ( x i ) i ∈ I is a summable family of elements of a Hausdorff A belian gr ou p G , then for every neighbo rhood V of zer o, x i ∈ V for all but finitely many i ∈ I . Proposition 3 Let G be the Cartesian pr oduct of a fa mily ( G λ ) λ ∈ L of Hausdo rf f Abelia n gr ou ps ( G has the p r od uct topology). Then s is the sum o f a family ( x i ) i ∈ I of elemen ts of G if, a nd only if, π λ ( s ) is the sum of ( π λ ( x i )) i ∈ I for each λ ∈ L (wher e π λ is the canonica l compo nent fr om G onto G λ ). Proposition 4 If φ is a continuous homomorph ism fr o m a Hausdorff Abelian gr oup G 1 to a Hausd orf f Abelian gr oup G 2 , and if ( x i ) i ∈ I is a summable family of elements of G 1 , then ( φ ( x i )) i ∈ I is summable in G 2 , and X i ∈ I φ ( x i ) = φ X i ∈ I x i ! . 3 The statement The objective of this paper is to prove th e following result, and to observe s ome its conseque nces. Theorem 5 Let K be an Hausd orf f topological field (it might be non- discr ete!), and X b e any set. Then, the topologica l dua l ( K X ) ′ of K X , un der th e pr o duct topology , is isomorphic (as a K -vector space) to K ( X ) . In o rder to illustrate the scop e of th is result, let us ass ume for an instant that K is a discrete field, and that V is an Hausdo rf f topolog ical K -vector space. The topology on V is said to b e linear if, and on ly if, it has a neighbo rhood basis of zer o consisting of vector subspaces open in the topology . Proposition 6 (??) Let V be a K -vector space with a line ar topology . Then the following con ditions ar e equivalen t: 1. V is complete, and all its open subspace s are of finite codimension. 2. V is an inver se limit of discr ete finite-d imensional vector spaces, with the in verse limit topology . 3. V is isomorphic, as a topological vec tor spa ce , to the algebraic dual W ∗ of a discr ete vector space W , with the topology of simple con v er gence. Equ ivalently , V is isomorph ic to K X , with the pr odu ct topology , for some set X . A topological vector space with the above equiv alent pro perties is ca lled linearly compact . This kind o f spaces is not so important fo r this paper , but w e o btain a character ization of their top ological du als as a Rigidity of the topological dual of spaces of formal series with r espect to pr oduct topologies 5 minor conseq uence o f our main result: we get that the topolog ical d ual o f any linearly compac t vector space is isomorphic to some K ( X ) . W e can also dedu ce a n i mmediate corollary that is a partial recipro cal of ou r main result. Corollary 1 Let K be a topological field, an d X be a set. Let us a ss ume that K X has the pr o duct to pology . Suppo se that X is in finite. Then, ( K X ) ′ is isomorphic to K ( X ) if, and only if , K is Hausdo rf f. Proof: If K is Hausdorff, then acco rding to T heorem 5 ( K X ) ′ is isom orphic to K ( X ) . No w , let R be a n indiscrete topolog ical ring . T hen, with respect to the tri vial topology on R , ( R X ) ′ = ( R X ) ∗ . Because X is infin ite, the field K cannot be indiscrete. But ring topolog ies o n a field (and in par ticular field topolog ies) may be either Hausdo rf f or the indiscrete one (see ? ). ✷ 4 The proof of Theorem 5 Lemma 7 Let R be a commuta tive ring with unit, and X be a set. Let us define Φ : R ( X ) → R R X p 7→  Φ( p ) : R X → R f 7→ h p, f i  . (3) Then, for every p ∈ R ( X ) , Φ( p ) ∈ ( R X ) ∗ , Φ is R -linear and one-to-o ne . Proof: The first and secon d p roperties are obvious. Let p ∈ R ( X ) such that Φ( p ) = 0 , then fo r every f ∈ R X , Φ( p )( f ) = 0 , and in par ticular , for every x ∈ X , 0 = Φ( p )( δ x ) = h p, δ x i = p ( x ) , in s uch a way that p = 0 . ✷ Lemma 8 A ss ume that R is a topologica l ring (Hausd orf f o r not) and that R X has the pr oduct topology . Then for every p ∈ R ( X ) , Φ( p ) is continuous, i.e. , Φ( p ) ∈ ( R X ) ′ . Proof: It is clear since Φ( p ) is a finite sum o f (scalar multiples) of projectio ns. ✷ Lemma 9 Let us suppose that R is an Hausdo r ff top ological ring, X is a set, and that R X has the p r od uct topology . F or every f ∈ R X , the family ( f ( x ) δ x ) x ∈ X is summable with sum f . Proof: It is suf ficien t to pr o ve th at for every x 0 ∈ X th e f amily ( h δ x 0 , f ( x ) δ x i ) x ∈ X = ( f ( x ) δ x ( x 0 )) x ∈ X is summable in R with sum h δ x 0 , f i = f ( x 0 ) which is immediate. ✷ Lemma 10 Und er the same assumptio ns a s Lemma 9, if ℓ ∈ ( R X ) ′ , then Y ℓ = { x ∈ X : ℓ ( δ x ) is in vertible in R } is finite. 6 Laur ent P oinsot Proof: Since ℓ is con tinuous (and linear), an d ( f ( x ) δ x ) x ∈ X is summable with sum f , then ( f ( x ) ℓ ( δ x )) x ∈ X is summable in R with sum ℓ ( f ) for e very f ∈ R X . Let us define f : X → R x 7→  ℓ ( δ x ) − 1 if x ∈ Y ℓ , 0 R otherwise . (4) Then, ( f ( x ) ℓ ( δ x )) x ∈ X is summable with sum ℓ ( f ) in R . Accordin g to properties of summability , for e v- ery neighb orhood U of 0 R in R , f ( x ) ℓ ( δ x ) ∈ U f or all but finitely many x ∈ X . Since R is assumed Hau s- dorff, the re is s ome neighborhoo d U of 0 R such that 1 R 6∈ U . If Y is n ot finite, then 1 R = f ( x ) ℓ ( δ x ) 6∈ U for every x ∈ Y which is a contradiction . ✷ Lemma 11 Let K be an Ha usdorff topological fi eld, X be a set, and a ss ume that K X is equipped wi th the pr odu ct topology . Let ℓ ∈ ( K X ) ∗ . If ℓ ∈ ( K X ) ′ , then ℓ ( δ x ) = 0 for all but finitely many x ∈ X . Proof: According to Lemma 1 0, the set { x ∈ X : ℓ ( δ x ) is inv ertible i n K } = { x ∈ X : ℓ ( δ x ) 6 = 0 } is finite. ✷ Lemma 12 Und er the same assumptio ns a s Lemma 11, Φ is onto. Proof: Let ℓ ∈ ( K X ) ′ . Let us define p ℓ : X → K x 7→ ℓ ( δ x ) . (5) A priori p ℓ ∈ K X . But according to Lemma 11, actually p ℓ ∈ K ( X ) . Let f ∈ K X . W e have Φ( p ℓ )( f ) = h p ℓ , f i = X x ∈ X p ℓ ( x ) f ( x ) = X x ∈ X ℓ ( δ x ) f ( x ) = ℓ ( f ) and then Φ( p ℓ ) = ℓ . ✷ Now it is easy to con clude th e proo f o f Theorem 5 since it follows directly from Lemmas 7 and 12. Remark 1 1. Th e algeb r a ic dua l of K X may be distinct fr om K ( X ) . Indeed, let ( e i ) i ∈ I be an a lg e- braic b asis o f K X (the e xistence of such a basis r eq uir es the axiom of choice for sets X of arbitr ary lar ge ca r dinal numb er). Ther efore , every map f ∈ K X may be ( uniquely) written a s a finite linea r combinatio n X i ∈ I f i e i , with f i ∈ K for each i ∈ I . Let u s consider the map ℓ : K X → K such tha t ℓ ( f ) = X i ∈ I f i . Clearly , ℓ is a linea r form, that is, an element o f the algebraic du al ( K X ) ∗ of K X . The family ( δ x ) x ∈ X is linearly independent in K X . Thus we may consider th e algebraic basis o f K X that extends ( δ x ) x ∈ X . Now , the corr esponding functional ℓ ha s a non zer o value for e ac h δ x . Ther e for e, if X is infi nite , then ℓ does not be long to the ima ge of Φ , or , in o ther terms, ℓ 6∈ ( K X ) ′ . In p articular , when e ver K is an Hausdorff topo lo gical fi eld, K X has th e pr o duct top ology , an d X is infinite, then ℓ is disco ntinuous at zer o ( and thus on the whole K X ). Rigidity of the topological dual of spaces of formal series with r espect to pr oduct topologies 7 2. A field topology may be either Ha usdorff or the indiscr ete one. I ts seems to r emain too little choice. Actually it is kno wn (see ?? ) that every infi nite field K has 2 2 | K | non-h omeomorphic topologies (wher e | X | denotes the cardinal number of a set X ). 3. Let R be a discrete ring, and X b e a set. Let X ∗ be the fr ee monoid on the alphabet X , ǫ b e the empty w or d , and | ω | be the length of a wor d ω ∈ X ∗ . Let us defin e M ≥ n = { f ∈ R X : ν ( f ) ≥ n } , n ∈ N , where ν ( f ) = inf { n ∈ N : ∃ ω ∈ X ∗ , | ω | 6 = n, a nd f ( ω ) 6 = 0 } for every non -zer o f ∈ R X (the infimum being taken on N ∪ {∞} , with ∞ > n for every n ∈ N , then ν (0) = ∞ ). The set R X , seen as the R -algebra R hh X ii of formal p ower series in noncommuta tive variables, may b e topologized (as a topologica l R -a lg ebra – see ? – an d, ther efor e, as a topological R -modu le) by the decr easing filtr ation of ideals M ≥ n : this is an e xa mple of the so-called Krull topolo gy (see ? ), and it is the usual top ology considered for formal p ower series in combinatorics a nd algebra; in case X is reduced to an element x , we r ecover the usu al M -adic topology of K [[ x ]] , where M = h x i is the p rincipal ideal generated by x . Wh enever X is finite, this Krull topology coincides (iii) with the pr odu ct topology with a discr ete R . According to The or em 5, in case wher e X is fi nite and R is a topological field K , the topo logical dual of K hh X ii (which is also a lin early compa ct space) is the space of polynomia ls K h X i in nonco mmutative variab les. 4. T ake a ny mon oid with a zer o (see ? ) with the fi nite deco mposition pr ope rt y (see ?? ), an d let R be a ring. Let u s consider the tota l contracted R -algebra R 0 [[ M ]] of the monoid with zer o M (see ? ) that c onsists, as a R -module, of { f ∈ R M : f (0 M ) = 0 R } , wher e 0 M is the zer o of M , while 0 R is the zer o o f the ring R . It is clear tha t R 0 [[ M ]] ∼ = R M ∗ (as R -algebras), with M ∗ = M \ { 0 M } . Now , let us a ss ume th at K is an Ha usdorff topological field. The pr oduct topology on K M induces the pr oduct top ology on K M ∗ , that corresponds to that of K 0 [[ M ]] . The to pological dual of K M ∗ being K ( M ∗ ) it is easy to check that ( K 0 [[ M ]]) ′ is isomorphic to the (usual) contracted a lgebr a (see ? ) K 0 [ M ] of the monoid with zer o M . 5. Th eor em 5 ma y be applied for th e discr ete to pology on K ∈ { R , C } , b ut also for the u sual topolo- gies of R and C , in such a wa y tha t the top ological dual sp aces ( R X ) ′ or ( C X ) ′ for bo th discr ete topology an d the topology induced b y the (usual) absolute va lues o n R , C ar e identical since iso- morphic to R ( X ) or C ( X ) . Notice that K X is a F r ´ echet space (see ? ), real o r co mple x depending on whether K = R o r K = C , when is co nsider ed the pr o duct topology r elative to the ab s olute value, and as such allows fu nctional analy s is like, for instance, Banach-Steinhau s, o pen map a nd closed graph theorems that do not hold in the case of the same space with the pr odu ct to pology relative to a discr ete K . 5 Consequ ence on contin uous endomor phisms As explained in the In troduction, th e rigidity of th e dual space with respect to the chan ge of produ ct topolog ies forces co ntinuous linear maps ( with respect to any of th ose topolog ies) to be represented by “ row-finite ” matrices. Let K be an Hausdorff topolog ical field, and X , Y be two sets. W e suppo se that K Z has the pro duct topolog y for Z ∈ { X , Y } . The set of all linear m aps (resp. continu ous linear map s) from K X to K Y is (iii) Notice that if X is infinite both topologies are distinct: for instance, let ( x n ) n ≥ 0 be a sequence of distinct elements of X , then this family is easil y seen summable in the product topology with R discrete, while it does not con ver ge i n the Krull topology . 8 Laur ent P o insot denoted by Ho m K ( K X , K Y ) (resp . L ( K X , K Y ) ). W e denote by K Y × ( X ) the set o f all maps M : Y × X → K such that for each y ∈ Y , the set { x ∈ X : M ( y , x ) 6 = 0 } is finite. Recall th at if p ∈ K ( X ) , then its support is given b y supp ( p ) = { x ∈ X : p ( x ) 6 = 0 } . (6) Let φ ∈ L ( K X , K Y ) . W e d efine the following map: M φ : Y × X → K ( y , x ) 7→ h δ y , φ ( δ x ) i . (7) Lemma 13 F or each φ ∈ L ( K X , K Y ) , M φ ∈ K Y × ( X ) , and the map φ 7→ M φ is into. Proof: F o r e very x ∈ X , the map K X → K f 7→ h δ y , φ ( f ) i (8) is an element of ( K X ) ′ (because it is the co mposition of φ and the projection onto K δ y ). According to Theorem 5, there is one and only on e p φ,y ∈ K ( X ) such th at for e very f ∈ K X , h p φ,y , f i = h δ y , φ ( f ) i . In particular, fo r e very x ∈ X ,  p φ,y ( x ) if x ∈ supp ( p φ,y ) 0 R otherwise = X z ∈ X p φ,y ( z ) δ x ( z ) = h p φ,y , δ x i = h δ y , φ ( δ x ) i . (9) Therefo re { x ∈ X | h δ y , φ ( δ x ) i 6 = 0 } = supp ( p φ,y ) , and then M φ ∈ K Y × ( X ) . Suppose that M φ = M φ ′ , then for every ( y , x ) ∈ Y × X , h δ y , φ ( δ x ) i = h δ y , φ ′ ( δ x ) i . Then, by bilinearity , φ ( δ x )( y ) − φ ′ ( δ x )( y ) = h δ y , φ ( δ x ) − φ ′ ( δ x ) i = 0 . Since this last equ ality h olds for e very y ∈ Y , φ ( δ x ) = φ ′ ( δ x ) for every x ∈ X . Now , let f ∈ K X , since f = X x ∈ X f ( x ) δ x (sum of a summ able family), by continuity , φ ( f ) = X x ∈ X f ( x ) φ ( δ x ) = X x ∈ X f ( x ) φ ′ ( δ x ) = φ ′ ( f ) . ✷ Theorem 14 The sets L ( K X , K Y ) a nd K Y × ( X ) ar e eq uipotent. Mo r e p r ecisely the map φ 7→ M φ of Lemma 13 is onto. Proof: Let M ∈ K Y × ( X ) . Let us define ψ M : K X → K Y by ψ M ( f ) = ψ M ( X x ∈ X f ( x ) δ x ) = X y ∈ Y X x ∈ X M ( y , x ) f ( x ) ! δ y . (Clearly the second sum on x ∈ X has o nly finitely many non zero term s since M ∈ K Y × ( X ) , and therefor e is defined in K .) The map ψ M is linear . T o see this, s ince ψ M ( f ) ∈ K X , it is suf ficien t to prove Rigidity of the topological dual of spaces of formal series with r espect to pr oduct topologies 9 that fo r every λ ∈ K , f , g ∈ K X , and every y ∈ Y , ψ M ( λf + g )( y ) = λψ M ( f )( y ) + ψ M ( g )( y ) . This equality to prove is equiv alent to the follo wing: ψ M ( λf + g )( y ) = λψ M ( f )( y ) + ψ M ( g )( y ) ⇔ h δ y , ψ M ( λf + g ) i = λ h δ y , ψ M ( f ) i + h δ y , ψ M ( g ) i ⇔ h δ y , ψ M ( λf + g ) i = h δ y , λψ M ( f ) + ψ M ( g ) i ⇔ X x ∈ X M ( y , x )( λf ( x ) + g ( x )) = X x ∈ X ( λM ( y , x ) f ( x ) + M ( y , x ) g ( x )) . (10) But the last equality is obvious (since the su ms hav e only finitely many non zero terms). Let us prove that ψ M is continuo us. It is suf ficient to prove that fo r every y ∈ Y , ℓ M ,y : K X → K , defined by ℓ M ,y ( f ) = h δ y , ψ M ( f ) i = X x ∈ X M ( y , x ) f ( x ) , is continuous. This is the case sin ce ℓ M ,y is a finite sum of scalar multiples of pro jections. Th erefore ψ M ∈ L ( K X ) . Finally we prove that M ψ M = M . Let ( y , x ) ∈ Y × X , we have M ψ M ( y , x ) = h δ y , ψ M ( δ x ) i = h δ y , X y ′ ∈ Y X z ∈ X M ( y ′ , z ) δ x ( z ) ! δ y ′ i = X z ∈ X M ( y , z ) δ x ( z ) = M ( y , x ) . (11) The map φ 7→ M φ is thus onto, and, by Lemma 13, it is a bijection. ✷ Note 1 1. If X = Y , then den oting the set L ( K X , K X ) of all continuous endomorp hisms by L ( K X ) , we find that L ( K X ) and K X × ( X ) ar e equ ipotent. 2. I f X = Y = N , th en is r e co ver e d th e usual notion of r o w-finite matrices (see ? ) . 3. I f Y is reduced to a single element x , th en L ( K X ) = L ( K X , K ) ∼ = L ( K X , K { x } ) ∼ = K { x }× ( X ) ∼ = K ( X ) , which is our Theor em 5. 6 T opological duality and completion In this section , we c onstruct e x plicitly the canonical isomorphism between the top ological duals of K ( X ) and K X using the fact that the former is the completion of the later . Recall the fo llo win g definition: let K be a field with an Hausdo rf f (field) topology , and V be an Hau s- dorff topolog ical K -vector space. T he completion of V is a pair ( b V , i ) where b V is complete (Hausdo rf f ) topolog ical K -vector space and i : V → b V such that 1. T he map i is an isomorp hism of top ological K -vector space structures from V into b V , i.e. , i is both an (algebraic) isomor phism an d a ho meomorphism into b V (or in other terms, i is a bicontinu ous on e- to-one R -linear map, or i is a continuo us one-to -one R -linear map a nd its in verse i − 1 : i ( V ) → V is continuo us where i ( V ) has the subspace topolog y indu ced by b V ); 10 Laur ent P o insot 2. T he image i ( V ) is dense in b V ; 3. For any comp lete (Hausdorff) K -vector space W and any co ntinuous and lin ear map φ : V → W , there exists one and only one map b φ : b V → W such that b φ ◦ i = φ . Remark 2 Acco r ding to ? , K X is co mplete and sep ar ated ( wi th respect to the pr oduct top ology) if, a nd only if, K is itself a complete Hausdorff field. In particular , th e topological d ual of K X does n ot depend on whether or not K is a complete field. Now let us assume that K has a the discrete top ology (therefo re K is a comp lete Hausdorff topolo gical field). Clearly , K X is th e completio n of K ( X ) (the later b eing equipp ed with the initial topolo gy with respect to the obvious projec tions which coin cides with the subspace top ology ind uced by K X for its produ ct topolo gy). Acco rding to the definition of a c ompletion, taking K in place of W , it is clear th at there exists a canonical isomorphism Ψ between the topolog ical dual of K ( X ) and K X . The isomorphism Ψ : ( K ( X ) ) ′ → ( K X ) ′ ℓ 7→ b ℓ (12) has inverse Ψ − 1 ( ℓ ) = ℓ ◦ i = ℓ | K ( X ) for ℓ ∈ ( K X ) ′ . The isomorp hism Ψ may be g i ven a precise definition. Let ℓ ∈ ( K ( X ) ) ′ . Then we hav e Ψ( ℓ ) = b ℓ : K X → K f 7→ X x ∈ X f ( x ) ℓ ( δ x ) . (13) Indeed since f = X x ∈ X f ( x ) δ x (sum of a summable family), we ha ve Ψ( ℓ )( f ) = b ℓ ( f ) = b ℓ ( X x ∈ X f ( x ) δ x ) = X x ∈ X f ( x ) b ℓ ( δ x ) (since b ℓ is continuou s) = X x ∈ X f ( x ) ℓ ( δ x ) . (since δ x ∈ K ( X ) ) (14) According to previously intro duced no tations, Ψ ( ℓ )( f ) = b ℓ ( f ) = h p Ψ( ℓ ) , f i , where we recall that p Ψ( ℓ ) ∈ K ( X ) is d efined by p Ψ( ℓ ) ( x ) = Ψ( ℓ )( δ x ) = b ℓ ( δ x ) = ℓ ( δ x ) (since δ x ∈ K ( X ) ). Note th at p Ψ( ℓ ) = Φ − 1 (Ψ( ℓ )) = Φ − 1 ( b ℓ ) . The map Φ − 1 ◦ Ψ : ( K ( X ) ) ′ → K ( X ) is an isomorph is m (com- position of isom orphisms). The polynomial Φ − 1 (Ψ( ℓ )) ∈ K ( X ) for ℓ ∈ ( K ( X ) ) ′ is thus given by Φ − 1 (Ψ( ℓ ))( x ) = ℓ ( δ x ) for x ∈ X . W e can check that ℓ ( p ) = h Φ − 1 (Ψ( ℓ )) , i ( p ) i for any ℓ ∈ ( K ( X ) ) ′ , and p ∈ K ( X ) . Indeed , ℓ ( p ) = Ψ( ℓ )( i ( p )) = h Φ − 1 (Ψ( ℓ )) , i ( p ) i . Rigidity of the topological dual of spaces of formal series with r espect to pr oduct topologies 11 7 W eak topology Let K be a to pological field , ( V , τ ) be a topological K -vector space, and V ′ be its topolog ical dual. W e call V ′ -weak topo lo gy the weakest topolog y o n V such th at the elements of V ′ are contin uous. Let us denote b y τ w this topo logy . Since the elem ents of V ′ are co ntinuous (for τ ), we h a ve τ w ⊆ τ . It can be shown that ( V , τ w ) is also a K -topological vector s pace, separated if K is so a nd V ′ separates t he elements of V . The topolog ical d ual V ′ w of ( V , τ w ) is called weak dual of V . Corollary 2 Let ( K , τ ) be an Hausd orf f to pological fi eld, X be a set, and K X endowed with the pr oduct topology , denoted by π ( X, τ ) , of | X | copies of ( K , τ ) . Th e weak dual ( K X ) ′ w of ( K X , τ w ) is (iso morphic to) K ( X ) . Proof: An element of K ( X ) is o bviously a c ontinuous linear form ( via the isom orphism between ( K X ) ′ and K ( X ) of Theorem 5) by definition of th e K ( X ) -weak topology . Th erefore, K ( X ) ⊆ ( K X ) ′ w . Let u s prove that the p roduct and the we ak topo logies are actually e qual. Th e weak topology τ w is, by defin ition, the weakest topolo gy for which every linear form h p, ·i : K X → ( K , τ ) is contin uous ( p ∈ K ( X ) ). In particular, the pro jections h δ x , ·i are also continuo us. Ther efore τ w is stro nger than π ( X, τ ) (since the later is the weakest topology with this proper ty). So π ( X, τ ) ⊆ τ w . Let us prove the recipro cal in clusion. T o do so, it is sufficient to prove that the identity map id : ( K X , π ( X, τ )) → ( K X , τ w ) is continu ous, which is equiv alent (according to usual properties of initial top ology) t o the fact that for every p ∈ K ( X ) , the map h p, ·i : ( K X , π ( X, τ )) → ( K , τ ) f 7→ h p, f i (15) is contin uous, which is obviously th e case since th ese maps are sum of a fin ite numb er of (scalar multiples of) projections. Therefore a linear form ℓ : K X → ( K , τ ) is continu ous with respect to the weak topology if, an d only if, it is contin uous with resp ect to the p roduct topology , and ther efore is an elem ent of K ( X ) (accord ing to Theo rem 5). ✷ 12 Laur ent P o insot