RACSAM (2011) 105:39–70
DOI 10.1007/s13398-011-0003-0
SURVEY
On ealcompac opological ec o spaces
J. K¸akol ·M. López-Pellice
Recei ed: 7 Janua y 2010 / Accep ed: 7 May 2010 / Published online: 3 Feb ua y 2011
© The Au ho (s) 2011. This a icle is published wi h open access a Sp inge link.com
Abs ac This su ey pape collec s some o olde and qui e new concep s and esul s
om desc ip i e se opology applied o s udy ce ain in ini e-dimensional opological ec-
o spaces appea ing in Func ional Analysis, including F éche spaces, (LF)-spaces, and hei
duals, (DF)-spaces and spaces o con inuous eal- alued unc ions C(X)on a comple ely
egula Hausdo space X. Especially (LF)-spaces and hei duals a ise in many ields o
Func ional Analysis and i s applica ions, o example in Dis ibu ions Theo y, Di e en ial
Equa ions and Complex Analysis. The concep o a ealcompac opological space, al hough
o iginally in oduced and s udied in Gene al Topology, has been also s udied because o e y
conc e e applica ions in Linea Func ional Analysis.
Keywo ds Angelici y ·Bai e and (b-) Bai e-like ·Bo nological ·Bo el se ·C∗-embedded ·
Class G·(DF)space ·Dis inguished space ·F éche –U ysohn ·k-Space ·K-Analy ic ·
(Weakly) Lindelö (Σ)·Locally con ex space ·(Σ-)Quasi-Suslin space ·(S ongly)
ealcompac space ·(Compac ) esolu ion ·Talag and compac ·(Coun able) igh ness ·
T ans-sepa able ·Weakly compac ·(WCG) space ·Web-bounded (compac )
Ma hema ics Subjec Classi ica ion (2000) 54H05 ·46A04 ·46A50
Dedica ed o P o esso Manuel Valdi ia, excellen p o esso and ma hema ical esea che , on he occasion o
his 80 h bi hday.
The esea ch o he i s named au ho was (pa ially) suppo ed by Minis y o Science and Highe
Educa ion, Poland, G an no. NN201 2740 33 and o he bo h au ho s by he p ojec MTM2008-01502 o
he Spanish Minis y o Science and Inno a ion.
J. K¸akol (B
)
Facul y o Ma hema ics and In o ma ics, A. Mickiewicz Uni e si y, 61-614 Poznan, Poland
e-mail: [email p o ec ed]
M. López-Pellice
Depa amen o de Ma emá ica Aplicada and IUMPA, Uni e sidad Poli écnica de Valencia,
46022 Valencia, Spain
e-mail: [email p o ec ed].es
40 J. K¸akol, M. López-Pellice
1 In oduc ion
Fo a Tichono space (also named comple ely egula Hausdo space) Xby Cp(X)and
Cc(X)we deno e he space o con inuous eal alued maps on Xwi h he poin wise and he
compac -open opology, espec i ely. By Lp(X)we deno e he ∗-weak dual o Cp(X).I
F:= { ∈C(X): (X)⊂[0,1]}, hen in [0,1]F he subspace {( (x): ∈F):x∈X}
is homeomo phic o X.Weiden i yXwi h his subspace {( (x): ∈F):x∈X}and he
closu e o Xin [0,1]Fis he S one– ˇ
Cech compac i ica ion o X, deno ed by βX.Takingin o
accoun he es ic ions o βXo he coo dina e p ojec ions o [0,1]Fwe deduce ha each
∈F, and he e o e each uni o mly bounded ∈C(X)has a unique con inuous ex ension
o βX.
By he ealcompac i ica ion υXo Xwe mean he subse o βXsuch ha x∈υXi ,
and only i , each ∈C(X)admi s a con inuous ex ension o X∪{x}. F om egula i y i
ollows ha each ∈C(X)admi s a con inuous ex ension o υX. The e o e he closu e in
[0,1]C(X)o
{( (x): ∈C(X)) :x∈X}
is homeomo phic o υX.
By de ini ion Xis called ealcompac i X=υX. F om he con inui y o he coo dina e
p ojec ions i ollows ha Xis ealcompac i , and only i , Xis homeomo phic o a closed
subspace o a ca esian p oduc o eal lines. E e y me ic sepa able space is ealcompac .
Clea ly closed subspaces o a ealcompac space a e ealcompac and also each p oduc o
ealcompac spaces is ealcompac . The in e sec ion o a amily o ealcompac subspaces
o a space is ealcompac , because his in e sec ion is homeomo phic o he diagonal o a
p oduc .
The ollowing well-known cha ac e iza ion o ealcompac spaces will be used in he
sequel, see [16,28].
P oposi ion 1 A comple ely egula Hausdo space X is ealcompac i , and only i , o
e e y elemen x ∈βX X he e exis s h ∈C(β X),h(X)⊂]0,1], i.e. which is posi i e on X
and h(x)=0.
P oo Assume ha he condi ion holds. Then
X={h−1
y]0,1]:y∈βX X}.
As each h−1
y]0,1]is a ealcompac subspace o βX(since h−1
y]0,1]is homeomo phic o
(β X×]0,1])∩G(hy),
whe e G(hy)means he g aph o hy), hen Xis also ealcompac . Con e sely, i Xis eal-
compac and
x0∈βX X=βX υX,
hen he e exis s a con inuous unc ion :X→Rwhich canno be ex ended con inuously
o X∪{x0}.F om
(x)=max( (x), 0)+min( (x), 0)=1+max( (x), 0)−(1−min( (x), 0))
we know ha one o he unc ions g1(x)=1+max( (x), 0)o g2(x)=1−min( (x), 0)
canno be ex ended con inuously o X∪{x0}. So (*) he e exis s a con inuous unc ion
On ealcompac opological ec o spaces 41
g:X→[1,∞[ which canno be ex ended con inuously o X∪{x0}.Le 
hbe a con inuous
ex ension o he bounded unc ion h:= 1/g o βX.I 
h(x0)= 0, hen we ge a con adic ion
wi h (*). Hence 
h(x0)=0. 
Recall ha a uni o m space Xis called ans-sepa able [31,33] i e e y uni o m co e o
Xhas a coun able subco e . Sepa able uni o m spaces and Lindelö uni o m spaces a e ans-
sepa able; he con e se is no ue in gene al al hough e e y ans-sepa able pseudome ic
space is sepa able.
Clea ly a uni o m space is ans-sepa able i , and only i , i is uni o mly isomo phic o
a subspace o a uni o m p oduc o sepa able pseudome ic spaces. This implies ha e e y
uni o m quasi-Suslin space [71, Chap e 1, Sec . 4.2] is ans-sepa able. No e also ha ans-
sepa able spaces enjoy good pe manence p ope ies. In pa icula , he class o ans-sepa able
spaces is he edi a y, p oduc i e and closed unde uni o m con inuous images, see [56].
Fo a opological ec o space ( s, in b ie ) E he ans-sepa abili y means ha Eis iso-
mo phic o a subspace o he p oduc o me izable sepa able s. Thus, in pa icula i Eis a
locally con ex space (lcs, in b ie ), hen he weak dual (E,σ(E,E)) o Eis ans-sepa able.
I is easy o see ha a s Eis ans-sepa able i , and only i , o e e y neighbo hood
o ze o Uin E he e exis s a coun able subse No Esuch ha E=N+U,see o
example [30,46,57,58].
Also a s Eis ans-sepa able i , and only i , o each con inuous F-semino m pon E
he F-semino med space (E,p)is sepa able, o he associa ed F-no med space E/ke pis
sepa able.
The concep o ans-sepa able spaces has been used o s udy se e al p oblems bo h om
analysis and opology, o example while s udying he me izabili y o p ecompac se s in
uni o m spaces and in he class o lcs, we e e he eade o pape s [12,15,21,22,40,41,58,
64]. P is e [57] p o ed he ollowing:
P oposi ion 2 A lcs E is ans-sepa able i , and only i , o e e y neighbo hood o ze o U
in E i s pola U◦is σ(E,E)-me izable.
This ac has been applied by P is e [57] o show ha p ecompac se s in (DF)-spaces
a e me izable.
No e he ollowing link be ween ealcompac and ans-sepa able space, see also ela ed
esul s o his o ype in [32].
P oposi ion 3 A comple ely egula opological Hausdo space is ealcompac i , and only
i , he e exis s an admissible uni o mi y Non X such ha (X,N)is ans-sepa able and
comple e.
P oo I Xis ealcompac , hen i is homeomo phic o a closed subse o RC(X). Then he
induced uni o mi y in Xis admissible comple e and ans-sepa able. Con e sely, i Nis
a ans-sepa able, comple e admissible uni o mi y on X, hen(X,N)is isomo phic o a
closed subspace o a p oduc o me izable sepa able (by ans-sepa abili y) uni o m spaces.
The e o e Xis ealcompac . 
P oposi ion 1may sugges he ollowing concep which o iginally has been in oduced by
K¸akol and ´
Sliwa [39].
De ini ion 1 We shall say ha Xis s ongly ealcompac i o e e y sequence (xn)no
elemen s in βX X he e exis s ∈C(β X)which is posi i e on Xand anishes on some
subsequence o (xn)n.
42 J. K¸akol, M. López-Pellice
Clea ly e e y s ongly ealcompac space is ealcompac . I is known [78,Exe .1B.4],
ha i Xis locally compac σ-compac , hen βX Xis a ze o se in βX,soXis s ongly
ealcompac . Recall also ha a subse A⊂Xis said o be C-embedded ((C∗)-embedded) i
e e y eal- alued con inuous (bounded and con inuous) unc ion on Acan be ex ended o a
con inuous unc ion on he whole space X.
Fo s ongly ealcompac spaces we no e he ollowing p ope y. The p oo p esen ed
below om [39] uses an a gumen o Neg epon is [51] conce ning [27, Theo em 2.7].
P oposi ion 4 I X is s ongly ealcompac , hen e e y in ini e subse D o βX X con ains
an in ini e subse S which is ela i ely compac in βX X and C∗-embedded in βX.
P oo Le (xn)nbe an injec i e sequence in D(i.e. xn= xmi n= m)andle :βX→
[0,1]be a con inuous unc ion which is posi i e on Xand anishes on a subsequence o
(xn)n.Se
S={xn:n∈N}∩ −1{0},Yn={x∈βX:| (x)|≥n−1},n∈N,
and
X1=S∪
n
Yn.
No e ha he space X1is egula and σ-compac . Hence i mus be a no mal space. Bu since
Sis closed in X1, hensoi isC∗-embedded in X1. The e o e Sis C∗-embedded in βX1.
Bu X⊂X1⊂βX. This yields he equali y βX1=βX.
This ac implies ha i Xis a s ongly ealcompac space, hen e e y in ini e closed subse
o βX Xcon ains a copy o he space βN.
On he o he hand, [4, Example 1.11] Baumga ne and an Douwen p o ided a sep-
a able i s coun able locally compac ealcompac space X(hence s ongly ealcompac
by Theo em 1below) o which βX Xcon ains a disc e e coun able subse which is no
C∗-embedded in βX. This esul wi h [4, Theo em 1.2] can be used o dis inguish an exam-
ple o a locally compac ealcompac space Xsuch ha βX Xcon ains a sequence (xn)n
o which does no exis ∈C(β X)which is posi i e on Xand anishes on (xn)n.This
space p o ides an example o a locally compac ealcompac space which is no s ongly
ealcompac .
The space Qo a ional numbe s is no s ongly ealcompac bu applying [14] one ge s
ha βQ Qis a βω-space, i.e. i Dis a coun able disc e e subse o βQ Qand D( he closu e
in βQ Q) is compac , hen D=βD,soDis C∗-embedded in βQ. I is well-known [28]
ha βQ Qcon ains a coun able subse which is no C∗-embedded in βQ.
A il e ( il e basis) Fon a opological space Xis said o be unbounded i he e exis s a
con inuous eal- alued unc ion on Xwhich is unbounded on each elemen o F.Then
is said o be unbounded on F.
The ollowing gene al Theo em 1below was ob ained in [39], pa s 1,2,and [70]pa 3.
To ge p ope y 3 om Theo em 1we need he ollowing wo lemmas.
Lemma 1 A il e Fon a opological space X is unbounded i , and only i , he e exis s
x∈F∈FF υX, whe e he closu e is aken in βX.
P oo Se
K:= 
F∈F
F
On ealcompac opological ec o spaces 43
and assume by con adic ion ha K⊂υX. Bu hen o each con inuous eal- alued unc ion
on X he e exis s an open U ⊂βXsuch ha K⊂U and |U ∩Xis bounded. No e
ha he e exis s F∈Fcon ained in U . Indeed, o he wise he amily o se s
{F U :F∈F}
sa is ies he ini e in e sec ion p ope y which leads a poin in K U . This is a con adic ion.
Hence we p o ed ha he e exis s F∈Fwhich belongs o U . This shows ha Fis no
unbounded.
To p o e he con e se assume ha he e exis s x∈K υX.Since
υX=
∈C(X)
υ (X),
whe e
υ (X):= {x∈βX: β(x)=∞},
[28, P oblem 8B.3], hen he e exis s ∈C(X)whose ex ension β:βX→R∞,whe e
R∞:= R∪{∞}( he Alexand o one-poin compac i ica ion), has p ope y ha ∞(x)=
∞.Bu x∈F o each F∈F,so hisp o es ha is unbounded on F.
Lemma 2 Each unbounded il e basis Fon a opological space X is con ained in an
unbounded ul a il e Uon X.
P oo I M:= {M⊂X:∃F∈F;F⊂M}, henMis an unbounded il e on X.By
Lemma 1 he e exis s
x∈
F∈M
F υX.
Le Abe he amily o all il e s Gon Xcon aining Mand such ha x∈F∈GF.O de ing
Aby inclusion, and since he e exis s a maximal chain in Ai s union Uis an ul a il e on X
con aining Fsuch ha
x∈
F∈U
F.
Using Lemma 1one ge s ha Uis unbounded on X.
Clea ly ealcompac spaces o poin wise coun able ype need no be locally compac
spaces as he space RNshows, Example 1below. Recall ha a opological space Xis o
poin wise coun able ype [2] i each x∈Xis con ained in a compac se K⊂Xo coun able
cha ac e in X.
Fo s ongly ealcompac spaces he si ua ion is di e en . Now we a e eady o p o e he
ollowing cha ac e iza ion o s ongly ealcompac spaces.
Theo em 1 1. A opological space X is s ongly ealcompac i , and only i , i is ealcom-
pac and βX X is coun ably compac . Hence e e y locally compac ealcompac space
is s ongly ealcompac .
2. E e y s ongly ealcompac space o poin wise coun able ype is locally compac .
3. A ealcompac space X is s ongly ealcompac i , and only i , o each sequence (Fn)n
o unbounded il e s ( il e bases) he e exis s a con inuous eal- alued unc ion on X
and a subsequence (Fnk)ksuch ha is unbounded on each Fnk.

44 J. K¸akol, M. López-Pellice
P oo 1Assume ha Xis a s ongly ealcompac space. Le P⊂βX Xbe an in ini e se
and le (xn)nbe an injec i e sequence in P. The e exis s a con inuous unc ion
:βX→[0,1]
which is posi i e on Xand ze o on some subsequence (xkn)no (xn)n. Then we no e
{xkn:n∈N}⊂ −1(0)⊂X∗.
Hence
{xkn:n∈N}d⊂ −1(0).
No e also ha {xkn:n∈N}dis non-emp y, whe e Adis he se o all accumula ion poin s o
A. This shows ha Phas an accumula ion poin .
Nowwep o e hecon e se.Assume ha Xis ealcompac and e e y in ini e subse o
βX Xhas an accumula ion poin in βX X.
Le (xn)nbe a sequence in βX X.I P={xn:n∈N}is ini e, hen he ealcompac
p ope y o Ximplies ha he e exis s a con inuous unc ion :βX→[0,1]which is
posi i e on Xand ze o on a subsequence o (xn)n.I P={xn:n∈N}is in ini e, ake
p∈Pd X. Then he e exis s a con inuous unc ion
:βX→[0,1]
which is posi i e on Xand anishes on p. No e ha o e e y >0 hese P∩ −1([0, ))
is in ini e, since −1([0, )) is a neighbou hood o he poin p∈Pd. Le us conside wo
possible cases.
(1) The se P∩ −1(0)is in ini e. In ha case is posi i e on Xand ze o on some
subsequence o he sequence (xn)n.
(2) The se P∩ −1(0)is ini e. Since o e e y >0 hese P∩ −1([0, )) is in ini e,
hen he e exis s an injec i e sequence ( n)nin Psuch ha he sequence ( ( n))nis s ic ly
dec easing and con e ges o ze o. Le P0={ n:n∈N},s0=1andsk∈( ( k+1), ( k))
o all k∈N. Then he sequence (sk)kis dec easing and con e ges o ze o. Se
Fk= −1([sk,sk−1])
o k∈N.ThenFkis compac and k∈Fni , and only i , k=n. Mo eo e ,
X⊂ −1((0,1])=
k
Fk
and
P0∩Fk={ k},k∈N.
I (x)=c>0, hen x∈ −1((2−1c,1]).Since ( k)→0, one ge s ha x/∈Pd
0. Hence,
i x∈Pd
0, henx∈ −1(0). Hence x/∈kFk. We showed ha
Pd
0∩
k
Fk=∅.
Bu Xis ealcompac . Hence o e e y k∈N he e exis s a con inuous unc ion k:βX→
[0,1]which is posi i e on Xand ze o on k.Nex se
Tk
n= −1
k([n−1,1]), n,k∈N.
On ealcompac opological ec o spaces 45
Then
X⊂ −1
k((0,1])=
n
Tk
n
and k/∈Tk
n o all k,n∈N. Mo eo e
X⊂
k
Fk∩X⊂
k
n
Fk∩Tk
n,P0∩(Fk∩Tk
n)⊂P0∩Fk={ k}.
Bu k/∈Fk∩Tk
n,so
P0∩(Fk∩Tk
n)=∅
o all n,k∈N. Hence P0∩W=∅and
Pd
0∩W⊂Pd
0∩
k
Fk=∅,
whe e W=k,nFk∩Tk
n. The e o e we ha e P0∩W=∅. We showed ha he e exis s an
in ini e subse P0o Pand an in ini e sequence o compac se s (Kn)nsuch ha
X⊂
n
Kn⊂βX,
n
Kn∩P0=∅.
Fo e e y n∈Nle gn:βX→[0,1]be a con inuous unc ion such ha
gn|Kn=1,gn|P0=0.
Pu
g=
n
2−ngn.
The unc ion g:βX→[0,1]is con inuous, posi i e on Xand ze o on some subsequence
o he sequence (xn)n. This shows ha o e e y sequence (xn)nin βX X he e exis s a con-
inuous unc ion on βXwhich is posi i e on Xand anishes on some subsequence o (xn)n.
2Assume ha Xis a s ongly ealcompac space o poin wise coun able ype bu Xis no
locally compac . Then he e exis x0∈X o which does no exis a ela i ely compac neigh-
bou hood bu o x0 he e exis s a compac se Kwi h x0∈Kand which admi s a coun able
(dec easing) basis (Un)no neighbou hoods o K.Fo e e yn∈Nchoose xn∈(Un X),
whe e he closu e is aken in βX. No e ha
(β X K)∩{xn}d=∅.
Indeed, le x∈(β X K).Le V⊂βXbe an open neighbou hood o Ksuch ha x∈(β X V).
Then he e exis s n0∈Nsuch ha Un0⊂V∩X,soUn0⊂V.Since
{xn}d⊂Un0⊂V,
hen one ge s ha x∈βX {xn}d. Hence {xn}d⊂K. Clea ly {xn}dis non-emp y. This shows
howe e ha Xis no s ongly ealcompac . A con adic ion.
3Assume ha Xis s ongly ealcompac and ha each Fnis an unbounded il e basis
on X. Fo each n∈N he e exis s an accumula ion poin o Fn,sayxn∈βX υX.Since
Xis a ealcompac space, hen υX=X.Bu Xis s ongly ealcompac , so he e exis s a
subsequence (xnk)ko (xn)nand a posi i e con inuous unc ion g∈C(X)such ha g(x)≤1
46 J. K¸akol, M. López-Pellice
and gβ(xnk)=0 o allk∈N.Bu hen := g−1∈C(X). Hence is unbounded on each
Fnksince
xnk∈
F∈Fnk
F υ (X).
This p o es one di ec ion o he s a emen 3.
To p o e he con e se assume ha (xn)nis a sequence in βX X. Then o each n∈N
he e exis s a il e Fnon Xwhich con e ges o xnin he space βX.Bu
xn∈
F∈Fn
F,xn/∈X=υX.
This shows ha each Fnis unbounded on X. Bu by he assump ion he e exis s a subsequence
(Fnk)ko (F)nand ∈C(X)which is unbounded on each Fnk.Se
g(x):= (1+| (x)|)−1
o each x∈X. Clea ly he unc ion gis posi i e on Xand is con inuous and bounded.
The e o e he e exis s a con inuous ex ension gβo g o βXand clea ly gβ(xnk)=0. This
p o es ha Xis s ongly ealcompac . The p oo is comple ed. 
In [39] we p esen ed he ollowing example o a s ongly ealcompac space which is no
locally compac .
Example 1 The e is a s ongly ealcompac space no locally compac . The space RNis
ealcompac and i is no s ongly ealcompac .
P oo Le Pbe a coun ably and non-emp y subse o βN N. No e ha he subspace X:=
N∪Po βNis a Lindelö space. Hence i is a ealcompac space. On he o he hand,
since e e y coun ably and closed subse o βNis ini e, [78, p. 71], one ge s ha he space
βX X=(βN N) Pis coun ably compac . On he o he hand Xis no locally compac . Now
Theo em 1applies o deduce ha Xis s ongly ealcompac . The second s a emen ollows
di ec ly om Theo em 1.
The ollowing heo em om [39] desc ibes s ongly ealcompac spaces in e m o C(X)
and applies o cha ac e ize bo nological and Bai e-like spaces Cc(X) o locally compac
spaces X. Recall ha a lcs Eis Bai e-like [60] i o e e y inc easing sequence (An)no
absolu ely con ex closed se s co e ing E he e exis s m∈Nsuch ha Amis a neighbou hood
o ze o in E.
Theo em 2 (i) I X is a s ongly ealcompac space, hen Cc(X)is Bai e-like and bo -
nological.
(ii) Consequen ly i X is locally compac , hen Cc(X)is Bai e-like and bo nological i ,
and only i , X is ealcompac .
(iii) Le X be a space o poin wise coun able ype. Then Cc(X)is bo nological and Bai e-
like i , and only i , X is s ongly ealcompac .
We ecall a ew concep s which will be used in he sequel.
We shall say ha Xadmi s a compac esolu ion i he e is a amily {Kα:α∈NN}
o compac se s co e ing Xwi h Kα⊂Kβi α≤β.I Xis a lcs and Kαa e bounded,
i.e. abso bed by neighbou hoods o ze o o X, hen{Kα:α∈NN}is called a bounded
esolu ion.
On ealcompac opological ec o spaces 47
(a) A opological space Xis a Lindelö Σ-space i he e is an uppe semi-con inuous (usco)
map om a (nonemp y) subse Σ⊂NNwi h compac alues in Xwhose union is X,
whe e he se o in ege s Nis disc e e and NNhas he p oduc opology [2,45,49,69].
I he same holds o Σ=NN, henXis called K-analy ic.
(b) Xis Σ-quasi-Suslin i he e exis s a se - alued map T om Σ⊂NNin o Xco e ing
Xsuch ha i αn→αin Σand xn∈T(αn), hen(xn)nhas a clus e poin in T(α).I
Σ=NN, henXis called a quasi-Suslin space, [71].
No e ha hal o P oposi ion 5below is Lemma 29 o [19].
P oposi ion 5 I X is Σ-quasi-Suslin (quasi-Suslin), hen υX is Lindelö Σ(K-analy ic).
Hence X is Lindelö Σi , and only i , X is Lindelö and Σ-quasi-Suslin.
P oo Le TbeamaponΣ⊂NNas in (b). E e y T(α) is coun ably compac , so i s closu e
T(α) in υXis compac . The map
α→T(α)
is (usco), so
Z:= 
α∈Σ
T(α)
is Lindelö Σ.ThenZis Lindelo and he e o e Z=νZ.Since
X⊂Z⊂υX,
hen Z=υZ=υXis Lindelö Σ. The o he case goes simila ly o Σ=NN.
(c) Xis web-compac [52] i he e exis s a nonemp y subse Σ⊂NNand a amily {Aα:
α∈Σ}in Xsuch ha i
Cn1,...,nk:= {Aβ:β=(mk)∈Σ, mj=nj,j=1,...,k}
o α=(nk)∈Σ, hen
{Aα:α∈Σ}=X,
and i α=(nk)∈Σand xk∈Cn1,n2,...,nk, hen(xk)khas a clus e poin in X. All
Σ-quasi-Suslinspaces a e web-compac . By [52, Theo em 3] he space Cp(X)is angelic
i Xis web-compac .
(d) Xis web-bounding [52]i
X={Aα:α∈Σ}
such ha i α=(nk)∈Σand xk∈Cn1,n2,...,nk, hen(xk)kis i.e., ( (xk))kis bounded
o each in C(X).I X={Aα:α∈Σ}, henXis called s ongly web-bounding.
(e) A lcs Eis web-bounded i Eis co e ed by a amily {Aα:α∈Σ}o se s o Σ⊂NN
such ha i α=(nk)∈Σand
xk∈Cn1,n2,...,nk,
hen (xk)kis bounded. Since Aα⊂Cn1,n2,...,nk,k∈N, henAαa e bounded. I X
is σ-bounded, i.e. co e ed by a sequence (Bk)ko bounding se s, hen Cp(X)has a
bounded esolu ion. Indeed, o α=(nk)∈NNse
Aα=
k ∈C(X):sup
x∈Bk
| (x)|≤nk.
54 J. K¸akol, M. López-Pellice
P oposi ion 14 a he i s glance looks somewha echnical bu i co e s many conc e e
classes o opological ec o spaces, o example each (d )-space Cc(X)has i s weak dual
Lindelö Σas ∞-ba elled by [38, Co olla y 3.3] and ha ing a amily o se s as in (ii): I (Sn)n
is a undamen al sequence o bounded se s in Cc(X)se Aα:= nanSn o α=(an)∈NN.
F om he de ini ion o he p ope y o be in class Gi ollows ha Eadmi s a esolu ion
consis ing o σ(E,E)- ela i ely coun able compac se s. This implies ha (E,σ(E,E))
is web-compac in sense o O ihuela [52].Bu henby[52, Theo em 3] i ollows ha
Cp((E,σ(E,E)) is angelic. Hence
(E,σ(E,E)) ⊂Cp((E,σ(E,E))
is angelic ( his p ope y is a pa icula case o P oposi ion 8). This ac co e s many o impo -
an classes o spaces excep spaces Cp(X). Indeed, in [8] we p o ed ha Cp(X) o uncoun -
able spaces Xdoes no belong o class G. Ne e heless, i is known ha Cp(X,E)is weakly
angelic o any web-compac Xand lcs Ein class G,see[11, Theo em 8, Co olla y 1.8]. We
p o ide a di ec p oo o his ac .
P oposi ion 15 I X is a web-compac space and E is a lcs in class G, henC
p(X,E)is
weakly angelic. I E ∈Gis sepa able and Cp(X)is angelic, hen Cp(X,Eσ)is also angelic,
whe e Eσmeans (E,σ(E,E)).
P oo Le {Aα:α∈Σ}be a web-compac ep esen a ion o X.Se G:= Cp(X,E).By
P oposi ion 8i is enough o show ha (G,σ(G,G)) con ains a o al web-compac subse .
I (g ) →gin Cp(X,E), hen o each s∈Xand each x∈Ewe ha e ha
(xg (s)) →xg(s).
The e o e he map
δsx:Cp(X,E)→R
de ined by
δsx(g):= xg(s)
is con inuous. The se
Z:= {δsx:s∈{Aα:α∈Σ},x∈E}
is a o al subse o (G,σ(G,G)). Indeed, i ∈Cp(X,E)and
0=δsx( )=x (s)
o each s∈{Aα:α∈Σ}and x∈E, hen (s)=0 o each s∈{Aα:α∈Σ}.
Then by con inui y, (s)=0 o each s∈X. This implies ha =0. Hence Zis a o al
subse .
Le E={Bβ:β∈NN}be a G- ep esen a ion o E.Then
Z={Dαβ :(α, β) ∈Σ×NN},
whe e
Dαβ ={δsx:s∈Aα,x∈Bβ}.

On ealcompac opological ec o spaces 55
To p o e ha he σ(G,G)-closu e o Zis a web-compac subspace o (G,σ(G,G)) we
need o p o e ha i ((αn,β
n))n→(α, β) in Σ×NNand o each n∈N
δsnx
n∈Dαnβn
hen he sequence (δsnx
n)nhas an adhe en poin in (G,σ(G,G)).As
((αn,β
n))n→(α, β)
we ha e ha {sn:n∈N}is a ela i ely coun ably compac subse o Xand {x
n:n∈N}is
an equicon inuous subse o Bγ,beingγan elemen o NN ha e i ies ha βn≤γ o each
n∈N. Then he sequence (δsnx
n)nhas a subne
(δsn(d)x
n(d))d∈D
such ha (sn(d))d∈D→s∈Xand
(x
n(d))d∈D→x
in (E,σ(E,E)). F om equicon inui y i ollows ha (x
n(d))d∈D→xuni o mly on he
p ecompac subse s o E. Then he p oo will be inished i we p o e ha
(δsn(d)x
n(d))d∈D→δsx
in (G,σ(G,G)). In o he wo ds, we ha e o p o e ha o each ∈C(X,E)we ha e ha
lim
d∈Dx
n(d)[ (sn(d))]=x[ (s)].
Bu his equali y ollows om he ollowing ac s:
(*) limd∈Dx[ (sn(d))]=x[ (s)].
(**) As { (sn):n∈N}is a ela i ely coun ably compac subse o E, and he e o e
p ecompac , hen
lim
d∈Dx
n(d)[ (sn(d))]=lim
d∈Dx[ (sn(d))].
Now assume ha Eis sepa able. Then (E,σ(E,E)) is sepa able by Co olla y 1.I G
is a coun able and dense subse in (E,σ(E,E)), henξ:= σ(E,G)is a me izable locally
con ex opology on Ewi h ξ≤σ(E,E). The assump ions o he F emlin heo em om [26,
3.5] a e sa is ied: Cp(X)is angelic and Eξis me izable, so Cp(X,Eξ)is angelic, whe e
Eξ:= (E,ξ).No e ha Cp(X,Eξ)≤Cp(X,Eσ). Now angelic lemma [26, 3.1; see also
Theo em 5] applies o deduce ha Cp(X,Eσ)is angelic. 
Applying las P oposi ion 15 we p o ide he ollowing ex ension o main esul o [68].
Theo em 8 Le E be a sepa able lcs in class Gand le ξbe a opology on C(X,E)s onge
han he poin wise opology o C(X,E). The ollowing asse ions a e equi alen :
(i) (C(X,E), ξ) is K-analy ic.
(ii) (C(X,E), ξ) admi s a compac esolu ion.
(iii) (C(X,E), ξ) admi s a ela i ely coun ably compac esolu ion.
P oo Since each K-analy ic space admi s a compac esolu ion, i is enough o show ha
(iii) ⇒(i): I (C(X,E), ξ) admi s a ela i ely coun ably compac esolu ion {Kα:α∈
NN}, hen{Kα:α∈NN}is a bounded esolu ion in Cp(X,E)in he poin wise opology
τp.SinceCp(X)is isomo phic wi h a subspace o Cp(X,E), henCp(X)admi s a bounded
56 J. K¸akol, M. López-Pellice
esolu ion. By P oposi ion 6 he space Cp(X)is angelic. Now we apply P oposi ion 15 o
ge ha Cp(X,Eσ)is angelic. Hence he space Cp(X,E)is angelic and hen (C(X,E), ξ)
is also angelic (see Theo em 5). Finally no e ha by Lemma 3 he space (C(X,E), ξ) is
K-analy ic. 
Lemma 3implies ha i a opological space (X,ξ)admi s a weake me ic opology, hen
(X,ξ)is K-analy ic i (X,ξ)has a compac esolu ion. I u ns ou ha his si ua ion implies
ha (X,ξ)is e en analy ic, i.e. (X,ξ)is a con inuous image o he space NN. The ollowing
applicable esul was ob ained by Talag and; he p oo p esen ed below is a modi ica ion o
he p oo due o Cascales and Oncina, see [10, Co olla y 4.3], see also [59, Theo em 5.5.1]
and [61, Co olla y 1, p.105].
P oposi ion 16 Le (X,τ)be a K-analy ic space and le d be a me ic on X whose opology
is coa se han τ.Then(X,τ)is analy ic. E e y egula analy ic space X admi s a weake
me ic opology.
P oo Le {Kα:α∈NN}be a compac esolu ion on (X,τ) and {zn:n∈N}beadense
subse o (X,d).ByBd(z, )deno e he d-closed ball in (X,d)o cen e zand adius >0.
Fo β=(bn)∈NNle
Dβ:= 
n∈N
Bd(zbn,n−1).
Each se Dβis uni a y o oid. Fo y∈X he e exis s (α, β) ∈NN×NNsuch ha
Kα∩Dβ={y}.
Fo Kα∩Dβ=∅,wedeno ebyyαβ he elemen o Xsuch ha Kα∩Dβ=yαβ .I
T:= (α, β) ∈NN×NN:∅= Kα∩Dβ=yαβ ,
hen he map :T→Xde ined by ((α, β)) =yαβ is on o.
Le (α(p), β(p))pbe a sequence in T ha con e ges o (α, β)in NN×NNand le
(α(p), β(p))p(m)be a subsequence.
By K-analy ici y we deduce ha yα(p),β(p)p(m)has an adhe en poin y∈Kα.Since
β(p)con e ges o β=(bn)n∈NN, he sequence yα(p),β(p)p(m)is e en ually in each
Bd(zbn,n−1), hence i s adhe en poin ybelongs o Bd(zbn,n−1).
This shows ha
y∈Kα∩Dβ=yαβ .
We p o ed ha (α, β)∈T, i.e. Tis a closed subse o NN×NNand, he e o e, Tis a
Polish space. Mo eo e we p o ed ha yαβ is an adhe en poin o each subsequence o
yα(p),β(p)p. This implies i ially ha yα(p),β( p)con e ges o yαβ , i.e., (α(p), β(p))
con e ges o (α, β). The e o e is a con inuous mapping om he Polish space Ton o
(Y,τ). This p o es ha (Y,τ)is analy ic.
In o de o p o e he second pa assume ha Δ={(x,x):x∈X}is he diagonal o
he analy ic space X×X. Clea ly Δand (X×X) Δa e analy ic and, he e o e, hey a e
Lindelö .
I x= y he e exis wo closed neighbou hoods Fxand Fyo xand y, espec i ely, such
ha
Fx×Fy⊂(X×X) Δ.
On ealcompac opological ec o spaces 57
The Lindelö p ope y enables us o de e mine a sequence (xn,yn)nsuch ha
X×X Δ=
n
Fxn×Fyn.
The e o e Δis a Gδ-subse o X×Xsince Δ=nGn,whe e
Gn=(X×X) (Fxn×Fyn).
Fo each (x,x)∈Δand n∈N he e exis s an open se Ux,nin Xsuch ha
(x,x)∈Ux,n×Ux,n⊂Gn
As he space Xis comple ely egula we may suppose ha he e exis s a con inuous unc ion
x,n:X→[0,1]such ha
x,n(Ux,n)⊂1
2,1, x,n(X Ux,n)⊂0,1
2.
By Lindelö p ope y o Δ he amily Ux,n:x∈Xcon ains a sequence (Ux(i,n),n)isuch
ha
Δ⊂
i
Ux(i,n),n×Ux(i,n),n:= G∗
n
F om Δ=nG∗
ni ollows ha i x= ya e wo di e en poin s o X, hen he e exis s
n∈Nsuch ha (x,y)/∈G∗
n. Then om (x,x)∈G∗
ni ollows ha he e exis s j∈Nsuch
ha x∈Ux(j,n),n. This implies ha y/∈Ux(j,n),n,since(x,y)/∈G∗
n. By cons uc ion
x(j,n),n(x)= x(j,n),n(y).
Then Xendowed wi h he opology ha makes con inuous he coun able amily o unc-
ions { x(i,n),n:(i,n)∈N2}is me izable wi h he me ic de ined by he o mula
d(x,y)=2−i−n x(i,n),n(x)− x(i,n),n(y):(i,n)∈N2
Clea ly d(x,y)de ines a me ic opology weake hen τ.
3 Weakly Lindelö F éche locally con ex spaces
We know al eady ha he weak dual o a quasiba elled lcs in class Gis K-analy ic by
Theo em 7and P oposi ion 11. In pa icula his yields ha e e y e lexi e F éche space is
weakly K-analy ic. Ano he la ge class o lcs o which he weak opology is Lindelö is he
class o (WCG) F éche locally con ex spaces.
In [42] Khu ana p o ed ha e e y (WCG) F éche space is weakly K-analy ic. Fo he
case when Eis a (WCG) Banach space we e e o [65], see also [17,53].
Recall ha a Banach space Eis weakly compac ly gene a ed (WCG) i he e exis s a
weakly compac subse in Ewhose linea span is a dense subspace o E.AlcsEis said o
be (WCG) i he e exis s an inc easing sequence o σ(E,E)-compac subse s o Ewhose
union is dense in E.
In [53] O ihuela used he me hod o cons uc ing p ojec ions in (WCG) Banach spaces
de eloped by Valdi ia om [72–76] and by O ihuela and Valdi ia in [55], o p o ide a di ec
p oo showing ha he weak opology o a (WCG) Banach space is Lindelö . This me hod
applies o p o e [53] ha a dual Banach space is weakly Lindelö i , and only i , i s ∗-weak
58 J. K¸akol, M. López-Pellice
dual uni ball is a Co son compac space. Hence i Eis a dual Banach space which is weakly
Lindelö , hen he p oduc E×Eis weakly Lindelö , oo.
We e e he eade o [1,9,54] (and e e ences) conce ning weakly coun ably de e mined
(WCD) and weakly Lindelö de e mined (WLD) Banach spaces which p o ided la ge clas-
ses (o weakly Lindelö spaces) han he class o (WCG) Banach spaces.
We ecall he ollowing heo em due o Khu ana [42].
Theo em 9 Le E be a F éche space which admi s an inc easing sequence o σ(E,E)-
compac se s whose union is dense in E. Then σ(E,E)is K-analy ic. Mo eo e E is a Bo el
subse o (E,σ(E,E)),whe eE is he bidual o E.
P oo Since e e y me izable lcs Eis angelic in he weak opology σ(E,E), hen o p o e
ha (E,σ(E,E)) is a K-analy ic space i is enough o show ha (E,σ(E,E)) has a com-
pac esolu ion by Lemma 3.
In a na u al way, we iden i y (E,σ(E,E)) wi h a subspace o RE, endowed wi h he
p oduc opology. The e o e o each xin Ewe ha e x=(g(x))g∈E. Fo each ∈E he
mapping
P :RE−→ R
de ined by
P ((αg)g∈E):= α
e i ies in any poin x=(g(x))g∈E∈E
P (x)=P ((g(x))g∈E)= (x)
and he e o e, he es ic ion o P o Eis .
Le (Vn)nbe a base o closed absolu ely con ex neighbou hoods o 0 in E, such ha
(n+1)Vn+1⊂Vn
o each n∈N.Le Vnbe he closu e o Vnin RE. Then, gi en nand pin N, ∈V0
nand
zn+p∈Vn+p,weha e ha
P (zn+p)≤sup P (x):x∈Vn+p=sup P (x):x∈Vn+p
F om his i ollows ha o each ∈V0
nand zn+p∈Vn+pwe ha e
P (zn+p)≤sup | (x)|:x∈Vn
n+p≤1
n+p(1)
Le (An)nbe an inc easing sequence o weakly compac absolu ely con ex subse s o E
such ha nAn=His dense in E.SinceHis dense in Eand Vnis a neighbou hood o
ze o in Ewe ha e ha
E⊂H+Vn⊂H+Vn
o each n∈N.I
x∈H+Vn:n∈N
hen he e exis s a sequence
(x=hn+zn)n(2)
On ealcompac opological ec o spaces 59
wi h hn∈Hand zn∈Vn.Fixann∈N. Then o each ∈V0
nand each pand qin Nwe
ha e by P de ini ion and (2) ha
 (hn+p−hn+q)=P (x−zn+p)−P (x−zn+q)=P (zn+q)−P (zn+p)
and he e o e om (1) we deduce ha
 (hn+p−hn+q)≤P (zn+p)+P (zn+q)≤2
n.
o each ∈V0
n. The uni o mi y implies ha he sequence (hs)sis Cauchy in he F éche
space Eand he e o e i has a limi h∈E. Then lims→∞ hs=hand om (1) i ollows ha
P (x)=lim
s→∞ P (hs+zs)=lim
s→∞  (hs)+P (zs)= (h)=P (h)
implying x=hin RE.Sincex=h∈E, hen
E=H+Vn:n∈N
and hen we no e ha
E=Am+Vn:m∈N:n∈N.
The e o e, Eadmi s a esolu ion Kα:α∈NNwi h
Kα=Amn+Vn:n∈N
o each α=(mn)∈NN.
In RE he se s Amn+Vn o n∈N, a e closed.
Claim. The closed se Kαis bounded in RE.
Indeed, i ∈E, hen he e exis s nin Nsuch ha ∈V0
nand we ha e
sup {| (x)|:x∈Kα}≤sup P (x):x∈Amn+Vn
≤sup | (x)|:x∈Amn+sup P (x):x∈Vn
=sup | (x)|:x∈Amn+sup P (x):x∈Vn
≤kmn+1,
whe e
kmn:= sup | (x)|:x∈Amn.
This p o es ha he closed se Kαis compac in REand i is also compac in (E,σ(E,E)).
We p o ed ha {Kα:α∈N}is a compac esolu ion in (E,σ(E,E)).
Finally, since Amn+Vnis closed in RE, hen
Amn+VnE
is closed in (E,σ(E,E)) and so
H+Vn∩E
is a Bo el se in (E,σ(E,E).Since
E=H+Vn∩E :n∈N
we deduce ha Eis a Bo el se in (E,σ(E,E)).

60 J. K¸akol, M. López-Pellice
I is known [71] ha (*) e e y lcs which is a Bai e space and addi ionally K-analy ic mus
be a F éche space, i.e. a me izable and comple e lcs. This combined wi h Theo em 9yields
he ollowing
P oposi ion 17 Le E be a (WCG) Bai e lcs. Then E is a F éche space i , and only i ,
(E,σ(E,E)) is K-analy ic.
P oo Assume ha Eis a F éche space. By Theo em 9 he space Eis weakly K-analy ic.
Con e sely, i (E,σ(E,E)) is K-analy ic, hen (E,σ(E,E)) admi s a compac esolu ion.
By (*) he space Eis me izable. Since Ehas a σ(E,E)-compac esolu ion and he o iginal
opology o Ehas a basis o neighbou hoods consis ing o σ(E,E)-closed se s, hen he
space Eadmi s a comple e esolu ion as well. Bu e e y me izable Bai e lcs which has a
comple e esolu ion is comple e, see [34, Theo em] and i s e ined e sion in [15, Theo em 3.5
and Co olla y 3.6]. 
P oposi ion 17 applies also o ge he well-known ac s a ing ha a sepa able space
X:= RAwi h uncoun able Ais no K-analy ic (since Xis (WCG) and Bai e bu no me -
izable).
I Eis a Banach space, hen he Mackey dual (E,μ(E,E)) is no me izable, excep
he case when Eis e lexi e. I is well-known ha (E,μ(E,E)) is a comple e lcs. I Bis
he dual uni ball in he dual Eo E, hen one may expec ha some cases (di e en om
e lexi i y in gene al case) may p o ide me izabili y o (B,μ(E,E)|B).
In [62] Schlüch e mann and Wheelle in oduced s ongly weakly compac ly gene a ed
((SWCG) sho ly) Banach spaces. A Banach space is (SWCG) i he space (B,μ(E,E)|B)
is me izable, see also [63].
The ollowing Theo em 10 ( om [62, Theo em 2.1]) cha ac e izes (SWCG) Banach spaces
in e m o some densi y condi ion. Theo em 10 shows also ha e e y (SWCG) Banach space
is (WCG).
In [62, Theo em 2.5] i is p o ed ha e e y (SWCG) Banach space is weakly sequen ially
comple e. Hence he space c0al hough is a (WCG) space is no (SWCG).
Theo em 10 The ollowing condi ions a e equi alen o a Banach space E wi h a closed
uni balls B ⊂E and B⊂E.
(i) (B,μ(E,E)|B)is me izable.
(ii) The e exis s a sequence (Kn)no weakly compac absolu ely con ex subse s o E such
ha o e e y weakly compac se L ⊂E and e e y >0 he e exis s n ∈Nsuch ha
L⊂Kn+B.
(iii) The e exis s a weakly compac absolu ely con ex se K ⊂E such ha o each weakly
compac se L ⊂E and e e y >0 he e is n ∈Nsuch ha L ⊂nK +B.
Assume now ha he space Eis a sepa able (SWCG) Banach space. Then clea ly he
space (E,μ(E,E)) is sepa able. Since (E,μ(E,E)) is sepa able, hen Bis sepa able as
well.
Indeed, le F(E)be he se o all absolu ely con ex neighbou hoods o ze o in μ(E,E)
and le Um∈F(E),m∈N, such ha (B∩[Um+Um])mis a basis o neighbou hoods o
ze o in B. By sepa abili y he e exis s a coun able se Bmsuch ha
E⊂Bm+Um,
and hen he e exis s in Ba coun able subse Cmsuch ha
B⊂Cm+Um+Um.
On ealcompac opological ec o spaces 61
Since E=nnBand each nBis me izable sepa able and comple e, hen he space
(E,μ(E,E)) is analy ic. The e o e we ha e
P oposi ion 18 Le E be a (SWCG) Banach space. Then (E,μ(E,E)) is analy ic i , and
only i , E is sepa able.
Le (S,Σ,μ) is a ini e measu e space. L1(μ, E)deno es a Banach space o Bochne
in eg able unc ions on Sin o a Banach space E.In[62, Theo em 3.2] Schlüch e mann and
Whelle p esen ed pa ial esul s o whe he X(SWCG) implies ha L1(μ, E)is Talag and
[67], see also Dies el [13], p o ed ha L1(μ, E)is (WCG) i Eis a (WCG) Banach space.
I Eis a sepa able Banach space, hen he Mackey dual (E,μ(E,E)) is a sepa able bu
he s ong dual (E,β(E,E)) need no be sepa able. Clea ly (E,β(E,E)) is analy ic i ,
and only i , (E,β(E,E)) is sepa able. Theo em 10 and P oposi ion 18 may sugges he
ollowing ques ion:
Le Ebe a sepa able Banach space. Is i ue ha he Mackey dual (E,μ(E,E)) o Eis
an analy ic space?
Fo he Mackey dual o Cp(X)wep o edin[36] he ollowing gene al ac sugges ed
by [18], whe e Fe ando p o ed ha he Mackey dual o Cp[0,1]is no analy ic bu weakly
analy ic.
Theo em 11 The Mackey dual o Cp(X)is analy ic i , and only i , X is coun able.
P oo Se X:= (X,τ)and assume ha he Mackey dual o Cp(X)is analy ic. Suppose, by
con adic ion, ha Xis uncoun able. Fo x∈X he unc ional
δx:Cp(X)−→ R
de ined by δx( )= (x)is linea and con inuous. Deno e by Lp(X)and Lμ(X) he dual o
Cp(X)wi h he weak dual opology σ=σ(Cp(X),Cp(X)) and wi h he Mackey opology
μ=μ(Cp(X),Cp(X)), espec i ely. Se Y={δx:x∈X}.Themap
δ:(X,τ)−→ (Y,σ|Y)
de ined by x→δxis a homeomo phism and he se Yis closed in Lp(X),see[2, P oposi ion
0.5.9]. Hence Yis also closed in Lμ(X). Thus (Y,μ|Y)is analy ic. Le γbe he opology
on Xsuch ha δis a homeomo phism be ween (X,γ) and (Y,μ|Y).Since(X,γ) is an
uncoun able analy ic space, i con ains a se Ahomeomo phic o he Can o se , see [59].
Clea ly
γ|A=τ|A.
Le (xn)n⊂Abe a sequence such ha xn= xm o n= m ha con e ges o some
x0∈(A {xn:n∈N}).
I is easy o see ha o e e y closed subspace Go (X,τ)and e e y x∈(X G) he e exis s
∈C(X,I)wi h (x)=1 such ha G∩supp =∅.Pu
Xn={xk:k>n}∪{x0}
o n∈N. Clea ly Xnis closed in Xand xn∈ Xn o n∈N. The e o e we can cons uc
induc i ely a sequence ( n)n⊂C(X,I), such ha n(xn)=1and
supp n∩Xn∪{supp k:1≤k<n}=∅.
62 J. K¸akol, M. López-Pellice
Then x0∈ {supp k:k∈N})and
supp n∩supp m=∅
o all n,m∈Nwi h n= m.
Deno e by C∗(X) he Banach space o all bounded eal- alued con inuous unc ions on
Xwi h he sup no m ·.Le g∈C∗(X).Fo k∈Nwe pu
αk=|g( k)|/g( k)
i g( k)= 0, and αk=1, o he wise. Then |αk|=1and
αkg( k)=|g( k)|
o k∈N.
Le n∈Nand Sn=n
k=1αk k.ThenSn∈C∗(X)and Sn=1. Thus
n

k=1
|g( k)|=
n

k=1
αkg( k)=|g(Sn)|≤g
o n∈N,so
∞

k=1
|g( k)|≤g.
Hence g( k)→0. I ollows ha he sequence ( n)ncon e ges weakly o 0 in C∗(X). Thus
he se
F0={0, 1,− 1, 2,− 2, ...}
is weakly compac in C∗(X). By he K ein–Smulian Weak Compac ness Theo em [47,
Theo em 2.8.14] he closed con ex hull Fo F0in C∗(X)is weakly compac . Clea ly
Fis he closed absolu ely con ex hull o he se { k:k∈N}in C∗(X). The opology o
he poin wise con e gence in C∗(X)is weake han he weak opology o C∗(X),soFis
compac in (C∗(X), ). Hence Fis compac in Cp(X), since he injec ion map
(C∗(X), ) −→ Cp(X)
is con inuous. Thus he unc ional
pF:Lμ(X)−→ [ 0,∞),
de ined by
pF(g)=sup{|g( )|: ∈F},
is a con inuous semino m. Since ( n)n⊂Fwe ha e
pF(δxn)≥| n(xn)|=1
o n∈N. I is easy o see ha (x0)=0 o all ∈F,sopF(δx0)=0. I ollows ha
δxn→ δx0in (Y,μ|Y),soxn→ x0in (X,γ); a con adic ion.
Assume now ha he space Xis coun able. I Cp(X)is ini e-dimensional, hen he Mackey
dual Lμ(X)o Cp(X)is ini e-dimensional; so i is analy ic. I Cp(X)is in ini e-dimensional,
hen Cp(X)is a me izable lcs isomo phic o a dense subspace o RN,soLμ(X)is algeb ai-
cally isomo phic o ϕ, he s ong dual o RN.Bu ϕwi h he s onges locally con ex opology
is he sum o an inc easing sequence o ini e-dimensional Banach spaces, so i is an analy ic
space. I ollows ha Lμ(X)is analy ic, oo. 
On ealcompac opological ec o spaces 63
Theo em 11 and i s p oo yields also he ollowing
Co olla y 2 The s ong dual o Cp(X)is analy ic i , and only i , X is coun able.
Recall ha Lp(X)is analy ic i , and only i , Xis analy ic by [2, P oposi ion 0.4.13]. Thus
Theo em 11 p o ides many o conc e e non analy ic lcs whose weak opology is analy ic.
Co olla y 3 Le X be an uncoun able analy ic space. Then he Mackey dual Lμ(X)o
Cp(X)is weakly analy ic bu no analy ic.
The Mackey dual (E,μ(E,E)) o a Banach space has been s udied also in [43,63].
In [63] he au ho s p o ed among he o he s ha i Eis a sepa able (SWCG) Banach space,
hen (E,σ(E,E)) (which is clea ly analy ic) is an ℵ0-space, i.e., i has a coun able pseudo-
base.
A collec ion Po subse s o a opological space Eis called a pseudobase i o any open
se U⊂Eand compac K⊂U he e exis s P∈Pwi h K⊂P⊂U. Recall also ha e e y
ℵ0-space is sepa able and Lindelö and e e y closed se is a Gδ-se , [48], [63, Theo em 4.1].
In [43] Ki k s udied he Mackey dual o spaces C(K)wi h compac K.
On he o he hand, by Ba and Hie meye [3,2.6](seealso[62], [63, p. 274] and [63,
Theo em 4.2]) he e exis s a sepa able Banach space E o which (E,σ(E,E)) is no an
ℵ0-space. I is known also [48], [63, Theo em 4.1], ha a egula opological space is bo h an
ℵ0-space and a k-space i , and only i , i is a quo ien o a sepa able me ic space. The e o e
i seems o be na u al o ask when o a Banach space E he space (E,σ(E,E)) is a k-space.
Recall ha a Hausdo space Xis a k-space i a se A⊂Xis closed in Xi , and only i ,
A∩Kis closed in K o each compac se K⊂X. We shall need he ollowing ac due o
G o hendieck [29, p. 134].
Lemma 5 Le A ⊂Ebe μ(E,E)-compac o a Banach space E. Then e e y σ(E,E)-
con e gen sequence in E con e ges uni o mly on A.
Nex P oposi ion p o es ha o e e y in ini e-dimensional Banach space E he space
(E,σ(E,E)) is ne e a k-space.
P oposi ion 19 I E is a Banach space o which (E,σ(E,E)) is a k-space, hen E is
ini e-dimensional.
P oo Le γbe he opology on Eo uni o m con e gence on μ(E,E)-compac se s. Then
clea ly σ(E,E)≤γ.Sinceσ(E,E)and γha e he same sequen ially compac se s by
Lemma 5, hen he bo h opologies ha e he same compac se s ( ecall ha σ(E,E)and γ
a e angelic).
Assume ha (E,σ(E,E)) is a k-space, hen we ha e σ(E,E)=γ.Le (xn)nbe a
null-sequence in he no m opology o E.Since{0}∪{xn:n∈N}is μ(E,E)-compac ,
hen he sequence (xn)nhas ini e-dimensional linea span. This yields ha E(hence also
E) is ini e-dimensional. 
4 F éche –U ysohn spaces in class G
In his sec ion we p o e ha e e y F éche –U ysohn as well as e e y Bai e lcs in class Gis
me izable, see [8]. Recall ha a opological space Xis F éche –U ysohn i o each A⊂X
and each x∈A he e exis s a sequence in Awhich con e ges o x.
70 J. K¸akol, M. López-Pellice
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