© Hindawi Publishing Corp.
AN ASCOLI THEOREM FOR SEQUENTIAL SPACES
GERT SONCK
(Received 10 November 1999 and in revised form 27 December 1999)
Abstract.Ascoli theorems characterize “precompact” subsets of the set of morphisms between two objects of a category in terms of “equicontinuity” and “pointwise precompact-ness,” with appropriate definitions of precompactness and equicontinuity in the studied category. An Ascoli theorem is presented for sets of continuous functions from a sequen-tial space to a uniform space. In our development we make extensive use of the natural function space structure for sequential spaces induced by continuous convergence and de-fine appropriate concepts of equicontinuity for sequential spaces. We apply our theorem in the context ofC∗-algebras.
2000 Mathematics Subject Classification. 54A20, 54C35, 54E15.
1. Introduction. Ascoli theorems characterize “precompact” subsets of the set of morphisms between two objects of a category in terms of “equicontinuity” and “point-wise precompactness,” with appropriate definitions of precompactness and equicon-tinuity in the studied category. Such general theorems are inspired by the classical Ascoli theorem, proved by G. Ascoli (and independently by C. Arzelà) in the 19th century (see [3, 4]). It characterizes compactness of sets of continuous real-valued functions on the interval[0,1]with respect to the topology of uniform convergence. Since then, many related theorems have been proved, for example, characterizing compactness of sets of continuous functions from a topological to a uniform space (see [6]), of uniformly continuous functions from a merotopic to a uniform space (see [5]) of continuous functions between topological spaces (see [14,17]). As was pointed out by Wyler in [24], it is clear that the setting for Ascoli theorems requires natu-ral function space structures; the existence of nice function spaces is guaranteed by Cartesian closedness of the considered topological construct. Around 1980, Dubuc [8] and Gray [12] both proposed a general theory for Ascoli theorems in a categori-cal setting, but as neither of them seems to be entirely satisfactory, Wyler suggested that more examples should be constructed in order to guide the general theory. Wyler himself developed new examples of Ascoli theorems for sets of continuous functions between limit spaces and of uniformly continuous functions from a uniform conver-gence space to a pseudo-uniform space (see [24]).
in particular, it was shown that the constructLis Cartesian closed. The natural func-tion spaces available in the setting ofLare extensively used in our development of the theory. We define appropriate concepts of equicontinuity and even continuity inL, and study the relations between these concepts. Further, we investigate relations between the different sequential function space structures (pointwise convergence, continu-ous convergence, and uniform convergence) and look at the induced structures on equicontinuous and evenly continuous sets. In this way, we obtain two versions of an Ascoli theorem for sets of continuous functions from anL-space toa uniform space. Finally, we apply our theorem in an example in the context ofᏯ∗-algebras.
The setNis the set of nonnegative integers and MONsthe set of all strictly increasing
mappings fromNtoN. Ifξis a sequence in a setX, we often writeξnforξ(n), and
the sequence itself is denoted byξn. The Fréchet-filter ofξonXis denoted byᏲ(ξ), that is, the filter generated by the sets{ξn; n≥m}withm∈N. A sequence ofξis
always of the formξ◦s, withs∈MONs. Ifx∈X, ˙xis the constant sequencexinX.
A convergence on the setXis a setᏸ⊂XN×Xsatisfying
(x,x)˙ ; x∈X⊂ᏸ,
(ξ,x)∈ᏸ ⇒ ∀s∈MONs:(ξ◦s,x)∈ᏸ. (1.1)
Then(X,ᏸ)is called anL-space or sequential (convergence) space. As usual, such a space will often be denoted by its underlying set only. If(X,ᏸ)is anL-space,ξ∈XN andx∈X,ξ (X,ᏸ→) x(orξ→X x,ξᏸ→ xor simplyξ→x) means(ξ,x)∈ᏸ; we say that ξᏸ-converges tox(or simplyX-converges tox or converges tox) and thatxis an ᏸ-limit point ofξ. IfXandY areL-spaces, a functionf:X→Y is called continuous inx∈Xif
ξ X→x ⇒f◦ξ Y→f (x), (1.2)
and continuous if it is continuous at each point ofX. The set of all continuous func-tions fromXtoY is denoted byC(X,Y ); it is a subset ofF(X,Y ), the set of all func-tions from the setXtothe setY. The construct of allL-spaces and continuous maps as morphisms is denoted byL. It is a well-fibred topological construct. A source
fi:(X,ᏸ) →Xi,ᏸii∈I (1.3)
inLis initial if and only if
ξ ᏸ→x⇐⇒ ∀i∈I:fi◦ξ ᏸ→i fi(x). (1.4)
Often, we will work inL∗, the bireflective subconstruct ofLwith as objects allL-spaces (X,ᏸ)satisfying the Urysohn-axiom:
∀ξ∈XN,∀x∈X:∀s∈MON
s,∃t∈MONs,ξ◦s◦t X→x
⇒ξ X→x (1.5)
(ᏸis then called anL∗-structure onX). Objects ofL∗are calledL∗-spaces or Urysohn sequential (convergence) spaces. For example, if(X,ᐁ)is a uniform space, withᏮa base forᐁ, anL∗-structure onXis defined by
(it is the convergence of sequences in the topology on X induced by ᐁ, and it is independent of the choice of the baseᏮ). In the following, if a uniform space(X,ᐁ) is considered as anL∗-space, X will always be endowed with the above-mentioned L∗-structure.
If(X,ᏸ)is anL-space, a pretopological structureP(ᏸ)onXis defined (see [15]) by the closure-operator
ᏼ(X) →ᏼ(X):A →x∈X; ∃ξ∈AN, ξ →x. (1.7)
For a subsetAo f anL-space(X,ᏸ), clᏸA(or simply clA) always means the closure of
Ain the pretopological space(X,P(ᏸ))andAis called closed in(X,ᏸ)ifAis closed in(X,P(ᏸ)). Fo rx∈X,ᐂᏸ(x)(orᐂ(x)) is the neighborhood filter ofxin(X,P(ᏸ)). A
subsetDo f anL-space(X,ᏸ)is called dense if it is dense in(X,P(ᏸ)), that is, if each point inXis a limit point of a sequence inD. AnL-space(X,ᏸ)is called separable if there is a countable dense subset in (X,ᏸ), that is, if (X,P(ᏸ)) is separable. A neighborhood covering system (or shortly ncs) of anL-space(X,ᏸ)is a neighborhood covering system of the pretopological space(X,P(ᏸ)), that is, a setσ of subsets of Xthat contain a neighborhood of each point ofX.
A bornology (see [13]) of a setXis a subsetαofᏼ(X)with (i) A∈α,A⊂A⇒A∈α,
(ii) finite unions of sets inαare inα, (iii) all finite subsets ofXare inα.
A set with a bornology is called a bornological set. Bornological sets are objects of a topological construct Born. A morphismf :(X,α)→(X,β)in Born is a mapping f:X→Y withf (α)⊂β.
2. Compactness and precompactness forL-spaces. AnL-spaceX is called com-pact if each sequence inXhas a convergent subsequence. IfAis a subset ofX, thenA is called compact if the subspaceAofXis a compactL-space, that is, if each sequence inAhas a subsequence that converges inXto a point ofA.
Proposition2.1. IfX is anL-space,ξ a sequence inX that converges tox∈X, thenξ(N)∪{x}is compact inX.
We now define the concept of precompactness inLwhich, according to Wyler (see [24]), should be a functorL→Born preserving the underlying sets and mappings.
Definition2.2. A subsetAo f anL-space is called precompact if each sequence in Ahas a subsequence that converges to a point ofX.
3. Uniformizable, regular, andR0spaces. The following definitions are inspired by the concepts ofR0-limit spaces in [21] and uniformizable limit spaces in [24].
Definition3.1. AnL-spaceXis called anR0-space if it satisfies the condition
ξ X→x, x˙ X→y ⇒ξ X→y, (3.1)
and it is called uniformizable if it satisfies
ξ X→x, η X→x, η X→y ⇒ξ X→y. (3.2)
Uniformizable spaces clearly areR0-spaces. The full subcategories ofLwith as objects allR0-spaces (resp., all uniformizable spaces) are bireflective inL.
In [11] a notion of regularity forL-spaces is defined. We formulate this definition here in the context ofL∗-spaces.
Definition3.2. LetXbe anL∗-space. Takeξ∈XN,x∈X, andΞn ∈(XN)N. We say thatΞnlinksξandxif for eachk∈N, the sequenceΞkisX-converging toξk
and for eachf∈N, the sequenceΞn(f (n))isX-converging tox; in this caseξand xare said tobe linked.
Definition3.3. AnL∗-spaceXis called regular ifξisX-converging tox if and only ifξandxare linked, for allξ∈XNandx∈X.
4. Function spaces inL. We first introduce someL-structures on function spaces. (1) IfXis a set andY is anL-space, theL-structureπof pointwise convergence on F(X,Y )is the product structure inLonF(X,Y ), that is, for a sequencefninF(X,Y ) andf∈F(X,Y ), we have
fn π→f⇐⇒ ∀x∈X:fn(x) π→f (x). (4.1)
IfY is anL∗-space, then sois(F(X,Y ),π).
(2) IfXandY areL-spaces, theL-structureΓ of continuous convergence onC(X,Y ) is defined by
fn→Γ f⇐⇒∀x∈X, ∀ξ∈XN, ∀s∈MONs:
ξ X→x ⇒fs(n)ξn Y→f (x)
. (4.2)
IfXis anL∗-space, then
fn →Γ f⇐⇒ ∀x∈X, ∀ξ∈XN:
ξ X→x ⇒fsξn Y→f (x)
. (4.3)
Again, ifY is anL∗-space, then sois(C(X,Y ),Γ).
(3) IfX is a set and(Y ,ᐁ)is a uniform space withᏮa base forᐁ, and ifA⊂X,
the sets
structure of uniform convergence onA, and denote it bysᐁX,Y,A, or simplysᐁ,A. Fo r a
sequencefninF(X,Y )andf∈F(X,Y ), we have
fn sᐁ,A→f⇐⇒ ∀B∈Ꮾ,∃k∈N,∀n≥k,∀x∈A,f (x),fn(x)∈B. (4.5)
Remark4.1. Note that
A⊂A⊂X ⇒s
ᐁ,A⊂sᐁ,A. (4.6)
If A= X, we write sᐁ instead ofsᐁ,A, and call sᐁ the sequential structure of
uni-form convergence. If nowσ ⊂ᏼ(X), we definesσX,Y (orsσ) as the supremum inL∗
of all the structures sᐁ,A on F(X,Y ), with A∈σ; sσ is called the structure of
uni-form convergence on the sets ofσ. For a sequencefninF(X,Y )andf∈F(X,Y ),
we have
fn sσ→f⇐⇒ ∀A∈σ:fnsᐁ,A→f . (4.7)
If forA⊂X,rAis the restriction map
F(X,Y ) →F(A,Y ):f →f|A, (4.8)
it is easily seen that
rA:
F(X,Y ),sX,Y σ
→F(A,Y ),sᐁA,Y
A∈σ (4.9)
is an initial source inL∗. Even more, each map
rA:
F(X,Y ),sᐁX,Y,A
→F(A,Y ),sᐁA,Y
(4.10)
is initial. Finally, remark that, if σ = {A}with A⊂ X, then sσ =sᐁ,A, and that, if σ= {{x}; x∈X}, thensσ=π.
For the above-defined function convergencesπ,Γ, and soforth, we use the same symbols for their induced convergences on subsets of their definition sets, that is, ifX andY both carryL-structures, we useπfor the subspace structure of the sequential structure of pointwise convergence on the subsetC(X,Y )ofF(X,Y ).
We recall that L and L∗ are Cartesian closed topological constructs and that, for L-spaces X and Y, (C(X,Y ),Γ) is the corresponding power-object (see [1]): f:X×Z→Y is continuous if and only iff∗:Z→(C(X,Y ),Γ)is continuous where f∗(z)(x)=f (x,z).
We now investigate limits of sequences of continuous functions in the sequential structures of uniform convergence.
Proposition4.2. LetXbe anL-space,(Y ,ᐁ)a uniform space,x∈XandA∈ᐂ(x). Iffnis a sequence inF(X,Y)of continuous functions inx,f∈F(X,Y), andfn sᐁ,A→f, thenf is also continuous inx.
Corollary4.3. LetXbe anL-space and let(Y ,ᐁ)be a uniform space. Ifσ is an ncs ofX, thenC(X,Y )is closed in(F(X,Y ),sσ).
Theorem4.4. (a)IfXandY areL-spaces,Γ ⊂πonC(X,Y ).
(b)IfXis a set and(Y ,ᐁ)is a uniform space,sσ⊂πonF(X,Y ), for all coversσ of X.
(c)IfXis a set and(Y ,ᐁ)is a uniform space,sᐁ⊂sσ onF(X,Y ), for allσ⊂ᏼ(X).
(d)IfXis a set and(Y ,ᐁ)is a uniform space,sᐁ⊂πonF(X,Y ).
(e)IfXis anL-space and(Y ,ᐁ)is a uniform space,sᐁ⊂Γ onC(X,Y ).
(f)IfXis anL-space and(Y ,ᐁ)is a uniform space,sσ⊂Γ onC(X,Y ), for all neigh-borhood covering systemsσ ofX.
(g)IfXis a compactL-space and(Y ,ᐁ)is a uniform space,sᐁ=Γ onC(X,Y ).
(h)IfXis anL-space and(Y ,ᐁ)is a uniform space,sσ=Γ onC(X,Y ), ifσ is the collection of all compact subsets ofX.
5. Even continuity. In [14], Kelley defined even continuous mappings between topological spaces, and Poppe [19] generalized this notion to sets of continuous map-pings between limit spaces. The following is straightforward transcription of these definitions toL-spaces.
Definition5.1. IfXandY areL-spaces, a subset ofC(X,Y )is called evenly con-tinuous if for allx∈X, y∈Y,ξ∈XN, andfn ∈HN, the conditionsξ X→x and
fn(x)→Y yimplyfn(ξn)Y→ y.
It is easily seen that a subset of an evenly continuous set inC(X,Y )is still evenly continuous and that a finite union of evenly continuous subsets is still evenly contin-uous. This does not mean that the evenly continuous subsets ofC(X,Y )always form a bornology onC(X,Y ); the next proposition clarifies when they do.
Proposition5.2. The following statements are equivalent: (a)TheL-spaceY is anR0-space.
(b)For allL-spacesXthe evenly continuous subsets ofC(X,Y )form a bornology on C(X,Y ).
Proposition 5.3. For a fixed R0-space Y, even continuity defines a functor ECY:Lop→Born, withECY(X)=(C(X,Y ),{evenly continuous subsets}).
6. Equicontinuity. In this section,Xis anL-space and(Y ,ᐁ)is a uniform space.
Definition6.1. A subsetHofC(X,Y )is called equicontinuous at a pointxofX if for allU∈ᐁand all sequencesξ X-converging tox,
∃k∈N,∀n≥k,∀f∈H:f (x),fξn∈U. (6.1)
The subsetHis called equicontinuous inA⊂XifHis equicontinuous at each point xofA, andHis called equicontinuous ifHis equicontinuous inX.
Proposition 6.2. The subsets of C(X,Y ) that are equicontinuous in x form a bornology onC(X,Y ).
Remark6.4. Tosee that an evenly subset ofC(X,Y )need not be equicontinuous, consider the following example, based on an example of Poppe in [20]. LetXbe theL∗ -space with the underlying setQ∩[0,1]and with the sequential structure induced by the usual metric onQ. Further, letY be the uniform subspace ofRwith as underlying set all strictly positive rational numbers. Then take an irrational numberi0∈[0,1] and a strictly decreasing sequencernof rationals in[0,1], converging inRtoi0. Now for alln∈Ndefine a continuous function
fn:X →Y:x →rn+nx. (6.2)
Finally putH= {fn; n∈N}.
We first show that H is not equicontinuous at 0. Ifξ is the sequence in X with ξn=1/nforn∈N0andξ0=0, thenξ→x. Fo rn∈N0we now have
fn
ξn−fn(0) =1. (6.3)
Toprove thatHis evenly continuous, takex∈X,y∈Y,ξ∈XN, and a sequence
fs(n)inH(s is a functionN→N) withξ→Xxandfs(n)(x)Y→ y. Ifx=0, the
con-ditionrs(n)Y→ yclearly implies that the set{rs(n); n∈N}is finite, and sothat also {fs(n); n∈N}is finite. Ifx >0, we have for allnsuch thats(n)≥(y+1−i0)/x
fs(n)(x)=rs(n)+s(n)x > i0+y+1−i0=y+1, (6.4)
and sothe set {fs(n); n∈ N} is finite, otherwise the sequence fs(n)(x) has an
infinite number of terms greater than y+1, which contradicts the convergence
fs(n)(x) Y→ y.
Soin each case the set{fs(n);n∈N}is finite, and so evenly continuous. This means
thatfs(n)(ξn)Y→ y. Sowe have proved thatHis evenly continuous.
Note that
H(0)=rn; n∈N (6.5)
is not precompact inY. This is not surprising, for the following proposition shows thatHwould be equicontinuous in 0 ifH(0)were tobe precompact inY, sinceHis evenly continuous.
Proposition6.5. LetH⊂C(X,Y )andx∈X. IfHis evenly continuous andH(x) is precompact inY, thenHis equicontinuous inx.
Proof. SupposeH is not equicontinuous in x. Then there existsU ∈ᐁ and a sequenceξwithξX→ xsuch that
∀k∈N,∃n≥k,∃f∈H:f (x),fξn∈U. (6.6)
This gives a subsequenceξ◦s(s∈MONs)and a sequencefninHwith
∀n∈N:fn(x),fnξs(n)∈U. (6.7)
Because H(x) is precompact there would exist a subsequence ft(n) of fn(t∈
MONs)such thatft(n)(x)converges inY toa pointy. No w H is evenly
continu-ous and so
becauseξ◦s◦tX→ x. By the last twoconvergences we can findk∈Nwith
ft(k)(x),ft(k)ξs(t(k))∈U (6.9)
yielding a contradiction.
The following proposition reduces equicontinuity of a set of functions to an or-dinary continuity of a suitable function, as one can do in the context of topological spaces [14]. First we need a few notations and remarks.
Forx∈X, ˆxis the evaluation map inx
ˆ
x:F(X,Y ) →Y:f →f (x). (6.10)
IfHis endowed with the structureπof pointwise convergence, it is easily seen that all functions ˆx|H:H→Y are continuous. So we have a function
λH:X →C(H,Y ):x →xˆ|H. (6.11)
Proposition6.6. IfH⊂C(X,Y )andx∈X, the following are equivalent: (a)His equicontinuous inx,
(b)λH:X→(C(H,Y ),SᐁH,Y):x→xˆ|His continuous inx.
Proof. First remark that for a sequenceξinX, we have
λH◦ξ s H,Y ᐁ
→λH(x)⇐⇒ξˆn|Hs H,Y ᐁ →xˆ|H
⇐⇒ ∀U∈ᐁ,∃k∈N,∀n≥k,∀f∈H:xˆ|H(f ),ξˆn|H(f )∈U
⇐⇒ ∀U∈ᐁ,∃k∈N,∀n≥k,∀f∈H:f (x),fξn∈U.
(6.12)
NowλHis continuous inxif and only if, for all sequencesξconverging toxinX, we
have
λH◦ξ s H,Y ᐁ
→λH(x), (6.13)
and with the equivalences above, this is precisely the condition forHtobe equicon-tinuous inx.
7. Convergence on evenly continuous and equicontinuous sets
Proposition7.1. LetXandY be L-spaces. Iffnis a sequence inC(X,Y )such that{fn; n∈N}is evenly continuous, and iff∈C(X,Y ), then we have
fnπ→f⇐⇒fn Γ→f . (7.1)
Corollary7.2. IfXandY areL-spaces, then the closures inC(X,Y )of an evenly continuous subset ofC(X,Y )for the sequential structuresπandΓ coincide.
Proposition7.3. LetXandYbeL-spaces withY a regularL∗-space. IfH⊂C(X,Y ) is evenly continuous andf∈F(X,Y ), then
Proof. Takex∈X,ξ∈XN withξ→X xand letfn ∈HNbe a sequence inHwith
fnπ→f. For alln∈N, we put
Ξn:N →Y:k →fk+n+1ξn. (7.3)
We now prove thatΞlinksf◦ξandf (x)inY. For alln∈N,Ξnis a subsequence offk(ξn)k, and
fkξnk Y→fξn, (7.4)
sowe haveΞn Y→ f (ξn).
Take a functionF∈NN. Put, for alln∈N,T (n)=Ξn(F(n)), sothatTis a sequence inY. Ifs∈MONs, we inductively definet:N→Nas follows:
t(0)=0, ∀n≥1 :t(n)=Fst(n−1)+st(n−1)+1. (7.5) Then clearlytalsois a strictly increasing functionN→N. No wft(n+1)is a subse-quence offnandft(n+1) π→ f, and
ft(n+1)(x)Y→f (x). (7.6)
Further,ξ◦s◦t X→ xand so
ft(n+1)ξs(t(n)) Y→f (x) (7.7)
is evenly continuous. Now
T◦s◦t(n)=Ξs(t(n))Fst(n) =fF(s(t(n)))+s(t(n))+1ξs(t(n)) =ft(n+1)ξs(t(n)),
(7.8)
and soT◦s◦t→Y f (x). This provesT →Y f (x). Regularity ofYnow givesf◦ξ Y→ f (x), and sofis continuous.
Proposition7.4. LetXbe a compactL-space, and(Y ,ᐁ)a uniform space. If{fn; n∈N}is an evenly continuous subset ofC(X,Y ),f∈F(X,Y ), then
fn π→f⇐⇒f∈C(X,Y ), fn sᐁ→f . (7.9) Proof. Iffnπ→ f, thenf is continuous byProposition 7.3. Further,fn→Γ fby Proposition 7.1and finallyfn sᐁ→f byTheorem 4.4.
Corollary7.5. IfXis a compactL-space, and(Y ,ᐁ)is a uniform space, then the closuresF(X,Y )of an evenly continuous subset ofC(X,Y )in the sequential structures πandsᐁcoincide.
Proposition7.6. LetXbe anL-space,x∈X, and let(Y ,ᐁ)be a uniform space. IfH is a subset ofC(X,Y ), thenHis equicontinuous inxif and only ifclπHis equicontinuous inx.
8. An extension theorem. In this section,Xis anL-space and(Y ,ᐁ)is a uniform space. Recall thatξ∈YNis a Cauchy-sequence if and only if
Proposition8.1. Letξ→X xand letfnbe a sequence inC(X,Y)with{fn; n∈N} equicontinuous inx. If for allk∈N,fn(ξk)n is a Cauchy-sequence in(Y ,ᐁ), then fn(x)also is a Cauchy-sequence in(Y ,ᐁ).
Proof. TakeU∈ᐁ. Choose a symmetricV∈ᐁwithV3⊂U. Equicontinuity inx gives ak0∈Nwith
∀j≥k0,∀n∈N:fn(x),fnξj∈V . (8.2)
Because nowfn(ξk0)nis a Cauchy-sequence, we havek∈Nwith ∀m,n≥k:fmξk0
∈V. (8.3)
Then we have form,n≥k
fm(x),fn(x)∈V3⊂U. (8.4)
This proves the proposition.
Corollary 8.2. LetD be a dense subset ofX, fn a sequence in C(X,Y ) with H= {fn; n∈N}equicontinuous. Suppose thatH(x)is precompact inY for allx∈X. If, for alld∈D, fn(d) is a Cauchy-sequence in Y, then fn(x) is a convergent sequence inY for allx∈X.
Theorem8.3. SupposeDis a dense subset ofX, fna sequence inC(X,Y )with H= {fn; n∈N}equicontinuous andH(x)precompact inY for allx∈X. Iff:D→Y is a function with
∀d∈D:fn(d) →f (d), (8.5)
thenf has a continuous extensionf˜: X→Y withfn Γ→f.˜
Proof. This theorem is an immediate consequence of the previous proposition.
9. An Ascoli theorem
Proposition9.1. LetXandY beL-spaces withY anL∗-space, which isR0. IfHis precompact in(C(X,Y ),Γ), thenHis evenly continuous.
Proof. Takex∈X, y∈Y , ξ∈XN, andfn∈HN with ξX→xandf
n(X)→Y y. If s∈MONs, thenfs(n)has aΓ-convergent subsequencefs(t(n))→Γ f, withf∈C(X,Y )
andt∈MONs. Then, becauseξ◦s◦t X→x, we have
fs(t(n))ξs(t(n)) Y→f (x). (9.1)
Furthermore,
fs(t(n))(x) Y→y, fs(t(n))(x) Y→f (x). (9.2)
So, becauseY isR0,
fs(t(n))ξs(t(n)) Y→y. (9.3)
This proves
fnξn Y→y (9.4)
Theorem 9.2. LetX andY beL-spaces with Y anL∗-space, which isR0. If H is precompact in (C(X,Y ),Γ), thenH is evenly continuous and for all x ∈X, H(x) is precompact inY.
Proof. The proof follows fromProposition 9.1and the continuity of ˆ
x:C(X,Y ),Γ→Y , (9.5)
andH(x)=x(H)ˆ .
Theorem9.3. LetXbe anL-space and(Y ,ᐁ)a uniform space. IfHis precompact in(C(X,Y ),Γ), thenH is equicontinuous and for allx∈X,H(x)is precompact inY. Further, ifXis a separableL-space, these two conditions are also sufficient forHto be precompact in(C(X,Y )Γ).
Proof. The first part of the theorem follows fromTheorem 9.2andProposition 6.5. For the second part, supposeH is equicontinuous andH(x) is precompact in Y for allx∈X, and letD= {dn; n∈N}be a dense subset ofX. Take a sequence fninH. Becausefn(d0)is a sequence inH(d0), it has a convergent subsequence fs0(n)(d0)(s0∈MONs); denote the limit point of this subsequence byf (d0). No w
suppose, by induction, thatsk∈MONsand
fsk(n)
dkn Y→f
dk. (9.6)
Thenfsk(n)(dk+1)has a convergent subsequencefsk◦s(n)(dk+1). Putsk+1=sk◦ s
and denote by f (dk+1) a limit point of fsk◦s(n)(dk+1). This defines a function
f:D→Y. No w fo r allk∈Nthere is at∈MONs such that for alln≥k,
fsn(n)
dk=fsk◦t(n)
dk. (9.7)
This means thatfsn(n)(dk)n≥kis a subsequence offsk(n)(dk), and thus
fsn(n)
dk Y→fdk. (9.8)
Sofsn(n)is a subsequence offnwithfsn(n)(d) Y
→f (d)for alld∈D. The exten-sion theorem (Theorem 8.3) then gives a continuous extenexten-sion ˜f :X→Y off with
fsn(n)
Γ
→f˜. This proves thatHis precompact in(C(X,Y ),Γ).
Corollary9.4. LetXbe a compact and separableL-space and(Y ,ᐁ)a uniform space. Then a subsetH ofC(X,Y )is precompact in (C(X,Y ),sᐁ)if and only ifH is
equicontinuous and for allx∈X,H(x)is precompact inY.
Proof. The proof follows fromTheorem 9.3andTheorem 4.4(g).
We can easily rewriteTheorem 9.3in another form. Therefore, ifF is a set,Xand Y areL-spaces, a mapΦ:X×F→Y is called a dual map if each functionΦf:X→Y: x→Φ(x,f )forf∈F belongs toC(X,Y ). For such a dual map, we putΓΦ, the initial L-structure, onFfor the mapF→C(X,Y ):f→Φf. IfY is anL∗-space,Γ∗-space,ΓΦis anL∗-structure onF. Finally, forH⊂F, we setΦH= {Φf; f∈H}.
allx∈X,ΦH(x)is precompact inY. Further, ifXis separable andclΓΦΦH⊂ΦF, then these two conditions are also sufficient forHto be precompact in(F,ΓΦ).
10. An example. We give an example of an application of our Ascoli theorem in the context ofᏯ∗-algebras. Some experience with the fundamental parts of the theory ofᏯ∗-algebras is needed and can be found in [7,18]. The example is based on an application of another Ascoli theorem for “pseudotopological classes” by McKennon in [16].
Let A be a Ꮿ∗-algebra, which we may and will regard as a subalgebra of its en-veloping von Neumann algebra Aν. Let Q
A = {x ∈Aν; ∀a,b∈A: a∗xb∈A} be
the set of quasi-multipliers of A and SA the set of all states of A (i.e., the set of
all positive linear functionals on Awith norm 1). Each linear functional ϕ:A→C
has a unique linear extension ϕν :Aν →Cthat is continuous for the weak
topol-ogy on Aν, and forx∈Q
A the map ˆx:A→C:ϕ→ϕν(x)belongs to the bidual
ofA. OnSA we place the L∗-structure introduced by the weak∗-topology onA
rel-ativized to SA. In this case, we can show that Φ:SA×QA →Cis a dual map.
Fur-thermore, norm-bounded sequences in QA converge in ΓΦ if and only if they con-verge in the quasi-strict topology onQA, that is, the topology onQA induced by all
semi-normsQA→C:x→ a∗xafora∈A. If theC∗-algebraA is separable, then SA is also separable. And finally, for a norm-bounded subset H of QA, the
condi-tion clΓ ΦΦH ⊂ΦF is satisfied. Sowe can applyTheorem 9.5 and get the following theorem.
Theorem10.1. IfAis a separableᏯ∗-algebra, then for each norm-bounded subset HofQAthe following are equivalent:
(1)His precompact in the sequential structure induced by the quasi-strict topology onQA.
(2){xˆ|SA;x∈H}is equicontinuous inC(SA,C)and for allϕ∈SAthe set{ϕν(x);x∈
H}is precompact inC.
References
[1] J. Adámek, H. Herrlich, and G. E. Strecker, Abstract and Concrete Categories, Pure and Applied Mathematics, John Wiley and Sons, New York, 1990.MR 91h:18001. Zbl 695.18001.
[2] P. Alexandroff and P. Urysohn,Une condition nécessaire pour qu’une classe (L) soit une classe (D), C. R. Acad. Sci. Paris177(1923), 1274–1276 (French).
[3] C. Arzelà,Sull’ integrabilita della equazioni differenziali ordinarie, Mem. Accad. Bologna
5(1895-1896), no. 5, 257–270 (Italian).
[4] G. Ascoli,Le curve limite di una varietà data di curve, Atti della R. accademia dei Lincei. Memorie della Classe di scienze fisiche, matematiche e naturali3(1884), no. 18, 521–586.
[5] H. L. Bentley and H. Herrlich,Ascoli’s theorem for a class of merotopic spaces, Conver-gence Structures 1984 (Bechynˇe, 1984), Akademie-Verlag, Berlin, 1985, pp. 47–53.
MR 87g:54062. Zbl 576.54020.
[6] N. Bourbaki,Topologie Générale, Hermann, Paris, 1967 (French).
[7] J. Dixmier, C∗-algebras, North-Holland Mathematical Library, vol. 15, North-Holland Publishing, New York, 1977, translated from the French by Francis Jellett.
[8] E. J. Dubuc,Concrete quasitopoi, Applications of Sheaves (Proc. Res. Sympos., Durham, 1977), Lecture Notes in Mathematics, vol. 753, Springer, Berlin, 1979, pp. 239–254.
MR 81e:18010. Zbl 423.18006.
[9] M. Fréchet,Sur quelques points de calcul fonctionnel, Rend. Circ. Mat. Palermo22(1906), 1–74 (French).
[10] ,Les Espaces Abstraits[Abstract Spaces], Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics], Éditions Jacques Gabay, Sceaux, 1989 (French), reprint of the 1928 original.MR 93g:01098.
[11] R. Friˇc and V. Koutník,Sequential convergence spaces: iteration, extension, completion, enlargement, Recent Progress in General Topology (Prague, 1991), North-Holland, Amsterdam, 1992, pp. 199–213.CMP 1 229 126. Zbl 811.54005.
[12] J. W. Gray,Ascoli’s theorem for topological categories, Categorical Aspects of Topology and Analysis (Ottawa, 1980), Lecture Notes in Mathematics, vol. 915, Springer, Berlin, 1982, pp. 86–104.MR 83k:54007. Zbl 498.18006.
[13] H. Hogbe-Nlend,Théorie des Bornologies et Applications[Theory of Bornologies and Ap-plications], Lecture Notes in Mathematics, vol. 213, Springer-Verlag, Berlin, 1971 (French).MR 58#30002. Zbl 225.46005.
[14] J. L. Kelley,General Topology, D. Van Nostrand Company, Toronto, 1955.MR 16,1136c. Zbl 066.16604.
[15] E. Lowen-Colebunders and G. Sonck,Exponential objects and Cartesian closedness in the construct PrTop, Appl. Categ. Structures1(1993), no. 4, 345–360.MR 94m:54029. Zbl 796.54019.
[16] K. McKennon,Ascoli’s theorem and the pure states of aC∗-algebra, Kyungpook Math. J.
28(1988), no. 1, 23–34.MR 90e:46048. Zbl 677.46044.
[17] N. Noble,Ascoli theorems and the exponential map, Trans. Amer. Math. Soc.143(1969), 393–411.MR 40#1978. Zbl 229.54017.
[18] G. K. Pedersen, C∗-algebras and their Automorphism Groups, London Mathemati-cal Society Monographs, vol. 14, Academic Press, London, 1979.MR 81e:46037. Zbl 416.46043.
[19] H. Poppe,Stetige Konvergenz und der Satz von Ascoli und Arzelà, Math. Nachr.30(1965), 87–122 (German).MR 32#6400. Zbl 129.37901.
[20] ,Stetige Konvergenz und der Satz von Ascoli und Arzelà. II, Monatsb. Deutsch. Akad. Wiss. Berlin8(1966), 259–264 (German).MR 34#6711. Zbl 148.16503.
[21] W. Robertson,Convergence as a nearness concept, Ph.D. thesis, Carleton University, On-tario, 1975.
[22] C. T. Scarborough and A. H. Stone,Products of nearly compact spaces, Trans. Amer. Math. Soc.124(1966), 131–147.MR 34#3528. Zbl 151.30001.
[23] P. Urysohn,Sur les classes (L) de Frèchet, L’Enseign. Math.25(1926), 77–83 (French). [24] O. Wyler,Ascoli theorems for Cartesian closed categories, Categorical Topology (Toledo,
Ohio, 1983), Sigma Ser. Pure Math., vol. 5, Heldermann, Berlin, 1984, pp. 599–617.
MR 87e:18004. Zbl 555.18004.
Gert Sonck: Vrije Universiteit Brussel, Department Wiskunde, Pleinlaan2, B-1050 Brussel, Belgium
