On realcompact topological vector spaces

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. 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