Bounded Linear Maps between (LF) spaces. Angela A. Albanese. Aplicaciones lineales acotadas entre espacios (LF)

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1 RACSAM Rev. R. Acad. Cien. Seie A. Mat. VOL. 97 (1), 2003, pp Análisis Mateático / Mateatical Analysis Bounded Linea Maps between (LF) spaces Angela A. Albanese Abstact. Caacteizations of pais (E, F ) of coplete (LF) spaces suc tat evey continuous linea ap fo E to F aps a 0 neigbouood of E into a bounded subset of F ae given. Te case of sequence (LF) spaces is also consideed. Tese esults ae siila to te ones due to D. Vogt in te case E and F ae Fécet spaces. Te eseac continues wo of J. Bonet, A. Galbis, S. Önal, T. Tezioğlu and D. Vogt. Aplicaciones lineales acotadas ente espacios (LF) Resuen. Se dan caacteizaciones de paes (E, F ) de espacios (LF) tales que toda aplicación de E en F que aplica un intevalo de 0 de E en un subconjunto acotado de F. Se considea tabién el caso de una sucesión de espacios (LF). Los esultados son siilaes a los obtenidos po D. Vogt paa el caso en que E y F son espacios de Fécet. Esta investigación continua el tabajo de J. Bonet, A. Galbis, S. Önal, T. Tezioğlu and D. Vogt. Te poble of te caacteization of tose pais of locally convex spaces E and F suc tat evey continuous linea ap fo E to F aps a 0 neigbouood into a bounded subset of F (denoted by L(E, F ) = LB(E, F )) as been extensively consideed in te liteatue wit diffeent puposes (e.g., see [1, 4, 5, 6, 10, 9, 12, 13]). Pais of Fécet spaces E and F fo wic te identity olds ave been copletely caacteized by D. Vogt in [12]. Fo pais of baelled (DF) spaces a siila esult as been povided by A. Galbis in [6] (fo fute caacteizations see also [1, 4]). It is wot noting tat te teoy of pais of Fécet spaces between wic evey continuous linea ap is bounded tuned to be a poweful tool in te study of te topological stuctue of Fécet spaces: fo exaple, it is stongly elated to te ipotant topological invaiants (DN) and (Ω) intoduced by D. Vogt (see [12]). On te ote and, in [2, 3] J. Bonet and P. Doaǹsi used suc a teoy to claify te elation between te vaious notions of vecto valued eal analytic functions. Motivated by tese facts, we continue ee te eseac on tis topic, in paticula giving a coplete caacteization of te pais of (LF) spaces E and F suc tat te identity L(E, F ) = LB(E, F ) olds. Ou caacteization is siila to te one given in [12] fo te case of Fécet spaces. Te aticle is divided in tee sections. In section 1 we collect soe geneal esults on L(E, F ) = LB(E, F ) wit E and F locally convex spaces. In section 2 we caacteize te pais of (LF) spaces E and F fo wic L(E, F ) = LB(E, F ). As consequences, we deive siila esults fo ote cases, fo exaple wit E (LF) space and F (LB) o DF space, wit E (LB) o Fécet space and F (LF) space, etc. Finally, in section 3 we apply ou esults to concete sequence (LF) spaces. Pesentado po José Bonet. Recibido: 10 de Octube de Aceptado: 9 de Octube de Palabas clave / Keywods: Locally convex spaces, (LF) spaces, sequence spaces, bounded linea aps Mateatics Subject Classifications: 46A03, 46A13, 46A04, 46A45, 46A32, 47L05, 46M10 c 2003 Real Acadeia de Ciencias, España. 13

2 A. A. Albanese Ou notation is standad. If E is a locally convex space, te faily of all bounded absolutely convex sets in E is denoted by B(E). If B B(E), E B denotes spanb equipped wit te gauge functional of B as a no. Te inclusion ap E B E is clealy linea and continuous. Te faily of all absolutely convex 0 neigbouoods in E is denoted by U(E). If E and F ae locally convex spaces, a linea ap T : E F is called bounded if tee is U U(E) so tat T (U) B(F ). Of couse, evey bounded ap is continuous. Te space of all linea and continuous aps (te space of all linea and bounded aps, esp.) fo E into F is denoted by L(E, F ) (by LB(E, F ), esp.). Clealy LB(E, F ) L(E, F ). L b (E, F ) denotes te space L(E, F ) endowed wit te topology of unifo convegence on te bounded sets of E. We efe te eade to [7, 8, 11] fo ote undefined notations and fo te geneal teoy of locally convex spaces. 1. Geneal Results In te sequel (E n ) n always denotes a sequence of locally convex spaces. Put E = n E n (te locally convex diect su of (E n ) n ), te ap i : E E, x (xδ n ) n, is an isoopis onto its ange fo evey N. On te ote and, te ap p : E E, (x n ) n x, is an ooopis onto and p i = id E fo evey N. Poposition 1 Let (E n ) n be a sequence of locally convex spaces and let F be a Fécet space. Ten te (i) L( n E n, F ) = LB( n E n, F ); (ii) L(E n, F ) = LB(E n, F ), n N. PROOF. (i) (ii). Let N and T L(E, F ). Ten T p L( n E n, F ) = LB( n E n, F ); ence tee is U U( n E n ) suc tat T (p (U)) B(F ). Since p is an ooopis onto, p (U) is also a 0 neigbouood in E and ence T LB(E, F ). (ii) (i). Let T L( n E n, F ). Ten T i n L(E n, F ) = LB(E n, F ) fo all n N; ence, fo eac n N, tee is U n U(E n ) suc tat T (i n (U n )) B(F ). Since F is a Fécet space, we can find a sequence of scalas (λ n ) n suc tat n λ n T (i n (U n )) = T ( n λ n i n (U n )) is also a bounded set in F, wee Γ( n λ n i n (U n )) U( n E n ). Ten te esult follows. Poposition 2 Let (E n ) n be a sequence of locally convex spaces and let F be a Fécet space. Ten te (i) L(F, n E n ) = LB(F, n E n ); (ii) L(F, E n ) = LB(F, E n ), n N. PROOF. (i) (ii). Let N and T L(F, E ). Ten i T L(F, n E n ) = LB(F, n E n ); ence, tee is U U(F ) suc tat i (T (U)) is a bounded set of n E n. Since i is an isoopis into, T (U) is also a bounded set in E and te esult follows. (ii) (i). Let T L(F, n E n ). Ten: N T (U ) n E n, (1) wee (U ) is a deceasing basis of 0-neigbouoods in F. Suppose tat (1) is not tue. Ten, fo eac N, tee is x U suc tat T (x ) n E n. Clealy, (x ) conveges to 0 in F and ence (T (x )) conveges to 0 in n E n too. Tus (T (x )) n E n fo soe N, obtaining a contadiction. 14

3 Bounded Linea Maps between (LF) spaces By (1), T (F ) n E n and ence T L(F, n E n ). Now, fo eac n p n T L(F, E n ) = LB(F, E n ) and ence tee is n N suc tat B n = p n (T (U n )) is a bounded set of E n. Put U = n U n, ten U U(F ) and T (U) n B n. Tis copletes te poof. Rea 1 It is clea fo te above poofs tat iplication (i) (ii) in Popositions 1 and 2 always olds fo any locally convex space F. But te convese geneally does not old as te following exaple sows. Let F = (l p ) (N), 1 p, and let E = (ω) (N). Clealy, te inclusion ap F E is linea and continuous, but not bounded. On te ote and, L(F, ω) = LB(F, ω) and L(l p, (ω) (N) ) = LB(l p, (ω) (N) ). Next, put E = n E n, fo eac N te ap j : E E, x (xδ n ) n, is an isoopis onto its ange and te ap : E E, (x n ) n x, is an ooopis onto suc tat j = id E. Poposition 3 Let (E n ) n be a sequence of locally convex spaces and let F be a DF space. Ten te (i) L( n E n, F ) = LB( n E n, F ); (ii) L(E n, F ) = LB(E n, F ), n N. PROOF. (i) (ii). Let N and T L(E, F ). Ten T L( n E n, F ) = LB( n E n, F ); ence tee is U U( n E n) suc tat T ( (U)) B(F ). Since is an ooopis onto, (U) is also a 0 neigbouood of E and te esult follows. (ii) (i). Let T L( n E n, F ). Ten: N T ({0} n> E n ) B, (2) wee (B ) is a fundaental inceasing sequence of bounded sets of F. Suppose tat (2) is not tue. Ten, fo eac N tee is x {0} n> E n suc tat T (x ) B. Clealy, (x ) conveges to 0 in n E n and ence (T (x )) conveges to 0 in F too. Since (B ) is a fundaental syste of bounded subsets of F, (T (x )) B fo soe N, obtaining a contadiction. Condition (2) iplies tat T ({0} n> E n) = {0}, being it a bounded subspace of F. On te ote and, fo eac n, T j n L(E n, F ) = LB(E n, F ) and ence tee is U n U(E n ) suc tat T (j n (U n )) B(F ). Put U = n U n n> E n, ten U is a 0 neigbouood of n E n and T (U) is a bounded set of F because U = U n E n j n (U n ) + {0} E n n n> n n> and ence T (U) n T (j n (U n )), wee n T (j n(u n )) is a bounded subset of F. Ten te esult follows. Poposition 4 Let (E n ) n be a sequence of locally convex spaces and let F be a DF space. Ten te (i) L(F, n E n) = LB(F, n E n); (ii) L(F, E n ) = LB(F, E n ), n N. 15

4 A. A. Albanese PROOF. (i) (ii). Let N and T L(F, E ). Ten j T L(F, n E n) = LB(F, n E n); tus tee is U U(F ) suc tat j (T (U)) B( n E n). Since j is an isoopis into, T (U) is also a bounded set of E and te esult follows. (ii) (i). Let T L(F, n E n). Ten, fo eac n N, n T L(F, E n ) = LB(F, E n ) and ence tee is U n U(F ) suc tat n (T (U n )) B(E n ). Since F is a DF space, tee is a sequence of scalas (λ n ) n suc tat U = n λ n U n is also a 0 neigbouood of F. On te ote and, fo eac n N n (T (U)) is bounded in E n, iplying tat T (U) is a bounded subset of n E n. Ten te esult follows. Rea 2 It is clea fo te above poofs tat iplication (i) (ii) in Popositions 3 and 4 always olds fo any locally convex space F. In geneal te convese is not tue as te following exaple sows. Let F = (l p ) N and E = (l q ) N, 1 p < q. Clealy, te inclusion ap F E is linea and continuous, but not bounded. On te ote and, L(F, l q ) = LB(F, l q ) and L(l p, E) = LB(l p, E). Let (E n, i n+1,n ) n be a sequence of locally convex spaces and linea and continuous inclusion aps i n+1,n : E n E n+1. Let E := ind n E n, i.e. E = n E n and it is endowed wit te finest locally convex topology fo wic all te inclusion aps i n : E n E ae continuous. Ten Poposition 5 Let E = ind n E n be an inductive liit of locally convex spaces and let F be a Fécet space. Ten: (i) L(E n, F ) = LB(E n, F ), n N L(E, F ) = LB(E, F ); (ii) E is egula and L(F, E n ) = LB(F, E n ), n N L(F, E) = LB(F, E). PROOF. (i). Let T L(E, F ). Ten, fo eac n N, T i n L(E n, F ) = LB(E n, F ) and ence tee is U n U(E n ) suc tat T (i n (U n )) B(F ). Since F is a Fécet space, tee is a sequence of scalas (λ n ) n fo wic T ( n λ n i n (U n )) = n λ n T (i n (U n )) is also a bounded set of F. On te ote and, Γ( n λ n i n (U n )) U(E) and ten T LB(E, F ). (ii). Let T L(F, E). Ten: N T (F ) E and T : F E is continuous. (3) To pove (3) it suffices to epeat te sae poof of Gontendiec Floet s factoization teoe [11, ]. Fo te sae of copleteness, we give ee te poof of (3). Let F be te filte geneated by (T (U )), wee (U ) is a deceasing basis of 0 neigbouoods in F. Fo eac n N let F n be te filte geneated by a basis of 0 neigbouoods in E n. Let (x ) be a F convegent sequence in E. Ten tee is (y ) F so tat T (y ) = x and (y ) is also a convegent sequence in F. Since F is a Fécet space, tee is an unbounded sequence of positive scalas (λ ) suc tat (λ y ) is bounded in F. It follows tat (λ x ) = (T (λ y )) is a bounded set of E. Since E is a egula inductive liit, tee is n N suc tat (λ x ) E n and bounded ee. Tus (x ) conveges to 0 in E n and (x ) is a F n convegent sequence. By [11, ], tee is N suc tat F is coase tan F and ten (3) follows. By (3), T L(F, E ) = LB(F, E ) and ence T LB(F, E). Rea 3 Te convese of Poposition 5 (i) does not old in geneal as te following exaple sows. Fo eac n N we define E n := <n ω n s, wee s is te Fécet space of all apidly deceasing sequences. Put E = ind n E n. By [11, 8.7.2] E is a dense topological subspace of ω N. If T L(E, s), tee is ten a unique linea continuous extension S L(ω N, s). Teefoe S, and ence T, is bounded because ω N is a quojection and s as a continuous no. On te ote and, it is plain tat L(E n, s) LB(E n, s) fo evey n N. 16

5 Bounded Linea Maps between (LF) spaces Let (E n, n,n+1 ) n be a sequence of locally convex spaces and n,n+1 : E n+1 E n linea and continuous aps. Let E := poj n E n. Ten all te aps : E E, (x n ) n x, ae continuous. We ave Poposition 6 Let E = poj n E n be a pojective liit of locally convex spaces and let F be a DF space. Ten: L(F, E n ) = LB(F, E n ), n N L(F, E) = LB(F, E). PROOF. Let T L(F, E). Ten, fo eac n N n T L(F, E n ) = LB(F, E n ) and ence tee is U n U(F ) suc tat B n = n (T (U n )) B(E n ). Since F is a DF space, tee is a sequence of scalas (λ n ) n suc tat U = n λ n U n is a 0 neigbouood in F, wee T (U) n λ nb n ; tus T (U) is bounded in E. 2. Te case of (LF) spaces In tis section we conside te equality L(E, F ) = LB(E, F ) in case E is a egula (LF) space and F is a coplete (LF) space o a DF space. We find a caacteization siila to te one given by Vogt [12] wen E and F ae Fécet spaces. Applications of ou esults fo paticula cases will be given. Let (E, i +1, ) be a sequence of Fécet spaces and i +1, : E E +1 linea and continuous inclusion aps. Let E = ind E so tat all inclusion aps i : E E ae continuous and i +1 i +1, = i. We always assue tat, fo eac N, E = poj (E, p ) is te educed pojective liit of te sequence (E, p ) of Banac spaces and linea and continuous aps s,+1 : E +1 E wit dense ange. Let s : E E be te canonical pojection suc tat s,+1 s +1 = s. Put U = {x E : p (s (x)) 1}. Ten (U ) is a basis of 0 neigbouoods in E. Witout loss of geneality, we can also pose tat U U +1 fo evey and N. Consequently, fo eac and N tee is a linea and continuous ap i +1, : E E +1 suc tat te following diaga coutes E s E s 1, E 1 i +1, i +1, i 1 +1, s +1 1, E +1 E E +1 s +1 We denote by p te gauge functional of U (te pola of U wit espect to te duality < E, E >). Let (F, j +1, ) be anote sequence of Fécet spaces and linea and continuous inclusion aps j +1, : F F +1. Let F = ind F so tat all inclusion aps j : F F ae continuous and j +1 j +1, = j. We always assue tat, fo eac N, F = poj (F, q ) is te educed pojective liit of te sequence (F, q ) of Banac spaces and linea and continuous aps t,+1 : F +1 F wit dense ange. Let t : F F be te canonical pojection so tat t,+1 t +1 = t. Put V = {x F : q (t (x)) 1}. Ten (V ) is a basis of 0-neigbouoods in F. We obseve tat: Rea 4 (i) Fo eac N we define J +1, : L b (E, F ) L b (E, F +1 ) by J +1, (T ) := j +1, T. Clealy, J +1, is an injective, linea and continuous ap. We can ten conside te inductive liit ind L b (E, F ) of te sequence (L b (E, F )) of locally convex spaces. If fo any N we define J : L b (E, F ) L b (E, F ) by J (T ) := j T, J is also an injective, linea and continuous ap and J +1 J +1, = J. Tus tee is an injective, linea and continuous ap J : ind L b (E, F ) L b (E, F ). On te ote and, if fo any N we define I,+1 : L b (E +1, F ) L b (E, F ) by I+1,(T ) := T i +1,, I,+1 is linea and continuous. We can ten conside te pojective liit poj L b (E, F ) of te pojective sequence (L b (E, F ), I,+1). Tus te ap I : L b (E, F ) poj L b (E, F ) defined by I (T ) := (T i ) is an isoopis onto. Moeove, if J +1, : poj L b (E, F ) (4) 17

6 A. A. Albanese poj L b (E, F +1 ) is te ap given by J +1,((T ) ) := (j +1, T ), J +1, is injective, linea and continuous and te following diaga is coutative I L b (E, F ) poj L b (E, F ) J +1, J +1, I L b (E, F +1 ) +1 poj L b (E, F +1 ). Consequently, we can conside te inductive liit ind poj L b (E, F ) and te ap I : ind poj L b (E, F ) ind L b (E, F ) given by ((I ) 1 ) (i.e., I((T ) ) := (I ) 1 ((T ) ) if (T ) poj L b (E, F )), wic is an isoopis onto. Now, given any and in N, fo eac N we define S, +1, : L b(e, F ) L b (E +1, F ) by S, +1, (T ) := T s,+1. Since s,+1 as dense ange, S, +1, is injective, linea and continuous. We can ten conside te inductive liit ind L b (E, F ) of te inductive sequence (L b (E, F ), S, +1, ). If fo N we define S, : L b (E, F ) L b (E, F ) by S, (T ) := T s, S, is also an injective, linea and continuous ap. Moeove, S, +1 S, +1, = S,. Ten tee is an injective, linea and continuous ap S, : ind L b (E, F ) L b (E, F ). Also, if I,,+1 : L b(e +1, F ) L b (E, F ) is te ap given by I,,+1 (T ) := T i +1,, is linea and continuous and tuns to be coutative te following diaga because of (4) I,,+1 L b (E 1, F ) I 1,,+1 L b (E +1 1, F ) S, +1, L b (E, F ) I,,+1 S +1, +1, L b (E +1, F ) S, L b (E, F ) I,+1 S +1, L b (E +1, F ). Teefoe te ap I,+1 : ind L b (E +1, F ) ind L b (E, F ), given by I,+1(T ) := T i +1, fo T L b (E +1, F ), is well defined, linea and continuous and suc tat te following diaga also coutes S ind L b (E, F ), L b (E, F ) I,+1 I,+1 S ind L b (E +1, F ) +1, L b (E +1, F ). Tis eans tat te ap S : poj ind L b (E, F ) poj L b (E, F ) defined by S ((T ) ) := (S, (T )) is linea and continuous. Since S, is injective fo evey N, S is also injective. Moeove, if J +1, : poj ind L b (E, F ) poj ind L b (E, F +1 ) is te ap suc tat J +1,((T ) ) = (j,+1 T ), ten J +1, is injective, linea and continuous and tuns to be coutative te following diaga J poj ind L b (E, F +1 ) S +1 poj L b (E, F +1 ) +1, J +1, S poj ind L b (E, F ) poj L b (E, F ). Te ap S : ind poj ind L b (E, F ) ind poj L b (E, F ), given by S((T ) ) := S ((T ) ) if (T ) poj ind L b (E, F ), is ten well-defined, injective, linea and continuous. Finally, given, and N, if fo any N we define T,,+1 : L b(e, F +1 ) L b (E, F ) by T,,+1 (T ) := t,+1 T, T,,+1 is also linea and continuous. Let poj L b (E, F ) be te pojective liit of te pojective sequence (L b (E, F ), T,,+1 ). Ten te ap T, : L b (E, F ) L b (E, F ) defined by T, (T ) = (t T ) is an isoopis onto. 18

7 Bounded Linea Maps between (LF) spaces Now, poceeding exactly as we did befoe, one sows tat we can conside te space ind poj ind poj L b (E, F ) and define in a canonical way an injective, linea and continuous ap T fo tis space to Tus we ave poved tat te ap ind poj ind L b (E, F ). R := J I S T : ind poj ind poj L b (E, F ) L b (E, F ) is well defined, injective, linea and continuous. (ii) Suppose tat F is a egula (LF) space. Ten, as it is easy to see, te iage of te ap R above constucted is te space LB(E, F ) of all linea bounded aps fo E into F. (iii) Suppose tat E is a egula (LF) space and F is a coplete (LF) space. Let (((n, j)) j ) n be a countable set of inceasing sequences of positive integes and let ((n)) n be anote inceasing sequence of positive integes. Ten te space H := {T L(E, F ) : T j,n := q (n) j (T (x)) < + n, j N} is a Fécet space wit espect to te topology geneated by te sequence ( j,n ) j,n of seinos. To see tis, it suffices to sow tat te inclusion ap H L b (E, F ) is continuous. Let B B(E) and V U(F ). Since E is a egula (LF) space, tee is n 0 N suc tat B E n0 and bounded ee. Now, V F (n0) is a 0 neigbouood in F (n0) and ence V F (n0) µv (n 0) j 0 fo soe j 0 N and µ > 0. Since B is a bounded set of E n0, we find a λ > 0 suc tat B λu n 0 (n 0,j 0 ). Ten W = {T H : T (U n0 (n 0,j 0 ) ) V (n0) j 0 } µ 1 λ{t L(E, F ) : T (B) V } = µ 1 λm, wee W and M ae 0 neigbouoods in H and in L b (E, F ), espectively. In fact, if T W, ten T (U n 0 (n ) V (n 0) 0,j 0) j 0 and ence T (B) λt (U n 0 (n ) λv (n 0) 0,j 0) j 0 λµ 1 V F (n0) λµ 1 V : teefoe T µ 1 λm. Since L b (E, F ) is coplete and te inclusion ap H L b (E, F ) is continuous, we obtain tat H is a Fécet space. Now, we ae eady to state and pove: Teoe 1 Let E = ind E be a egula (LF) space and let F = ind F be a coplete (LF) space. Ten te (i) L(E, F ) = LB(E, F ); (ii) fo eac sequence (((n, j)) j ) n of inceasing sequences of positive integes and fo eac inceasing sequence ((n)) n of positive integes N N N N j 0, n 0 N C > 0 : (5) x U q (T (x)) C ax q (n) j (T (x)), fo evey T L(E, F ) wit x U n q (n) (n,j) j (T (x)) < + fo eac j, n N. 19

8 A. A. Albanese PROOF. (i) (ii). Let ((n, j)) j ) n be a sequence of inceasing sequences of positive integes and let ((n)) n be anote inceasing sequence of positive integes. We conside te space H = {T L(E, F ) : T j,n = q (n) j (T (x)) < +, j, n N}. By Rea 4 (iii), H is a Fécet space wit espect to te topology geneated by te sequence ( j,n ) j,n of seinos and te inclusion ap H L b (E, F ) is continuous. On te ote and, by ypotesis L(E, F ) = LB(E, F ) ; tus, by Rea 4 (i) and (ii), te linea ap R: ind poj ind poj L b (E, F ) L b (E, F ) is a continuous algebaical isoopis onto. Endowed te space L(E, F ) wit te topology induced via te ap R, it follows tat te inclusion ap H L(E, F ) = ind as closed gap. Since H is a Fécet space and L(E, F ) = ind poj poj ind ind poj L b (E, F ) poj L b (E, F ) is a stictly webbed, we can apply te Wilde s closed gap teoe [7, 5.4.1] to conclude tat tis inclusion ap is continuous. Tus, by te Localization Teoe [7, 5.6.3], tee is N suc tat H poj ind poj L b (E, F ) and te ap H poj ind poj L b (E, F ) is also continuous. Teefoe fo eac N te ap H ind poj L b (E, F ), T T i, is continuous. Finally, by Gotendiec s factoization teoe [11, ], tee is N suc tat is also continuous, wee T s = T i. We ave sown tat H poj L b (E, F ) = L b (E, F ), T T, N N N N j 0, n 0 N C > 0 : x U q (T (x)) C ax q (n) j (T (x)), fo evey T H. Tis copletes te poof. (ii) (i). Let T L(E, F ). Ten T i n L(E n, F ) fo all n N. By Gotendiec s factoization teoe [11, ], we can find an inceasing sequence ((n)) n of positive integes suc tat T i n L(E n, F (n) ) fo all n N. Since T i n L(E n, F (n) ), we can find anote inceasing sequence ((n, j)) j of positive integes suc tat, fo eac j N, q (n) j (T (x)) = d nj < +. On te ote and, by assuption tee is N so tat (5) olds. Consequently, we get: 20 N N N j 0, n 0 N C > 0 : q(t (x)) C ax d nj < +. x U

9 Bounded Linea Maps between (LF) spaces Tis iplies tat T (U ) is a bounded subset of F fo all N. Since F is a etizable space, tee is a sequence of scalas (λ ) suc tat T ( λ U ) = λ T (U ) is bounded in F, wee Γ( λ U ) is a 0 neigbouood in E. Ten te poof is coplete. Wit obvious canges te pevious caacteization olds fo ote cases, e.g., taing E (LF) space o (LB) space and F Fécet space o coplete DF space. To obtain tis, it is enoug to do only soe eas. Conside fist te case wee E is a egula (LF) space and F is a Fécet space. Let F be a Fécet space. Assue tat F = poj (F, q ) is te educed pojective liit of te pojective sequence (F, q ) of Banac spaces and linea and continuous aps t,+1 : F +1 F wit dense ange. Let t : F F be te canonical pojection suc tat t,+1 t +1 = t. Put V = {x F : q (t (x)) 1}. Ten (V ) is a basis of 0 neigbouoods in F. Fo eac N put F = F and j +1, = j = id F. Clealy, F = ind F. Teefoe: Rea 5 (i) Let E = ind E be an (LF) space and let F be a Fécet space. By Rea 4 (i) and (ii), te ap R tuns to be a linea and continuous ap fo te space poj ind poj L b (E, F ) into te space L b (E, F ) and its iage is exactly te space LB(E, F ) of all linea bounded aps fo E into F. (ii) Let E be a egula (LF) space and let F be a Fécet space. Let ((n)) n be an inceasing sequence of positive integes. Poceeding exactly as we did in Rea 4 (iii), one sows tat te space H := {T L(E, F ) : T n = q n (T (x)) < +, n N} x U n (n) is a Fécet space wit espect to te topology geneated by te sequence ( n ) n of seinos and te inclusion ap H L b (E, F ) is continuous. Poceeding as we did to pove Teoe 1, we easily get: Poposition 7 Let E = ind E be a egula (LF) space and let F be a Fécet space. Ten te (i) L(E, F ) = LB(E, F ); (ii) fo eac inceasing sequence ((n)) n of positive integes and fo eac N x U N N n 0 N C > 0 : (6) q (T (x)) C ax x U n (n) q n (T (x)), fo evey T L(E, F ) wit x U n q n (T (x)) < + fo eac n N. (n) Next, conside te case wee E is a Fécet space and F is a egula (LF) space. Let E be a Fécet space. Assue tat E = poj (E, p ) is te educed pojective liit of te pojective sequence (E, p ) of Banac spaces and linea and continuous aps s,+1 : E +1 E wit dense ange. Let s : E E be te canonical pojection suc tat s,+1 s +1 = s. Put U = {x E : p (s (x)) 1}. Ten (U ) is a basis of 0 neigbouoods in E. Fo eac N let E = E and i +1, = i = id E. Clealy, E = ind E. We ave: Rea 6 (i) Let E be a Fécet space and let F = ind F be a egula (LF) space. By Rea 4 (i) and (ii), te ap R tuns to be a linea and continuous ap fo te space ind ind poj L b (E, F ) into te space L b (E, F ) and its iage is te space LB(E, F ) of all linea and bounded aps fo E into F. 21

10 A. A. Albanese (ii) Let E be a Fécet space and let F = ind F be a coplete (LF) space. Let ((n)) n be an inceasing sequence of positive integes and N. Poceeding as we did in Rea 4 (iii), one sows tat te space H := {T L(E, F ) : T n = qn(t (x)) < +, n N} x U (n) is a Fécet space wit espect to te topology geneated by te sequence ( n ) n of seinos and te inclusion ap H L b (E, F ) is continuous. Aguenting as we did to pove Teoe 1, we ten obtain: Poposition 8 Let E be a Fécet space and let F = ind F be a coplete (LF) space. Ten te (i) L(E, F ) = LB(E, F ); (ii) fo eac inceasing sequence ((n)) n of positive integes and fo eac N, 0 N N n 0 N C > 0 : (7) q 0 x U (T (x)) C ax qn(t (x)), x U (n) fo evey T L(E, F ) wit x U(n) q n(t (x)) < + fo eac n N. Suppose tat F is a coplete DF space. Let (B ) be a fundaental syste of absolutely convex closed bounded sets of F wit B B +1. Denote by q te gauge of B and by F te linea span of B. Ten (F, q ) is a noed space and te inclusion ap j : (F, q ) F is continuous. Actually, also te inclusion aps j +1, : F F +1 ae continuous. Let F i := ind (F, q ). Tus te inclusion ap F i F is continuous (algebaically F i = F ). Since F is a coplete DF space, (F, q ) is a Banac space and F i is a coplete (LB) space. Moeove, if E is a locally convex space, te inclusion ap L b (E, F i ) L b (E, F ) is linea and continuous and LB(E, F i ) = LB(E, F ). We ave: Rea 7 (i) Let E = ind E be an (LF) space and let F = ind F be a egula (LB) space (F is te inductive liit of te inductive sequence (F, j +1, ) of Banac spaces and linea and continuous inclusion aps j +1, : F F +1 so tat all inclusion aps j : F F ae continuous) o let F be a coplete DF space. By Reas 4 (i) and (ii), te ap R tuns to be a linea and continuous ap fo te space ind poj ind L b (E, F ) into te space L b (E, F ) and its iage is te space LB(E, F ). (ii) Let E = ind E be a egula (LF) space and let F = ind F be a coplete (LB) space o let F be a coplete DF space. Let ((n)) n and ((n)) n be two inceasing sequences of positive integes. Poceeding again as we did in Rea 4 (iii), one sows tat te space H = {T L(E, F ) : T n := q (n) (T (x)) < +, n N} x U n (n) is a Fécet space wit espect to te topology geneated by te sequence ( n ) n of seinos and te inclusion ap H L b (E, F ) is continuous. Repeating te poof of Teoe 1 wit obvious canges, we get: Poposition 9 Let E = ind E be a egula (LF) space. Let F = ind F be a coplete (LB) space o let F be a coplete DF space. Ten te (i) L(E, F ) = LB(E, F ); 22

11 Bounded Linea Maps between (LF) spaces (ii) fo eac pai of inceasing sequences (((n)) n, ((n)) n ) of positive integes x U N N, n 0 N C > 0 : (8) q (T (x)) C ax x U n (n) q (n) (T (x)), fo evey T L(E, F ) wit x U n q (n) (n) (T (x)) < + fo eac n N. It also olds a esult siila to te pevious ones in case E is a egula (LB) space and F a coplete (LF) o (LB) space. Let E = ind (E, p ) be te inductive liit of te sequence (E, i +1, ) of Banac spaces (U = {x E : p (x) 1} denotes te closed unit ball of E ) and linea and continuous inclusion aps i +1, : E E +1, wee all inclusion aps i : E E ae continuous. Ten: Rea 8 (i) Let E = ind E be an (LB) space and let F = ind F be a egula (LF) space (let F = ind F be a egula (LB) space o let F be a coplete DF space, esp.). By Reas 4 (i) and (ii), te ap R tuns to be a linea and continuous ap fo te space ind poj poj L b (E, F ) (fo te space ind poj L b (E, F ), esp.) into te space L b (E, F ) and its iage is te space LB(E, F ). (ii) Let E = ind E be a egula (LB) space and let F = ind F be a coplete (LF) space (let F be a coplete (LB) o DF space, esp.). Let ((n)) n be an inceasing sequence of positive integes. Ten (te space H = {T L(E, F ) : T j,n := x U n q (n) j (T (x)), j, n N} H = {T L(E, F ) : T n := x U n q (n) (T (x)), n N}, esp.) is a Fécet space wit espect to te topology geneated by te sequence ( j,n ) j,n (( n ) n, esp.) of seinos and te inclusion ap H L b (E, F ) is continuous. By epeating again te sae poof of Teoe 1 wit siple canges, we obtain: Poposition 10 Let E = ind E be a egula (LB) space and let F = ind F be a coplete (LF) space. Ten te (i) L(E, F ) = LB(E, F ); (ii) fo eac inceasing sequence ((n)) n of positive integes N N N j 0, n 0 N C > 0 : (9) q(t (x)) C x U ax q (n) j (T (x)), x U n fo evey T L(E, F ) wit x Un q (n) j (T (x)) < + fo eac j and n N. Poposition 11 Let E = ind E be a egula (LB) space and let F = ind F be a coplete (LB) space o let F be a coplete DF space. Ten te (i) L(E, F ) = LB(E, F ); (ii) fo eac inceasing sequence ((n)) n of positive integes N N n 0 N C > 0 : (10) q (T (x)) C x U ax q (n) (T (x)), x U n fo evey T L(E, F ) wit x Un q (n) (T (x)) < + fo eac n N. 23

12 A. A. Albanese 3. Te case of sequence (LF) spaces In tis section we conside concete sequence (LF) spaces wic wee intoduced by Vogt [13, 5] and we apply ou esults of 2. Let (a i, ) i,, be a atix wit non negative eal valued enties and wit te following popeties: Fo eac N put A = (a i, ) i, and i,, N a i, a+1 i,, i,, N a i,+1 a i,, i, N N a i, > 0. λ 1 (A ) = {x = (x 1, x 2,...) : p (x) = i x i a i, < +, N}. Ten λ 1 (A ) is a Fécet space wit fundaental syste of seinos (p ) and te inclusion ap i +1, : λ 1 (A ) λ 1 (A +1 ) is continuous fo evey N. Following [13, 5], we set E 1 = λ 1 (A ) endowed wit te finest topology fo wic all te inclusion aps i : λ 1 (A ) E 1 ae continuous, i.e. E 1 = ind λ 1 (A ) is an (LF) space. In te sequel, e i always denotes te i t unit vecto of E 1. Recall tat in [13, 5.14] it as been sown tat E 1 is a coplete (LF) space if, and only if, it is a egula (LF) space if, and only if, te atix (a i, ) i,, is of type (WQ), i.e. (a i, ) i,, satisfies te following condition: We ave: µ N n, N, K N N N, S > 0 i N : a i,n S(a i,µ + a N i,k). Teoe 2 Let F be a coplete (LF) space. Suppose tat E 1 is a egula (LF) space. Ten te following conditions ae equivalent: (i) L(E 1, F ) = LB(E 1, F ); (ii) fo eac sequence (((n, j)) j ) n of inceasing sequences of positive integes and fo eac inceasing sequence ((n)) n of positive integes fo evey y F (1) and i N. N N N N j 0, n 0 N C > 0 : (11) (a i,) 1 q(y) C ax (a (n,j) i,n ) 1 q (n) j (y), PROOF. (i) (ii). Let (((n, j)) j ) n be a countable set of inceasing sequences of positive integes. Let ((n)) n be anote inceasing sequence of positive integes. Since L(E, F ) = LB(E, F ), we can apply Teoe 1 to conclude tat condition (5) olds. Accodingly, we sow tat condition (11) olds too. Let y F (1) and let i N. We define T : E 1 F by T (x) := x i y. Clealy, T is a linea and continuous ap. Moeove, fo eac j and n N, we ave: q (n) j (T (x)) = (a (n,j) i,n ) 1 a (n,j) i,n x i q (n) j (y) (a (n,j) i,n ) 1 q (n) j (y) < +. 24

13 Bounded Linea Maps between (LF) spaces Teefoe, by (5) we obtain tat (a i,) 1 q(y) = q(t ((a i,) 1 e i )) x U C ax q (n) j (T (x)) C ax (a (n,j) i, 1 n n ) 1 q (n) j (y). 0 q (T (x)) Ten (ii) olds. (ii) (i). Let T L(E 1, F ). Ten T i n L(λ 1 (A n ), F ) fo evey n N; ence, by Gotendiec s factoization teoe [11, ] we can find an inceasing sequence ((n)) n of positive integes suc tat T i n L(λ 1 (A n ), F (n) ) fo evey n N. It follows tat, fo eac n N, we can find anote sequence ((n, j)) j of positive integes suc tat j N q (n) j (T (x)) = d nj < +. (12) By (ii), condition (11), accodingly tee is N suc tat N N N j 0, n 0 N C > 0 : (13) (a i, ) 1 q (y) C ax (a (n,j) i,n ) 1 q (n) j (y), fo evey y F (1) and i N. Tis iplies tat T (U ) F and bounded ee. Indeed, if x U T (x) = i x it (e i ); ence, by (12) and (13) we get: and x = i x ie i, ten q(t (x)) i i i x i q (T (e i )) = i x i a i, C ax ) x i a i, (a i, ) 1 q (T (e i )) (a (n,j) i,n ) 1 q (n) j (T (e i )) x i a i, C ax d nj = L p 1 n n (x) L 0 (because (e i ) i λ 1 (A 1 ), (T (e i )) i F (1) ). Tis eans tat T (U ) F and bounded ee. Since F is a etizable space, tee is a sequence of scalas (λ ) suc tat T ( λ U ) = λ T (U ) is also bounded in F, wee Γ( λ U ) is a 0 neigbouood of E 1. Now, te poof is coplete. Now, let (b j, ) j,, be anote atix wit nonnegative eal valued enties and wit te following popeties: j,, N b j, b+1 j,, j,, N b j,+1 b j,, j, N N b j, > 0. Fo eac N put B := (b j, ) j, and λ (B ) = {x = (x 1, x 2,...) : q(x) := x j b j, < +, N}. j Ten te space λ (B ) is a Fécet space wit fundaental syste of seinos (q ) and te inclusion ap j +1, : λ (B ) λ (B +1 ) is continuous fo evey N. Following [13, 5], we set E = 25

14 A. A. Albanese λ (B ) endowed wit te finest topology fo wic all te inclusion aps j : λ (B ) E ae continuous, i.e. E = ind λ (B ). As befoe, we denote by e j te j t unit vecto of E. Moeove, in [13, 5.14] it as been sown tat E is a coplete (LF) space if, and only if, it is a egula (LF) space if, and only if, te atix (b j, ) j,, is of type (WQ), i.e. (b j, ) j,, satisfies te following condition: We ave: µ N n, N, K N N N, S > 0 i N : b i,n S(b i,µ + b N i,k). Teoe 3 Let E = ind E be a egula (LF) space. Suppose tat E = ind λ (B ) is a coplete (LF) space. Ten te (i) L(E, E ) = LB(E, E ); (ii) fo eac sequence (((n, j)) j ) n of inceasing sequences of positive integes and fo eac inceasing sequence ((n)) n of positive integes N N N N j 0, n 0 N C > 0 : (14) b l,p (u) C ax b j 1 n n l,(n) p n (n,j) (u), 0 fo evey l N and u E wit p n (n,j) (u) < + fo eac j and n N. PROOF. (i) (ii). Let ((n, j)) j ) n be a countable set of inceasing sequences of positive integes. Let ((n)) n be anote inceasing sequence of positive integes. Since L(E, E ) = LB(E, E ), we can apply Teoe 1 to conclude tat condition (5) olds. Accodingly, we sow tat condition (14) olds too. Let u E wit p (n,j) (u) < + fo evey j and n N. Let l N. Conside te ap T : E E defined by T (x) := u(x)e l. Clealy, T is linea and continuous. In paticula, fo eac j and n N: Teefoe, by condition (5) we obtain tat: q (n) j (T (x)) = u(x) q (n) j (e l ) b l,p (u) = x U q (T (x)) C = u(x) b j x U n l,(n) (n,j) = p n (n,j) (u)bj l,(n) < + ax = C ax b j 1 n n l,(n) p n (n,j) (u). 0 q (n) j (T (x)) Tis coplete te poof. (ii) (i). Let T L(E, E ). Ten T i n L(E n, E ) fo evey n N. Since E n is a Fécet space and E is a (LF) space, we can apply Gotendiec s factoization teoe [11, ] to find an inceasing sequence ((n)) n of positive integes suc tat T i n L(E n, λ (B (n) )) fo evey n N. Tus, fo eac n N tee is anote inceasing sequence ((n, j)) j of positive integes suc tat j N q (n) j (T (x)) = d nj < +. (15) 26

15 Bounded Linea Maps between (LF) spaces By condition (14), accodingly tee is N suc tat N N N j 0, n 0 N C > 0 : (16) b l,p (u) C ax b j 1 n n l,(n) p n (n,j) (u), 0 fo evey l N and u E wit p n (n,j) (u) < + fo eac j and n N. Tis iplies tat T (U ) λ (B ) and bounded ee. To see tis we poceed as follows. Let g l : E K, x x l. Put u l = g l T. Ten T (x) = (u l (x)) l. Moeove, fo eac j and n N, by (15) we get: b j l,(n) p n (n,j) (u l) = l l b j l,(n) u l (x) = = q (n) j (T (x)) = d nj < + b j l,(n) u l(x) l wic assues us tat p n (n,j) (u l) < + fo all l N. Teefoe, we can apply (16) to obtain tat: q(t (x)) = x U x U u l (x) b l, l x U p (u l )b l l l = C ax = C ax C b j l,(n) p n (n,j) (u l) l d nj < +. p (x)p (u l )b l l ax b j 1 n n l,(n) p n (n,j) (u l ) 0 It follows tat T (U ) λ (B ) and bounded ee. Since λ (B ) is a etizable space, tee is a sequence of scalas (λ ) suc tat T ( λ U ) = λ T (U ) is also a bounded set of λ (B ), wee Γ( λ U ) is clealy a 0 neigbouood in E. Now, te poof is coplete. Consequently, it is easily to pove: Teoe 4 Suppose tat E 1 is a egula (LF) space and tat E is a nuclea (LF) space. Ten te (i) L(E 1, E ) = LB(E 1, E ); (ii) fo eac sequence (((n, j)) j ) n of inceasing sequences of positive integes and fo eac inceasing sequence ((n)) n of positive integes N N N N j 0, n 0 N C > 0 : (17) i, l N (a i,) 1 b l, C ax (a (n,j) i,n ) 1 b j 1 n n l,(n). 0 We now obseve tat, if te atix A = A = (a i ) i, fo evey N, ten te space E 1 tuns to be te Fécet space λ 1 (A), wee te sets defined by U := {x = (x 1, x 2,...) : i x i a i 1} fo a basis of 0 neigbouoods in λ 1 (A). Ten, aguenting as we did to pove Teoe 2 and using Poposition 8, we obtain: 27

16 A. A. Albanese Poposition 12 Let F be a coplete (LF) space. Ten te following condition ae equivalent: (i) L(λ 1 (A), F ) = LB(λ 1 (A), F ); (ii) fo eac inceasing sequence ((n)) n of positive integes and fo eac N, 0 N N n 0 N C > 0 : (18) (a i ) 1 q 0 (y) C ax (a (n) i ) 1 q n(y), fo evey y F (1) and i N. On te ote and, if (a i, ) i = (a i, ) i = a fo evey and N, te space λ 1 (A ) tuns to be te Banac space l 1 (a ) = {x = (x 1, x 2,...) : x = i x i a i, < + } and ence E 1 = 1 = ind l 1 (a ) is an (LB) space. Ten, aguenting again as we did to pove Teoe 2 and using Poposition 10, we get: Poposition 13 Let 1 = ind l 1 (a ) be a egula (LB) space and let F = ind F be a coplete (LF) space. Ten te (i) L( 1, F ) = LB( 1, F ); (ii) fo eac inceasing sequence ((n)) n of positive integes N N N j 0, n 0 N C > 0 : (19) (a i, ) 1 q(y) C ax (a i,n ) 1 q (n) j (y), fo evey y F (1) and i N. Clealy, we can obtain esults siila to te ones of Popositions 7, 9 and 11 in case E = E 1 and F is a Fécet space o in case E = E 1 and F is a coplete (LB) o DF space, o in case E = 1 and F is a coplete (LB) o DF spaces espectively. To pove tis facts it suffices to aguent as we did in te poof of Teoe 2 and to use Poposition 7, 9 and 11, espectively. Actually, one gets: Poposition 14 Let F be a Fécet space. Suppose tat E 1 is a egula (LF) space. Ten te following conditions ae equivalent: (i) L(E 1, F ) = LB(E 1, F ); (ii) fo eac inceasing sequence ((n)) n of positive integes and fo eac N N N n 0 N C > 0 : (20) (a i,) 1 q (y) C ax (a (n) i,n ) 1 q n (y), fo evey y F and i N. Poposition 15 Let F be a coplete (LB) o DF space. Suppose tat E 1 is a egula (LF) space. Ten te (i) L(E 1, F ) = LB(E 1, F ); 28

17 Bounded Linea Maps between (LF) spaces (ii) fo eac pai (((n)) n, ((n)) n ) of inceasing sequences of positive integes N N N n 0 N C > 0 : (21) (a i,) 1 q (y) C ax (a (n) i,n ) 1 q (n) (y), fo evey y F (1) and i N. Poposition 16 Let 1 = ind l 1 (a ) be a egula (LB) space and let F be a coplete (LB) o DF space. Ten te (i) L( 1, F ) = LB( 1, F ); (ii) fo eac inceasing sequence ((n)) n of positive integes N N n 0 N C > 0 : (22) (a i, ) 1 q (y) C ax (a i,n ) 1 q (n) (y), fo evey y F (1) and i N. Next, if te atix B = B = (b j ) j, fo all N, ten te space E tuns to be a Fécet space, i.e. E = λ (B), wee te sets so defined V = {x = (x 1, x 2,...) : x j b j 1} j fo a basis of 0 neigbouoods in λ (B). Ten, a poof siila to te one of Teoe 3 togete wit Poposition 7 gives: Poposition 17 Let E be a egula (LF) space. Ten te (i) L(E, λ (B)) = LB(E, λ (B)); (ii) fo eac inceasing sequence ((n)) n of positive integes and fo eac N N N n 0 N C > 0 : (23) b j p (u) C ax b n j p n (n) (u), fo evey u E wit p n (n) (u) < + fo all n N and j N. On te ote and, if (b j, ) j = (b j, ) j = b fo all N, te space λ (B ) tuns to be te Banac space l (b ) = {x = (x 1, x 2,...) : x = j x j b j, < + } and ence te space E = = ind l (b ) is an (LB) space. Ten, by epeating te sae aguent used in te poof of Teoe 3 and by using Poposition 9, we get: Poposition 18 Let E be a egula (LF) space. Suppose tat is a coplete (LB) space. Ten te (i) L(E, ) = LB(E, ); (ii) fo eac pai (((n)) n, ((n)) n ) of inceasing sequences of positive integes N N, n 0 N C > 0 : (24) b j, p (u) C ax b j,(n) p n (n) (u), fo evey j N and u E wit p n (n) (u) < + fo all n N. 29

18 A. A. Albanese Finally, we can also obtain esults siila to te ones of Popositions 8, 10 and 11 in case E is a Fécet space and F = E o in case E is a egula (LB) space and F = E, o in case E is a egula (LB) space and F =, espectively. Te poof of tis facts is based upon te sae aguent used in te poving Teoe 3 and upon Popositions 8, 10 and 11, espectively. Actually, we ave: Poposition 19 Let E be a Fécet space. Suppose tat E is a coplete (LF) space. Ten te following conditions ae equivalent: (i) L(E, E ) = LB(E, E ); (ii) fo eac inceasing sequence ((n)) n of positive integes and fo eac N, 0 N N n 0 N C > 0 : (25) b j, 0 p (u) C ax b n j,p (n) (u), fo evey j N and u E wit p (n) (u) < + fo all n N. Poposition 20 Let E be a egula (LB) space. Suppose tat E is a coplete (LF) space. Ten te (i) L(E, E ) = LB(E, E ); (ii) fo eac inceasing sequence ((n)) n of positive integes N N N j 0, n 0 N C > 0 : (26) b l,p (u) c ax b j 1 n n l,(n) p 0 n(u), fo evey u E and l N. Poposition 21 Let E be a egula (LB) space. Suppose tat is a coplete (LB) space. Ten te (i) L(E, ) = LB(E, ); (ii) fo eac inceasing sequence ((n)) n of positive integes N N n 0 N C > 0 : (27) b j, p (u) C ax b j,(n) p n(u), fo evey j N and u E. We also note tat, by cobining te vaious cases E 1, λ 1 (A) and 1 togete wit te cases E, and λ (B) (te latte spaces all ave to be taen nuclea), we can obtain esults siila to te one of Teoe 4. It is left to te eade te easy duty to find te appopiate caacteizations in evey possible and significant case. Acnowledgeent. Tese esults wee obtained duing a stay of te auto at te Univesidad Politécnica of Valencia. Se gatefully tans Pof. J. Bonet fo is ind ospitality and fo is elpful coents and suggestions. 30

19 Bounded Linea Maps between (LF) spaces Refeences [1] Bonet, J. (1987). On te identity L(E, F ) = LB(E, F ) fo pais of locally convex spaces, Poc. Ae. Mat. Soc., 99, [2] Bonet, J. and P. Doańsi, P. (1998). Real analytic cuves in Fécet spaces and tei duals, M. Mat., 126, [3] Bonet, J. and P. Doańsi, P. (2001). Paaete dependence of solutions of patially diffeential equations in spaces of eal analytic functions, Poc. Ae. Mat. Soc., 129, [4] Bonet, J. and Galbis, A. (1989). Te identity L(E, F ) = LB(E, F ), tenso poducts and inductive liits, Note Mat., 9, [5] Dieolf, S. (1985). On spaces of continuous linea appings between locally convex spaces, Note Mat., 5, [6] Galbis, A. (1988). Coutatividad ente liites inductivos y poductos tensoiales, PD Tesis, Valencia. [7] Jacow, H. (1981). Locally Convex Spaces, Teubne Stuttgat. [8] Meise, R. and Vogt, D. (1997). Intoduction to Functional Analysis, Claendon Pess, Oxfod. [9] Önal, S. and Tezioğlu, T. (1990). Unbounded linea opeatos and nuclea Köte quotients, Ac. Mat., 54, [10] Önal, S. and Tezioğlu, T. (1992). Subspaces and quotient spaces of locally convex spaces, Dissetationes Mat., 318, [11] Péez Caeas, P. and Bonet, J. (1987). Baelled Locally Convex Spaces, Not Holland Mat. Studies 131, Asteda. [12] Vogt, D. (1983). Fécetäue, zwiscen deuen jede stetige lineae Abbildung bescänt ist, J. Reine Angew. Mat., 345, [13] Vogt, D. (1992), Regulaity popeties of (LF) spaces, in Pogess in Functional Analysis, K. D. Biestedt, J. Bonet, J. Hovát, M. Maeste (eds.), Not Holland Mat. Studies 170, Asteda, A. A. Albanese Dipatiento di Mateatica E. De Giogi Univesità degli Studi Lecce Via Pe Anesano Lecce. Italy albanese@ilenic.unile.it 31