A quantitative approach to weak compactness in Fréchet spaces and spaces C(X)
Abstract
[EN] Let E be a Frechet space, i.e. a metrizable and complete locally convex space (lcs), E '' its strong second dual with a defining sequence of seminorms parallel to center dot parallel to(n) induced by a decreasing basis of absolutely convex neighbourhoods of zero U-n, and let H subset of E be a bounded set. Let ck(H) := sup{d(cluste(E '') (phi), E) : phi is an element of H-N} be the "worst" distance of the set of weak *-cluster points in E '' of sequences in H to E, and k(H) := sup{d(h, E) : h is an element of (H) over bar} the worst distance of (H) over bar the weak *-closure in the bidual of H to E, where d means the natural metric of E ''. Let gamma(n)(H) := sup {vertical bar lim(p) lim(m) u(p) (h(m)) - lim(m) lim(p) u(p) (h(m))vertical bar : (u(p)) subset of U-n(0), (h(m)) subset of H}, provided the involved limits exist. We extend a recent result of Angosto-Cascales to Frechet spaces by showing that: If x** is an element of (H) over bar, there is a sequence (x(p))(p) in H such that d(n)(x**, y**) <= gamma(n)(H) for each sigma (E '', E')-cluster point y** of (x(p))(p) and n is an element of N. Moreover, k(H) = 0 iff ck(H) = 0. This provides a quantitative version of the weak angelicity in a Frechet space. Also we show that ck(H) <= (d) over cap((H) over bar, C(X, Z)) <= 17ck(H), where H subset of Z(X) is relatively compact and C(X, Z) is the space of Z-valued continuous functions for a web-compact space X and a separable metric space Z, being now ck(H) the "worst" distance of the set of cluster points in Z(X) of sequences in H to C(X, Z), respect to the standard supremum metric d, and (d) over cap((H) over bar, C(X, Z)) := sup{f, C(X, Z), f is an element of (H) over bar}. This yields a quantitative version of Orihuela's angelic theorem. If X is strongly web-compact then ck(H) <= (d) over cap((H) over bar, C(X, Z)) <= 5ck(H); this happens if X = (E', sigma(E', E)) for E is an element of (sic) (for instance, if E is a (DF)-space or an (LF)-space). In the particular case that E is a separable metrizable locally convex space then (d) over cap((H) over bar, C(X, Z)) = ck(H) for each bounded H subset of R-X
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Else ie
Angos o He nández, C.; Kakol, JM.; López Pellice , M. (2013). A quan i a i e app oach o
weak compac ness in F éche spaces and spaces C(X). Jou nal o Ma hema ical Analysis
and Applica ions. 403(1):13-22. doi:10.1016/j.jmaa.2013.01.055.
A quan i a i e app oach o weak compac nesss in F ´eche
spaces and spaces C(X)✩
C. Angos oa, J. K¸akolb, M. L´opez-Pellice c
aDep o. de Ma em´a ica Aplicada y Es adis ica: Uni e sidad Poli ´ecnica de Ca agena, 30203
Ca agena, Spain
bFacul y o Ma hema ics and In o ma ics. A. Mickiewicz Uni e si y, 61-614 Pozna´n, Poland
cDep o. de Ma em´a ica Aplicada and IMPA. Uni e si a Poli `ecnica de Val`encia, E-46022 Valencia,
Spain
Abs ac
Le Ebe a F ´eche space, i.e. a me izable and comple e locally con ex space (lcs), E′′
i s s ong second dual wi h a de ining sequence o semino ms k · kninduced by a dec eas-
ing basis o absolu ely con ex neighbou hoods o ze o Un, and le H⊂Ebe a bounded
se . Le ck(H) := sup{d(clus E′′ (ϕ), E) : ϕ∈HN}be he “wo s ” dis ance o he se o
weak∗-clus e poin s in E′′ o sequences in H o E, and k(H) := sup{d(h, E) : h∈H} he
wo s dis ance o H he weak∗-closu e in he bidual o H o E, whe e dmeans he na u al
me ic o E′′ . Le γn(H) := sup |limplimmup(hm)−limmlimpup(hm)|: (up)⊂U0
n,(hm)⊂H,
p o ided he in ol ed limi s exis . We ex end a ecen esul o Angos o-Cascales o
F ´eche spaces by showing ha : I x∗∗ ∈H, he e is a sequence (xp)pin Hsuch ha
dn(x∗∗, y∗∗)≤γn(H) o each σ(E′′ , E′)-clus e poin y∗∗ o (xp)pand n∈N. Mo eo e ,
k(H) = 0 i ck(H) = 0. This p o ides a quan i a i e e sion o he weak angelici y
in a F ´eche space. We show ha ck(H)≤ˆ
d(H, C(X, Z)) ≤17ck(H),whe e H⊂ZX
is ela i ely compac and C(X, Z) is he space o Z- alued con inuous unc ions o a
web-compac space Xand a sepa able me ic space Z. I Xis web-compac and no mal
and Z:= R, we show ha ck(H)≤ˆ
d(H, C(X)) ≤12ck(H).A co esponding esul o
s ongly web-compac spaces Xis also ob ained wi h sha pe cons an s. This yields a
quan i a i e e sion o O ihuela’s angelic heo em o spaces Cp(X, Z) and applies also o
show: I Xis he weak∗-dual o a (DF )-space o an (LF )-space and H⊂RXis bounded,
hen ck(H)≤ˆ
d(H, C(X)) ≤5ck(H).
1. In oduc ion
Many classical esul s abou compac ness in unc ional analysis can be deduced om
sui able inequali ies abou dis ances o spaces o con inuous unc ions. This line o
✩The esea ch was suppo ed o he i s named au ho by he p ojec MTM2008-05396 o he Spanish
Minis y o Science and Inno a ion and by Fundaci´on S´eneca (CARM), g an 08848/PI/08, o he second
named au ho by Na ional Cen e o Science, Poland, g an no. N N201 605340 and o he second and
hi d au ho s by he p ojec MTM2008-01502 o he Spanish Minis y o Science and Inno a ion.
Email add esses: ca los.angos o@upc .es (C. Angos o), [email protected] (J. K¸akol),
mlopezpe@ma .up .es (M. L´opez-Pellice )
P ep in submi ed o Else ie Decembe 24, 2011
*Manusc ip
esea ch mo i a es a numbe o specialis s o s udy se e al quan i a i e coun e pa s o
some classical esul s. We e e o wo ks [9], [2], [3], [4], [11], [15], [16] also as a good sou ce
o e e ences. Especially esul s om [9] and [2], yielding se e al cha ac e iza ions o weak
compac ness o bounded se s in a Banach space, mo i a ed ou p esen pape . Pape s
ci ed abo e p o ided some ools which ha e been used o new quan i a i e e sions o
Gan mache ’s heo em abou weak compac ness o adjoin ope a o s in Banach spaces,
Ebe lein–G o hendieck’s heo em, G o hendieck’s cha ac e iza ion o weak compac enss
in eal Banach spaces C(K) := C(K, R), and he classical K ein-Smulyan’s heo em.
Theo em 1 below deals wi h he ollowing non-nega i e unc ions de ined on he amily
o bounded se s Hin a Banach space E, see [2, De ini on 1]:
(i) γ(H) := sup{| limmlimn m(xn)−limnlimm m(xn)|: ( m)m⊂BE′,(xn)n⊂H}
assuming ha he i e a ed limi exis ,
(ii) ck(H) := sup{d(clus E′′ (φ), E), φ ∈HN},
(iii) k(H) := ˆ
d(Hω∗
, E) = sup{d(x∗∗, E), x∗∗ ∈Hω∗
},
whe e dis he usual in dis ance o se s associa ed o he na u al no m in E′′.
Le Xbe a comple ely egula Hausdo space. I H⊂C(X)⊂RXis poin wise
bounded, he closu e HRX
is compac in he opology τpo poin wise con e gence in RX.
I ˆ
d(HRX
, C(X)) := sup{d( , C(X)) : ∈HRX
},whe e dis he s anda d sup emum
me ic, hen ˆ
d= 0 i HRX
⊂C(X) i His τp- ela i ely compac in C(X). The e o e
ˆ
d > 0 p o ides a measu e o non-τp-compac ness o Hin C(X).
The ollowing in e es ing esul [2, Theo em 2.3] mo i a ed ou wo k.
Theo em 1. Fo any bounded se Hin a Banach space Ewe ha e ck(H)≤k(H)≤
γ(H)≤2ck(H)≤2k(H).I x∗∗ ∈Hω∗
, he e exis s a sequence (xn)nin Hsuch ha
kx∗∗ −y∗∗k ≤ γ(H) o any clus e poin y∗∗ o (xn)nin E′′.His weakly ela i ely
compac in Ei one (equi alen ly all) o ck(H),k(H),γ(H)is ze o.
In he i s pa we show ha some echniques om abo e ci ed pape s can be used also
in he ame o F ´eche spaces, i.e., me izable and comple e lcs.
We p o ide quan i a i e cha ac e iza ions o weak-compac ness in a F ´eche space.
The app oxima ion Theo em 7 (ex ending [2, Theo em 3.2] o F ´eche spaces) is he
quan i a i e e sion o he weak angelici y o a F ´eche space.
Theo em 14 and Co olla y 15 p o ide a quan i a i e e sion o O ihuela’s angelic he-
o em [18, Theo em 3] showing ha ck(H)≤ˆ
d(H, C(X, Z)) ≤17ck(H),whe e H⊂ZX
is ela i ely compac and C(X, Z) is he space o Z- alued con inuous unc ions o web-
compac spaces Xand sepa able me ic space Z. I Xis web-compac and no mal and
Z:= R, hen ck(H)≤ˆ
d(H, C(X)) ≤12ck(H). A co esponding esul o s ongly
web-compac spaces Xis also ob ained wi h a mo e sha pe cons an s using he same
p oo s ha [5, Theo em 3.1 and Theo em 3.2], esul s o web-compac spaces equi e
ex a wo k. This yields also a quan i a i e app oach o he weak angelici y o any lcs in
he class G.
No a ion and e minology: Le Ebe a F ´eche space and le (Un)nbe a de-
c easing basis o absolu ely con ex neighbou hoods o ze o. By (E′, β (E′, E)) and
2
(E′′, β (E′′ , E′)) we mean he s ong dual o Eand (E′, β (E′, E)), espec i ely. In
(E′′, β (E′′ , E′)) he sequence o bipola s (U00
n)nis a dec easing basis o absolu ely con-
ex neighbou hoods o ze o. By khkn= sup |h(u)|:u∈U0
nwe deno e he semino m
in E′′ associa ed wi h U0
nand dnmeans he pseudome ic de ined by k.kn. The es ic-
ion o k.kn o E, also deno ed by k.kn, is he semino m de ined by Un. The opol-
ogy o Ecan be de ined by he F-no m d(x, y) := Pn2−nkx−ykn(1 + kx−ykn)−1
o x, y ∈E. The opology o he space (E′′, β(E′′, E′)) is de ined by he F-no m
d(x∗∗, y∗∗) := Pn2−nkx∗∗ −y∗∗kn(1 + kx∗∗ −y∗∗kn)−1 o all x∗∗, y∗∗ ∈E′′. Addi ion-
ally wi hou loss o gene ali y, we assume in his pape ha 2Un+1 ⊂Un o n∈N; and
his clea ly implies ha 2kx∗∗kn≤ kx∗∗kn+1 o n∈Nand each x∗∗ ∈E′′.
I His a bounded subse o E hen H0is a neighbou hood o ze o in (E′, β (E′, E))
and he bypola H00 is a compac subse o (E′′, σ (E′′, E′)). The e o e an E-bounded
subse His weakly ela i ely compac i and only i Hσ(E′′ ,E′)is con ained in E.
Nex concep s a e he na u al ex ensions o he gi en abo e:
γn(H) := sup
lim
plim
mup(hm)−lim
mlim
pup(hm)
: (up)⊂U0
n,(hm)⊂H
assuming he in ol ed limi s exis . Le
ckn(H) := sup dn(clus E′′ (ϕ), E) : ϕ∈HN
and
ck(H) := sup d(clus E′′ (ϕ), E) : ϕ∈HN
whe e clus E′′ (ϕ) := Tp{ϕ(m) : m > p}σ(E′′ ,E′)is he se o all clus e poin s in E′′ o
he sequence ϕ∈HNand dn(A, B) = in {dn(a, b) : a∈A, b ∈B}. Also de ine
kn(H) := sup dn(h, E) : h∈Hσ(E′′ ,E′),
and
k(H) := sup d(h, E) : h∈Hσ(E′′ ,E′).
We say ha H ε-in e changes limi s wi h a subse Bo E′i
sup
lim
plim
mup(hm)−lim
mlim
pup(hm)
: (up)⊂B, (hm)⊂H≤ε
whe e ε≥0 and he in ol ed limi s exis . Fo ε= 0 we say Hin e changes limi s wi h
B, see [14]. γn(H)≤ε(γn(H) = 0) means: H ε-in e changes (in e changes) limi s wi h
U0
n. No e ha
2γn(H)≤γn+1(H),2ckn(H)≤ckn+1(H),2kn(H)≤kn+1(H).
Hence supnγn(H)<∞, supnckn(H)<∞, supnkn(H)<∞i γn(H) = 0, ckn(H) = 0,
kn(H) = 0, n ∈N, espec i ely.
A space Xis angelic i e e y ela i ely coun ably compac se Ain Xis ela i ely
compac and o each x∈A he e is a sequence (xn)nin Acon e ging o x, see [13].
3
2. Fi s obse a ions and ema ks
Fo x∗∗ ∈E′′ we ha e d(x∗∗, E) = 0 i x∗∗ ∈Ei dn(x∗∗, E) = 0 o n∈N.Hence
P oposi ion 2. Fo a bounded subse Ho a F ´eche space E he se His weakly ela-
i ely compac i k(H) = 0 i kn(H) = 0 o all n∈N.
Mo eo e , om he de ini ions i ollows easily ha ckn(H)≤kn(H). To p o e mo e we
need he ollowing wo addi ional lemmas.
Lemma 3. Le Hbe a bounded subse o a F ´eche space Eand le h∈Hσ(E′′ ,E′).
Then o each n∈N he e exis s a ne (uβ)βin U0
n ha σ(E′, E)-con e ges o 0and
such ha o each ne (hα)αin H ha σ(E′′ , E′)-con e ges o hwe ha e dn(h, E) =
limβlimαuβ(hα). Consequen ly, he e exis sequences (hm)min Hand (up)pin U0
nsuch
ha dn(h, E) = limplimmup(hm)and limmlimpup(hm) = 0. Hence kn(H)≤γn(H).
P oo . The linea unc ional ude ined on he linea hull o Eand hby u(e+λh) =
λdn(h, E) o e∈E e i ies |u(e+λh)|=|λ|dn(h, E) = dn(λh, E) = dn(e+λh, E)≤
ke+λhkn.By he Hahn-Banach heo em uadmi s a linea ex ension o E′′, also named
u, such ha
|u(x∗∗)| ≤ kx∗∗kn
o each x∗∗ ∈E′′. Clea ly u∈U00
n0= (U0
n)00 and we ob ain a ne (uβ)βin U0
nsuch
ha
u(x∗∗) = lim
βuβ(x∗∗)
o each x∗∗ ∈E′′. In pa icula
dn(h, E) = u(h) = lim
βuβ(h),0 = d(e, E) = u(e) = lim
βuβ(e)
o each e∈Eso (uβ)βσ(E′, E)-con e ges o 0. I (hα)αis a ne in H ha σ(E′′, E′)-
con e ges o h, hen each uβ(h) is he limi o he ne (uβ(hα))αand
dn(h, E) = u(h) = lim
βlim
αuβ(hα),0 = lim
αu(hα) = lim
αlim
βuβ(hα).
By Lemma 2.1 o [8] he e exis sequences (hm)min Hand (up)pin U0
nsuch ha
dn(h, E) = lim
plim
mup(hm),0 = lim
mlim
pup(hm).
By dn(h, E) = limplimmup(hm)−limmlimpup(hm) we ha e kn(H)≤γn(H).
Lemma 4. Le (hα)αbe a ne in a bounded subse Ho a F ´eche space E. Le hbe
aσ(E′′, E′)-clus e poin o (hα)α. I ( β)βis a ne in U0
nsuch ha he in ol ed limi s
limβlimα β(hα)and limαlimβ β(hα)exis , hen
lim
βlim
α β(hα)−lim
αlim
β β(hα)
≤2dn(h, E).
Hence γn(H)≤2ckn(H) o each n∈N.
4
P oo . I (uβ)βis a ne in U0
n ha σ(E′, E)-con e ges o 0 and he in ol ed limi s in
limβlimαuβ(hα) exis , hen
lim
βlim
αuβ(hα)
≤dn(h, E).(1)
Indeed, o each ε > 0 le hε∈Ebe such ha
dn(h, hε)< dn(h, E) + ε.
By he hypo hesis limαuβ(hα) = uβ(h) and limβuβ(hε) = 0. Then
lim
βlim
αuβ(hα)
=
lim
βuβ(h)
=
lim
βuβ(h−hε)
≤dn(h, hε)< dn(h, E) + ε.
The inequali y |limβlimαuβ(hα)|< dn(h, E) + εis ue o each posi i e numbe ε, so
we ha e
lim
βlim
αuβ(hα)
≤dn(h, E).
To p o e he main inequali y pick aσ(E′, E)-clus e poin o ( β)β. By hypo hesis,
limαlimβ β(hα) exis s and (hα) is a clus e poin o ( β(hα))βso limβ β(hα) = (hα).
Then limα (hα) exis s and (h) is a clus e poin o ( (hα))αso limα (hα) = (h).
The e o e uβ:= 2−1( β− ) is a ne in U0
n ha σ(E′, E) con e ges o 0 and such
ha he in ol ed limi s in limβlimαuβ(hα) exis s, because by hypo hesis he limi s in
limβlimα β(hα) exis and limβlimα (hα) = limβ (h) = (h).Then
lim
βlim
α β(hα)−lim
αlim
β β(hα)
=
lim
βlim
α β(hα)−lim
α (hα)
=
=
lim
βlim
α β(hα)−lim
βlim
α (hα)
= 2
lim
βlim
α2−1( β− ) (hα)
≤2dn(h, E),
whe e he las inequali y ollows om (1). Hence γn(H)≤2ckn(H) o each n∈N.
P oposi ion 5. ckn(H)≤kn(H)≤γn(H)≤2ckn(H) o a bounded subse Ho a
F ´eche space Eand each n∈N. Then ck(H) = 0 i k(H) = 0.
P oo . The second and hi d inequali ies ollow om p e ious lemmas. The i s in-
equali y is ob ious.
P oposi ion 6. I His a bounded subse o a F ´eche space E, hen he ollowing con-
di ions a e equi alen :
(i) ck(H)=0,
(ii) k(H)=0,
(iii) His weakly ela i ely coun ably compac ,
(i ) His weakly ela i ely compac .
P oo . I is clea ha (ii)⇒(i )⇒(iii)⇒(i). The implica ion (i)⇒(ii) ollows om
P oposi ion 5.
5
3. App oxima ion by sequences and angelici y
The ollowing heo em ex ends he las pa o [2, Theo em 2.3] o F ´eche spaces.
Theo em 7. Le Hbe a bounded subse o a F ´eche space Ede ined by he sequence o
inc easing semino ms (k.kn)n. Le x∗∗ ∈Hσ(E′′ ,E′). The e exis s a sequence (xm)min
Hsuch ha i y∗∗ is a σ(E′′, E′)-clus e poin o (xm)m, hen kx∗∗ −y∗∗kn≤γn(H) o
each n∈N.
P oo . The p oo is based on he ollowing wo obse a ions.
Claim 1. Le Sbe a ini e subse o Hσ(E′′ ,E′)and le nand mbe wo na u al
numbe s. The e exis s a ini e subse Ln(m)⊂U0
nsuch ha o each x∗∈U0
n he e
exis s y∗∈Ln(m) sa is ying
sup {|s(x∗−y∗)|:s∈S}< m−1.
Indeed, le pbe he ca dinal numbe o S. Then he claim ollows om he ac ha
(x∗∗(x∗))x∗∗ ∈S:x∗∈U0
nis a bounded subse o a p oduc o p-copies o he scala
ield (o eal o complex numbe s).
Claim 2. The e exis s a sequence (xm)min Hand o each m∈N he e exis ini e
se s Ln(m)⊂U0
n o n= 1,2,··· , m, such ha i x∗∈U0
n, he e is y∗∈Ln(m) e i ying
sup {|s(x∗−y∗)|:s∈ {x∗∗, x1, x2,··· , xm−1}} < m−1,(2)
and
sup {|(x∗∗ −xm)(z∗)|:z∗∈Lm}< m−1(3)
o Lm=S{Ln(q) : 1 ≤n≤q≤m}.
Indeed, applying Claim 1 wi h S={x∗∗}and n=m= 1 we p o ide a ini e se
L1(1) ⊂U0
1such ha o each x∗∈U0
1 he e exis s y∗∈L1(1) such ha
sup {|s(x∗−y∗)|:s∈ {x∗∗}} <1−1.
Then o x∗∗ ∈Hσ(E′′ ,E′) he e exis s x1∈Hsuch ha o L1=L1(1) we ha e
sup {|(x∗∗ −x1)(z∗)|:z∗∈L1}<1−1.
Assume ha Claim 2 has been checked o a ixed m∈N. To comple e he p oo i is
enough o apply m+ 1 imes he Claim 1 wi h S={x∗∗, x1, x2,··· , xm}and we ob ain
he m+ 1 ini e se s Ln(m+ 1) ⊂U0
n,n= 1,2,··· , m + 1, such ha o each nand each
x∗∈U0
n he e exis s y∗∈Ln(m+ 1) sa is ying
sup {|s(x∗−y∗)|:s∈ {x∗∗, x1, x2,··· , xm}} <(m+ 1)−1,
and o x∗∗ ∈Hσ(E′′ ,E′) he e exis s xm+1 ∈Hsuch ha
sup {|(x∗∗ −xm+1)(z∗)|:z∗∈Lm+1}<(m+ 1)−1,
whe e Lm+1 =S{Ln(q) : 1 ≤n≤q≤m+ 1}.
6
Finally we show ha he sequence (xm)m om Claim 2 is as equi ed. Fix x∗∈U0
n.
F om (2) i ollows ha o each q > n he e exis s y∗
q∈Ln(q) such ha x∗∗(x∗−y∗
q)<
q−1and xj(x∗−y∗
q)< q−1 o each j < q. The e o e
x∗∗(x∗) = lim
qx∗∗(y∗
q) (4)
and
xj(x∗) = lim
qxj(y∗
q).(5)
By (3) we ha e x∗∗(z∗) = limmxm(z∗) o each z∗∈SmLm. In pa icula x∗∗(y∗
q) =
limmxm(y∗
q) o each y∗
q. This and (4) imply
x∗∗(x∗) = lim
qlim
mxm(y∗
q).(6)
Le y∗∗ be a σ(E′′, E) clus e poin o (xm)m. Fo he p e iously ixed x∗∈U0
n he e
exis s a subsequence (xm ) such ha y∗∗(x∗) = lim xm (x∗),and hen by (5) we ha e
y∗∗(x∗) = lim
lim
qxm (y∗
q).(7)
Since y∗
q∈Ln(q)⊂U0
n, we apply (6) and (7) o show |x∗∗(x∗)−y∗∗(x∗)| ≤ γn(H).Since
his holds o each x∗∈U0
n, we conclude kx∗∗ −y∗∗kn≤γn(H).
Co olla y 8 ([13, 3.10 (1)]). E e y F ´eche space Eis σ(E, E′)-angelic.
P oo . Le Hbe a ela i ely coun ably compac subse o (E, σ(E, E′)). Then ck(H) =
0 so by P oposi ion 6, His ela i ely compac in (E, σ(E, E′)) .Theo em 7 implies ha , i
x∈Hσ(E,E′) he e exis s a sequence (xp)pin Hsuch ha xis he unique σ(E, E′)-clus e
poin o he sequence (xp)pin (E, σ(E, E′)). This and σ(E, E′)-compac ness o Hσ(E,E′)
implies ha he sequence (xp)pcon e ges o xin (E, σ(E, E′)) and hen angelici y o
(E, σ(E, E′)) ollows.
4. App oxima ion by sequences in Cp(X)
Fo a opological space X, a me ic space (Z, d), we conside in ZX he s anda
sup emum me ic, ha we also deno ed be d ha we allow o ake he alue +∞, i.e.,
d( , g) = sup{d( (x), g(x)) : x∈X}.
Fo a ela i ely compac se H⊂(ZX, τp) de ine
ck(H) := sup
ϕ∈HN
d(clus ZX(ϕ), C(X, Z)),
whe e d(A, B) = in {d(a, b) : a∈A, b ∈B}. Clea ly each ela i ely coun ably compac
se in (ZX, τp) is ela i ely compac . Le ˆ
d(A, B) := sup{d(a, B) : a∈A}. Recall he
ollowing concep s.
(a)Xis a Lindel¨o Σ-space i he e is an uppe semi-con inuous map om a (nonemp y)
subse Ω ⊂NNwi h compac alues in Xwhose union is X, whe e he se o in ege s N
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is disc e e and NNhas he p oduc opology, see [1]. I he same holds o Ω = NN, hen
Xis called K-analy ic.
(b)Xis quasi-Suslin i he e exis s a se - alued map T om NNin o Xco e ing X
such ha i αn→αand xn∈T(αn), hen (xn)nhas a clus e poin in T(α), see [19].
(c)Xis web-compac [18] i he e exis s a nonemp y subse Σ ⊂NNand a amily
{Aα:α∈Σ}in Xwhose union Dis dense in Xand, i
Cn1,...,nk:= [{Aβ:β= (mk)∈Σ, mj=nj, j = 1,...,k}
o α= (nk)∈Σ wi h xk∈Cn1,n2,...,nk, hen (xk)khas a clus e poin in X. I X
is web-compac wi h D=X, we call Xs ongly web-compac . All quasi-Suslin spaces
a e s ongly web-compac . By [18, Theo em 3] he space Cp(X) is angelic i Xis web-
compac .
We need he ollowing wo echnical ac s om [5, Lemma 1] and [5, Lemma 3].
Lemma 9. Le Xbe a opological space, Za me ic space and H⊂ZXaτp- ela i ely
compac se . Then H2ǫ-in e changes limi s wi h ela i ely coun ably compac se s o X,
whe e ǫ:= ck(H)+ˆ
d(H, C(X, Z)).
Lemma 10. Le (Z, d)be a sepa able me ic space, Xbe a se and H⊂ZXwi h he
poin wise opology τpand ǫ≥0. Assume
(i) X=S{Aα:α∈Σ} o some amily o se s {Aα:α∈Σ}.
(ii) Fo each α= (nk)∈Σ he se H ǫ-in e changes limi s in Zwi h e e y sequence
(xn)nin X ha is e en ually in each Cn1...nk o k∈N.
Then o each ∈H( he closu e in ZX) he e exis s a sequence ( n)nin Hsuch
ha supx∈Xd( (x), g(x)) ≤ǫ o any clus e poin go ( n)nin ZX.
We p o e an ex ension o [5, Theo em 3.1] which yields o he quan i a i e e sion o
O ihuela’s angelic heo em [18, Theo em 3], see Co olla y 17. The i s Theo em 11
wo ks o s ongly web-compac spaces X. The o he one Theo em 14 deals jus wi h
web-compac spaces X.
Theo em 11. Le Xbe a web-compac space wi h a ep esen a ion D=S{Aα:α∈Σ}
wi h X=D. Le (Z, d)be a sepa able me ic space and H⊂ZXaτp- ela i ely compac
se . Then o each ∈H( he closu e in ZX) he e exis s a sequence ( n)nin Hsuch
ha
sup
x∈D
d( (x), g(x)) ≤2ck(H) + 2 ˆ
d(H, C(X, Z)) ≤4ck(H)
o any clus e poin go ( n)nin ZX.
P oo . Le ǫ:= ck(H) + ˆ
d(H, C(X, Z)) and le ˜
H={ |D: ∈H}. We p o e ha
condi ion (ii) in Lemma 10 holds o Dand ˜
H. Take α= (nk)∈Σ and le (xn)nbe a
sequence in D ha is e en ually in each Cn1...nk o k∈N. No e ha each subsequence o
(xn)nadmi s a subsequence (yk)ksuch ha yk∈Cn1...nk o k∈N. Indeed, o n1 he e
is m1∈Nsuch ha xn∈Cn1 o all n≥m1. Se y1:= xm1. By induc ion we ob ain a
subsequence (yk)ko (xn)nsuch ha yk∈Cn1...nk o each k∈N. The same p ocedu e
holds o any subsequence o (xn)n. Since Xis web-compac , e e y such sequence (yk)k
has a clus e poin in X. This means ha he se {xn:n∈N}is ela i ely coun ably
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