GLOBAL SCHAUDER DECOMPOSITIONS OF LOCALLY CONVEX SPACES

GLOBAL SCHAUDER DECOMPOSITIONS OF LOCALLY CONVEX SPACES

MILENA VENKOVA^∗

Abstract

We define global Schauder decompositions of locally convex spaces and prove a necessary and sufficient condition for two spaces with global Schauder decompositions to be isomorphic. These results are applied to spaces of entire functions on a locally convex space.

Given two spaces,EandF, with Schauder (or evenS-absolute) decom- positions, the existence of isomorphisms between the spaces forming the de- compositions does not imply thatEandF are isomorphic. In order to tackle this problem when the underlying decompositions consist of Banach spaces, P. Galindo, M. Maestre and P. Rueda defined in [12] a subclass ofS-absolute decompositions of Fréchet spaces: R-Schauder decompositions. To consider the corresponding problem whenEandF are locally convex spaces and the underlying decompositions are not necessarily Banach spaces, we were led to define global Schauder decompositions.

1. Introduction

In this section we give initial definitions and preliminary results.

First we introduce notation that will be used throughout the article. LetE denote a locally convex space over the complex numbersC, and letEdenote the space of all continuous linear functionals onE. WhenEis endowed with the strong topology (i.e. the topology of uniform convergence over the bounded subsets ofE), we denote it byE_β.

ForEa locally convex space we letP(ⁿE)denote the space of all continu- ousn-homogeneous polynomials onE. The topology onP(ⁿE)of uniform convergence over the compact (respectively bounded) subsets ofEis denoted byτ0(respectivelyτ_b). A third topology onP(ⁿE)can be defined in the fol- lowing way. A semi-norm p on P(ⁿE) is τ_w-continuous if for every zero

∗Supported by a Basic Research Grant of Science Foundation Ireland.

Received June 9, 2005.

neighbourhoodV inEthere exists a positive constantC(V )such that p(P )≤C(V )PV

for all P ∈ P(ⁿE). The topology generated by all such semi-norms is de- noted byτ_w. When n = 1, E_i := (P(¹E), τ_w) is the inductive dual of E, E_β :=(P(¹E), τ_b)is the strong dual ofEandE_c :=(P(¹E), τ0). By

n,s,π

E (respectively

n,s,ε

E) we denote the completed symmetricn-fold tensor product ofEendowed with the projective tensor topology (resp. the injective tensor topology).

For more definitions and properties of polynomials and holomorphic func- tions on locally convex spaces we refer the reader to [7] and [8], and for more information on locally convex spaces we refer the reader to [13] and [14].

Definition1.1. A sequence of subspaces{E_n}nof a locally convex space Eis aSchauder decompositionofEif:

• For eachxinEthere exists a unique sequence of vectors(x_n)_n,x_n∈E_n, such that

x= ∞ n=1

x_n := lim

m→∞

m n=1

x_n.

• The projections(un)^∞_n=1defined by

um

∞ n=1

xn

:=

m n=1

xn

are continuous.

The topology on eachE_nis induced by the topology onE. A Schauder decom- position{E_n}nof a locally convex spaceEisabsoluteif for eachp∈cs(E),

q ∞

n=1

x_n

:= ∞ n=1

p(x_n)

defines a continuous semi-norm onE.

The following definition is our main tool in this paper.

Definition1.2. A Schauder decomposition{E_n}^∞n=0of a locally convex spaceEis aglobal Schauder decompositionif for allr >0, allx=_∞

n=1x_n∈ Ewithxn∈Enfor eachn,

(1) r·x:=^∞

n=1

rⁿxn ∈E;

and for eachp∈cs(E),

(2) p_r

∞ n=1

x_n

:= ∞ n=1

rⁿp(x_n)

defines a continuous semi-norm onE.

In particular, takingr =1 we see that global Schauder decompositions are absolute.

Remark1.3. If{E_n}nis a global Schauder decomposition for the locally convex spaceE, there is a generating family of semi-normsp∈cs(E)of the form

(3) p

∞ n=1

x_n

= ∞ n=1

p(x_n).

Let q(x) := sup_np(x_n) wherep is a continuous semi-norm satisfying (3).

Since sup_np(x_n)≤_∞

n=1p(x_n), the semi-normqis continuous. Letq₂(x):= sup_n(2ⁿp(x_n)), from the inequality

q(x)≤p(x)=^∞

n=1

1

2ⁿ2ⁿp(xn)≤sup

n

(2ⁿp(xn)) ∞ n=1

1 2ⁿ

=q2(x)≤p2(x), it follows that the semi-norms{q(x)= sup_np(x_n)}generate the topology on E. Moreover, condition (2) in Definition 1.2 is equivalent to the condition that for eachq∈cs(E),

(4) q_r

∞ n=1

x_n

:=sup

n

(rⁿq(x_n))

defines a continuous semi-norm onE. Thus the locally convex topology ofE can be defined both byl₁-type or byc₀-type norms.

For completeness we will give the definitions for two other types of Schauder decompositions,S-absolute decompositions and R-Schauder decompositions.

LetS denote the set of all sequences(αn)^∞_n=1⊂Csuch that lim sup_n→∞|αn|^1/n

≤1.

Definition 1.4. A Schauder decomposition {E_n}n of a locally convex space E is an S-absolute decomposition if for all α = (α_n)_n ∈ S and x=_∞

n=1xn∈E, withxn ∈Enfor alln,

(5) α·x:=^∞

n=1

αnxn∈E

and, for eachp∈cs(E)and eachα=(αn)ninS,

(6) pα

∞ n=1

xn

:=^∞

n=1

|αn|p(xn)

defines a continuous semi-norm onE.

For results and applications of S-absolute decompositions we refer the reader to [7] and [8]. We will just mention that a Schauder decomposition of a barrelled locally convex space satisfying (5) is anS-absolute decomposition.

Definition 1.5. Let {E_n}n denote an absolute Schauder decomposition of the locally convex spaceE. We say that{E_n}n is a T.S.(=Taylor series) complete decomposition if for any sequence (xn)n with xn ∈ En for all n, _∞

n=1p(xn) <∞for allp∈cs(E)implies_∞

n=1xn ∈E.

T.S. completeness and conditions (1) and (5) are all completeness condi- tions with respect to a decomposition. In particular, theS-absolute Schauder decomposition of a sequentially complete locally convex space is T.S. com- plete. If{En}ⁿ is a T.S. complete and global Schauder decomposition, then it is anS-absolute decomposition.

Let us now consider the case whenEis a Fréchet space such that there is a sequence of Banach spaces{En}^∞n=0which is a Schauder decomposition of E. Let 0< R ≤ ∞. The decomposition{En}^∞_n=0isR-Schauder ([12]) if for every sequence(x_n)_n,x_n∈E_n, the seriesx=_∞

n=1x_nconverges inEif and only if lim sup_nx_n^1/nn ≤1/R.

IfEis a Fréchet space and{E_n}^∞n=0is an∞-Schauder decomposition of E consisting of Banach spaces, then it is a global Schauder decomposition ofE. Indeed, letA = {(rⁿ)n : r > 0}, consider the Köthe sequence space λ¹(A, (En)n). This is the Fréchet space

(xn)n ∈ ^∞n=1En :pr_∞

n=0xn

:= _∞

n=0rⁿx_nn< ∞for allr >0

, endowed with the topology generated by the semi-norms{p_r}r>0. Clearly,{E_n}^∞n=0forms a global Schauder decompos- ition ofλ¹(A, (E_n)_n). By ([12], Theorem 1)Eis topologically isomorphic to λ¹(A, (En)n), hence{En}^∞n=0forms a global Schauder decomposition ofE.

To show that the converse is not true, consider the Köthe matrixA^∗ = (nr)ⁿ

n : r > 0

and a sequence of Banach spaces {E_n}^∞n=0. The cor- responding Köthe sequence space λ¹(A^∗, (En)n) =

(xn)n ∈ ^∞n=1En : p^∗_r∞

n=0x_n

:= _∞

n=0(nr)ⁿx_nn < ∞ for all r > 0

endowed with the topology generated by the semi-norms{p^∗_r}r>0is a Fréchet space. It is easy to check that{E_n}^∞n=0is a global Schauder decomposition ofλ¹(A^∗, (E_n)_n). Let x=_∞

n=1x_n∈ ^∞n=1E_nsuch thatx_nn=1/nⁿ, then lim sup_nx_n^1/nn =0.

On the other hand, forr ≥1 the seriesp_r(x)=_∞

n=0(nr)ⁿx_nnis divergent

hencexdoes not belong toλ¹(A^∗, (En)n), i.e.{En}^∞n=0is not an∞-Schauder decomposition forλ¹(A^∗, (E_n)_n).

2. Application of Global Schauder Decompositions

Proposition2.1. Let E andF be locally convex spaces. Let {E_n}^∞n=0 and {Fn}^∞n=0 be T.S.-complete global Schauder decompositions for E andF re- spectively. For eachnlet

T_n:E_n−→F_n

be an isomorphism satisfying the following two conditions:

(A)For everyq ∈cs(F )there existp∈cs(E)and positive numberscand t such that

(7) q(T_n(x_n))≤ctⁿp(x_n) for everyx=_∞

n=0xninEand every positive integern.

(B)For everyp∈cs(E)there existq ∈cs(F )and positive numbersdand vsuch that

p(T_n⁻¹(yn))≤dvⁿq(yn) for everyy=_∞

n=0y_ninF and every positive integern.

ThenT =_∞

n=0Tnis an isomorphism betweenEandF.

Proof. Letq ∈cs(F ). By condition (A) there existp∈cs(E)and positive numberscandt such that

∞ n=0

q(T_n(x_n))≤c ∞ n=0

tⁿp(x_n)=cp_t(x) <∞

for everyx =_∞

n=0x_n∈E. ThusT is well defined and continuous. Lety = _∞

n=0yn ∈F, we will prove thatT is surjective. SinceTnis an isomorphism for everyn, there exist{xn}n,xn ∈En, such thatT (xn)=yn. Letp∈cs(E).

By condition (B) there existq∈cs(F )and positive numbersdandvsuch that ∞

n=0

p(xn)=^∞

n=0

p(T_n⁻¹(yn))≤d ∞ n=0

vⁿq(yn)=dqv(y) <∞.

Since{En}^∞n=0is T.S. complete,x =_∞

n=0xn ∈EandT (x)=y.

DefineS = _∞

n=0T_n⁻¹. Since the hypotheses are symmetric with respect toEandF, the above also proves thatSis well defined and continuous. It is easy to check thatSis the inverse ofT.

The converse proposition holds in a more general situation.

Proposition2.2. LetEandF be locally convex spaces, and let{En}^∞n=0

and{F_n}^∞n=0be their respective Schauder decompositions. Let T :E−→F

be an isomorphism such thatT (Em)⊆Fmfor every positive integerm. Then Tm:=T|Em −→Fmis an isomorphism for eachmandTmsatisfies conditions (A) and (B) of Proposition 2.1.

Proof. Lety_m∈F_m⊂F. SinceT is surjective there existsx =_∞

n=0x_n such thatT (x)=_∞

n=0T (xn)= ym. By hypothesisT (xn)∈Fnfor everyn, henceT_m(x_n)=0 form=nandy_m=T (x_m)=T_m(x_m), i.e.T_mis surjective.

SinceT is injectiveT_mis also injective. ThusT_mis a bijective mapping. The continuity ofT_mandT_m⁻¹follows from the continuity ofT andT⁻¹.

We now show that conditions (A) and (B) are satisfied. Letq∈cs(F ). Since T is continuous, there existp∈cs(E)andc >0 such thatq(T (x))≤cp(x) for everyx ∈E. In particular, forx =x_m∈E_mwe have

q_m(T_m(x_m))≤cp_m(x_m).

Hence inequality (7) is satisfied fort = 1. Condition (B) follows in a similar way from the continuity ofT⁻¹.

3. Stability Properties of Global Schauder Decompositions The following lemma is an adjustment of ([8], Lemma 3.31).

Lemma3.1. LetEbe a barrelled locally convex space and{E_n}^∞n=0be a Schauder decomposition ofEsatisfying condition(1), then{En}^∞n=0is a global Schauder decomposition ofE.

Proof. Letpbe a continuous semi-norm onE, and letr >0. The set

x∈E:p_r(x)=^∞

n=1

rⁿp(x_n)≤1

= ^∞

m=1

x =^∞

n=1

xn∈E: m

i=1

rⁱp(xi)≤1

is a barrel, and consequently a neighbourhood of zero inE. Thus p_r is con- tinuous for everyr >0.

Lemma3.2. Let Ebe a sequentially complete locally convex space and {E_n}^∞n=0 be a Schauder decomposition of E satisfying condition (2), then {E_n}^∞n=0is a global Schauder decomposition ofE.

Proof. Let x = _∞

n=1xn ∈ E andr > 0, denote sn := n

i=1rⁱxi. If p∈cs(E)then

p(s_n−s_m)= n i=m

rⁱp(x_i)=p_r n

i=m

x_i

−→0

asm, n→ ∞. Thus(s_n)_nis a Cauchy sequence, hence_∞

n=1rⁱx_i ∈E.

Proposition3.3. Let{E_n}n denote a global Schauder decomposition for the locally convex spaceE. Then{E_n}nis a global Schauder decomposition for the completionE.

Proof. Let x ∈ E, then there exists a net (xβ)β ⊂ E such that x = limβ→∞xβ. Sincexβ ∈ Ethere exist(xβ,n)n such thatxβ = _∞

n=0xβ,nfor everyβ. The nets(x_β,n)_βare Cauchy for everyn, hence there existsx_n∈E_nfor everynsuch thatx_n := limβ→∞x_β,n. Letp ∈cs(E)be from the generating family of continuous semi-norms satisfying (3). Givenε > 0 we can find β0>0 such that

∞ n=0

p(x_β,n−x_β,n) < ε

for allβ, β> β0. By passing to the limit inβand extendingpby continuity to the completion we get

∞ n=0

p(x_β,n−x_n)≤ε

whenβ > β₀. This implies that the series_∞

n=0x_nis convergent andx_β → _∞

n=0x_n. Since(x_β)_β has a unique limit,x=_∞

n=0x_n. The projections(u_n)_n defined by

u_m ∞

n=0

y_n

:= m n=1

y_n

where_∞

n=0yn ∈E, are linear and continuous and hence can be extended by uniform continuity to the completionE. Hence{En}nis a Schauder decom- position for the completionE.

Letr >0 and letpˆ ∈cs(E). Since{En}nis a global Schauder decomposi- tion forE, the mapping

ˆ pr

∞ n=0

yn

=^∞

n=0

rⁿp(yˆ n),

wherey=_∞

n=0yn∈E, defines a continuous semi-norm onE. Takingr =1 we get

∞ n=0

ˆ

p(x_n)≤ lim

β→∞

∞ n=0

ˆ

p(x_n−x_β,n)+ lim

β→∞

∞ n=0

ˆ p(x_β,n)

= lim

β→∞

∞ n=0

ˆ

p(x_β,n)= lim

β→∞pˆ1(x_β), hence _∞

n=0p(xˆ n) defines a continuous semi-norm on E. For an arbitrary r >0 we have

∞ n=0

rⁿp(xˆ _n)= ∞ n=0

ˆ

p(rⁿx_n)≤ lim

β→∞

∞ n=0

ˆ

p(rⁿx_β,n)= lim

β→∞pˆ_r(x_β).

Sincepˆ_r is a continuous semi-norm onE, the limit exists and is finite. Thus _∞

n=0rⁿp(xˆ n)defines a continuous semi-norm onE. An application of Lemma 3.2 completes the proof.

Proposition 3.4. If {En}ⁿ is a global Schauder decomposition for the locally convex spaceEthen{(E_n)_i}nis a global Schauder decomposition for the inductive dual ofE,E_i.

Proof. By ([14], 10.3)E_i =E_i, and by Proposition 3.3{E_n}nis a global Schauder decomposition forE. Hence we can assume thatE and allE_n are complete.

Letϕ∈E, we denoteϕ|En byϕ_n. By Remark 1.3 there exists a continuous semi-norm p such that |ϕ(x)| ≤ p(x) for any x ∈ E, with p(xn) → 0 as n → ∞, and p(x) = sup_np(x_n) for any x = _∞

n=1x_n ∈ E. Hence ϕ ∈ (E, p) and can be extended to E_p := (E, p)/p⁻¹(0). We will denote the extension ofptoEpagain byp, and let(Ep)n:=(En, p|En)/p|En−1(0).

Then

E_p= ∞

n=1

x_n:x_n ∈(E_p)_n, p|En(x_n)→0 asn→ ∞

,

and p_∞

n=1xn

= sup_np|En(xn). Let ϕ¯n denote the extension of ϕn in ((E_p)_n). Since{E_n}nis a Schauder decomposition ofE,ϕ=_∞

n=1ϕ¯_npoint- wise onE. Letpbe the dual semi-norm ofponEand letB_pbe the unit ball ofE_p. Then

p

ϕ− m n=1

¯ ϕ_n

= sup

x∈Bp

ϕ(x)− m n=1

¯ ϕ_n(x)

= sup

x∈Bp

^∞

n=m+1

¯ ϕ_n(x)

.

Letx ∈ Eand{λn}n ⊂ C,|λn| ≤ 1 for alln ∈ N. Since{En}n is an abso- lute decomposition andE is complete, _∞

n=1λ_nx_n ∈ E(see p. 189 of [8]).

This allows us to choose {λ_n}n so that λ_nϕ¯_n(x) = | ¯ϕ_n(x)|for all n. Since sup_n|λ_n|p(x_n)≤1 for allx∈B_p, it follows thatλ·x∈B_p. Hence

p

ϕ− m n=1

¯ ϕ_n

= sup

x∈Bp

∞ n=m+1

ϕ¯_n(x). Suppose sup_x∈Bp∞

n=m+1ϕ¯_n(x)does not tend to zero asm→ ∞. Then there existsδ >0 such that for allm∈Nwe can findx^(m)∈Bpwith

∞ n=m+1

ϕ¯n(x^(m))≥δ.

Letm=1 andx⁽¹⁾be the corresponding element ofBp. There existsm1>1

such that _m1

n=1

ϕ¯_n(x_n⁽¹⁾)≥ δ 2.

By induction we can build an increasing sequence{m_j}j∈N⊂Nand a sequence {x^{(j )}} ⊂B_psuch that

m_j+1

n=mj+1

ϕ¯_n(x_n^(j+1))≥ δ 2 for allj. Let

y_n=

⎧⎪

⎪⎨

⎪⎪

⎩

0 n≤m_j,

1

nx_n^(j+1) mj+1≤n≤mj+1, 0 n > m_j+1.

Sincep(x^{(j )})≤1 andp(x)=sup_kp(x_k), we have thatp(y_n)≤1/n, hence p(y_n) → 0 asn → ∞and(y_n)_n ⊂ B_p. This implies that_∞

n=1y_n ∈ E_p. As before we can choose {λn}n ⊂ C, |λn| ≤ 1 for all n ∈ N, so that ϕ(_∞

n=1λnyn)=_∞

n=1|ϕ(yn)|. However ∞

n=1

|ϕ(yn)| =^∞

n=1

1

nϕ¯n(x_n^(j+1))≥ δ 2

∞ n=1

1 n, i.e._∞

n=1|ϕ(y_n)|is divergent, a contradiction. Hencep

ϕ−m n=1ϕ¯_n

→0 andϕ= _∞

n=1ϕ_n ∈E_p. Since, by definition,E_i = ind

p∈cs(E)

(E, p)/p⁻¹(0) ,

the mappingE_p −→Eiis continuous, andϕ=_∞

n=1ϕninE_i. Moreover, by ([14], Proposition 10.3.4) the canonical surjectionE_i−→ (E_n)_i is open and continuous, henceE_iinduces the inductive topology on(E_n). Thus{(E_n)_i}n

is a Schauder decomposition for E_i. The above also shows that ϕ_n(x) = ϕ_n(x_n)=ϕ(x_n).

Next we show that{(E_n)_i}n is a global Schauder decomposition forE_i. Letϕ∈E_i,ϕ =_∞

n=1ϕ_n, and letr >0. Ifx∈Ethenr·x ∈Eand (r·ϕ)(x):=

∞ n=1

rⁿϕ_n(x_n)=ϕ ∞

n=1

rⁿx_n

=ϕ(r·x)

is well defined. Sinceϕis continuous there exists a continuous semi-normp onEsuch that|ϕ(x)| ≤p(x)for anyx∈E. Then

|(r·ϕ)(x)| ≤^∞

n=1

rⁿϕn(xn)≤pr(x).

Sincepr is a continuous semi-norm onE, this implies thatr ·ϕ ∈ E. An application of Lemma 3.1 completes the proof.

A proof for the following proposition can be obtained by modifying the proof of Proposition 3.4.

Proposition3.5.If{En}nis anS-absolute decomposition for the locally convex spaceEthen{(E_n)_i}nis anS-absolute decomposition forE_i.

Next we look at the strong dual of a locally convex space.

Proposition 3.6. If {En}n is a global Schauder decomposition for the locally convex spaceEthen{(En)_β}nis a global Schauder decomposition for E_β.

Proof. Letϕ= _∞

n=0ϕn ∈Ewhereϕn := ϕ|^En. By the continuity ofϕ there existsp∈cs(E)such that|ϕ(x)| ≤p(x)for allx=_∞

n=0x_n∈E. Let r >0, then

(8)

∞ n=1

rⁿϕn(xn)=^∞

n=1

ϕn(rⁿxn)≤pr

∞ n=1

xn

.

Let_∞

n=1rⁿϕ_n

(x):=limn→∞n

i=1rⁱϕ_i(x_i). By (8),_∞

n=1rⁿϕ_n∈E. The topology onE_β is generated by all semi-norms of the form

s(ϕ):=sup{|ϕ(x)|:x∈A},

for all A bounded subsets of E. Let B be a bounded set in E, r ·B := _∞

n=1rⁿx_n :x∈B

andp∈cs(E). Then sincep_r ∈cs(E), sup

x∈r·B

p(x)=sup

x∈B

p ∞

n=0

rⁿxn

=sup

x∈B

pr(x) <∞.

Hence the setr·Bis bounded inE. Therefore ^∞

n=m

rⁿϕn

B

=sup

x∈B

^∞

n=m

ϕn(rⁿxn) = sup

x∈r·B

^∞

n=m

ϕn

→0

asm → ∞. Hence{(E_n)_β}nis a Schauder decomposition forE_β satisfying (1). It remains to show that condition (2) is satisfied. LetB be a bounded set inEand let

˜ B:=

∞ n=1

λ_nx_n :x∈B, (λ_n)_n⊂Csuch that|λ_n| ≤1 for alln∈N

.

The setB˜ is bounded inE. Indeed, letp∈cs(E)satisfying (3). Then sup

x∈ ˜B

p(x)=sup

x∈B

∞ n=1

p(λ_nx_n)=sup

x∈B

∞ n=1

p(x_n) <∞.

This allows us to choose{λ_n}nso thatλ_nϕ_n(x)= |ϕ_n(x)|for alln. Then sup

x∈2rB˜

|ϕ(x)| =sup

x∈ ˜B

^∞

n=1

(2r)ⁿϕn(xn) =sup

x∈B

∞ n=1

(2r)ⁿ|ϕn(xn)| ≥(2r)ⁿϕnB

for alln. Letq(ϕ)=sup

x∈B

|ϕ(x)|, then

qr(ϕ)≤^∞

n=1

rⁿsup

x∈B

|ϕn(xn)| ≤^∞

n=1

1 2ⁿ sup

x∈2rB˜

|ϕ(x)| = ϕ(x)2rB˜

is continuous onE_β.

4. Global Schauder Decompositions of Spaces of Holomorphic Functions

In this sectionH(E)denotes the space of entire functions on a locally convex spaceE.

Proposition4.1. LetEbe a locally convex space. Then

(1) {(P(ⁿE), τ0)}^∞n=0is a global Schauder decomposition for(H(E), τ0).

(2) {(P(ⁿE), τ_w)}^∞n=0 is a global Schauder decomposition for(H(E), τ_δ) and(H(E), τ_w).

Proof. By ([8], Proposition 3.36),{(P(ⁿE), τ₀)}^∞n=0and{(P(ⁿE), τ_w)}^∞n=0

areS-absolute Schauder decompositions for(H(E), τ₀)and(H(E), τ_δ)re- spectively.

Letf =_∞

n=0 dⁿf (0)

n! ∈H(E)andr >0. IfK⊂Eis a compact balanced set, by the local boundedness off there exists a balanced openV ⊂ Esuch thatK⊂V and_∞

n=0^{dnf (0)}

n!

V <∞. Then (9) r·f1/rV ≤^∞

n=0

rⁿ dⁿf (0)

n!

1/rV

=^∞

n=0

dⁿf (0) n!

V

≤ fV. Hencer·f ∈H(E).

Since(H(E), τ_δ)is barrelled, by Lemma 3.1{(P(ⁿE), τ_w)}^∞n=0is a global Schauder decomposition for(H(E), τ_δ). By replacingV byKin (9) we get r ·fK ≤ frK for allf ∈ H(E). Hence{(P(ⁿE), τ₀)}^∞n=0 is a global Schauder decomposition for(H(E), τ₀). The proof that{(P(ⁿE), τ_w)}^∞n=0is a global Schauder decomposition for(H(E), τw)is similar (see also Propos- ition 3.36 of [8]).

Propositions 4.1 and 2.1 imply

Corollary4.2.LetEbe a locally convex space. Thenτδ =τw onH(E) if and only if for everyτδ-continuous semi-normqthere exist aτw-continuous semi-normpand positive numberscandtsuch that

(10) q dⁿf (0)

n!

≤ctⁿp dⁿf (0) n!

for everyf =_∞

n=0 dⁿf (0)

n! ∈H(E)and every positive integern.

LetEbe a locally convex space, denote

Hb(E)= {f ∈H (E):fA <∞for every bounded setA}. The functions inHb(E) are called holomorphic functions of bounded type.

When endowed withτb, the topology of uniform convergence over the bounded sets ofE,Hb(E)becomes a locally convex space. The proof of Proposition 4.1 can easily be modified to show the following:

Proposition4.3. LetEbe a locally convex space. Then{(P(ⁿE), τ_b)}^∞n=0

is a global Schauder decomposition for(Hb(E), τ_b).

In ([12], Examples 2 and 4) are given several examples of spaces of en- tire functions on a Banach space such that the corresponding polynomial sub- spaces are their∞-Schauder decompositions – and, thus, their global Schauder decompositions. All our results will apply to these spaces, and in particular Proposition 2.1 reduces in this special case to ([12], Theorem 9(ii)).

LetEbe a Banach space with a unit ballBE. In ([6]) the authors defined the spaceH^bI(E) of entire functions whose restrictions tonBE are integral for alln. Endowed with the system of semi-norms{p_n(f ) = f|nBEI}^∞n=1, HbI(E)is a Fréchet space and{(PI(ⁿE), · I)}^∞n=0is an∞-Schauder (and hence global) decomposition forHbI(E). Now consider the entire functions of bounded nuclear type onE, HNb(E) ([8], Definition 4.47). With the to- pology generated by the semi-norms{πn(f ) = f|nBEN}^∞n=1,HNb(E)is a Fréchet space and a short calculation shows that{(PN(ⁿE), · N)}^∞n=0is an

∞-Schauder decomposition forHNb(E). By ([5], Theorem 2) if1→

n,s,ε

E for some integernthenP^N(ⁿE)andP^I(ⁿE)are isometrically isomorphic. By ([12], Corollary 11) we obtain

Proposition4.4.LetEbe a Banach space such that

n,s,ε

Edoes not contain a copy of1for anyn∈N. ThenHbI(E)andHNb(E)are isomorphic.

Furthermore, by ([5], Proposition 3) we can replace “

n,s,ε

Edoes not contain a copy of1for anyn∈N” with the condition thatEhas RNP.

Now letEbe a locally convex space, let

(11) Gb(E):= {ϕ∈Hb(E)^∗:ϕisτ0-continuous

on the bounded subsets ofHb(E)}. When endowed with the topologyτ_gof uniform convergence on the bounded subsets ofHb(E),G_b(E)becomes a complete locally convex space. Let E be a locally convex space such that theτb-bounded sets ofHb(E)are locally bounded. By ([8], Lemma 3.25) ifB is a locally boundedτ_b-bounded subset ofHb(E), then it is relatively compact in(Hb(E), τ₀). This allows us to apply ([15], Theorem 1.1) (see also p. 115 of [2]), and we obtain thatG_b(E)_i = (Hb(E), τ_b^bor). We have proved the following proposition.

Proposition4.5. LetEbe a locally convex space such that theτ_b-bounded sets ofHb(E)are locally bounded. Then

G_b(E)_i =(Hb(E), τ_b^bor).

IfE is a bornological DF space then the τ_b-bounded sets of Hb(E) are locally bounded by ([10], Proposition 15). By ([10], Theorem 4),(Hb(E), τ_b)

is Fréchet, and hence ultrabornological, which impliesτb=τ_b^bor. Thus ifEis a bornological DF space,G_b(E)_i =(Hb(E), τ_b).

Proposition 4.6. Let E be a locally convex space. Then the sequence (P(ⁿE), τ_b^bor)

G_b(E)_∞

n=0is a global Schauder decomposition forG_b(E).

Proof. The spaceG_b(E)is a subspace of(Hb(E), τ_b^bor)since its elements areτ_b-continuous on the bounded sets ofHb(E)and hence areτ_b^bor-continuous.

Let(f_β)_β be a bounded net inH_b(E) which tends to 0 uniformly on every compact subsetK of E, and let r > 0. Since{(P(ⁿE), τb)}^∞n=0is a global Schauder decomposition for (Hb(E), τb)and rK is also a compact set, we

have _∞

n=0

rⁿ

dⁿf_β(0) n!

K

= ∞ n=0

dⁿf_β(0) n!

rK

−→0

asβ→ ∞. Hence_∞

n=0r^ndnf_n!^β(0)

βis also a boundedτ0-null net inHb(E).

Letϑ =_∞

n=0ϑ_n ∈G_b(E)whereϑ_n:=ϑ|P(ⁿE). Then (12)

∞ n=0

rⁿϑ_n

(f_β)=^∞

n=0

ϑ_n ∞

n=0

rⁿdⁿf_β(0) n!

−→0

asβ → ∞. This implies_∞

n=0rⁿϑ_n∈G_b(E)for everyr >0.

Now let pbe aτ_g-continuous semi-norm. Without loss of generality we may assume that

p(ϑ )=sup

f∈B|ϑ (f )|, whereBis a bounded subset ofHb(E). Let

B_n:=

dⁿf (0)

n! :f ∈B

,

then ∞ n=0

rⁿp_n(ϑ_n)= ∞ n=0

rⁿ ϑ_n

dⁿf (0) n!

Bn

= ∞ n=0

ϑ_n

dⁿf (0) n!

rBn

= sup

f∈rB|ϑ (f )|.

SincerB is also a bounded subset ofHb(E), the semi-norm _∞

n=0rⁿp_n is continuous. This completes the proof.