Grothendieck locally convex spaces of continuous vector valued functions

PACIFIC JOURNAL OF MATHEMATICS Vol. 120, No. 2, 1985 GROTHENDIECK LOCALLY CONVEX SPACES OF CONTINUOUS VECTOR VALUED FUNCTIONS FRANCISCO J. FRENICHE Le ^{X, E) be he space o con inuous unc ions om he com- ple ely egula Hausdo space X in o he Hausdo locally con ex space E, endowed wi h he compac -open opology. Ou aim is o cha ac e ize he ^(X, E) spaces which ha e he ollowing p ope y: weak-s a and weak sequen ial con e gences coincide in he equicon inu- ous subse s o ^(X, E)'. These spaces a e he e called G o hendieck spaces. I is shown ha in he equicon inuous subse s o E' he σ(E', E)- and β(E', ^-sequen ial con e gences coincide, i ^(X, E) is a G o hendieck space and X con ains an in ini e compac subse . Con- e sely, i X is a G-space and E is a s ic induc i e limi o F eche -Mon el spaces ^(X, E) is a G o hendieck space. The e o e, i is p o ed ha i £ is a sepa able F eche space, hen E is a Mon el space i and only i he e is an in ini e compac Hausdo X such ha , E) is a G o hendieck space. 1. In oduc ion. In his pape X will always deno e a comple ely egula Hausdo opological space, E a Hausdo locally con ex space, and &( X, E) he space o con inuous unc ions om Xin o E, endowed wi h he compac -open opology. When E is he scala ield o eals o complex numbe s, we w i e ^(X) ins ead ^( X, E). I is well known ha Ή{X, E) is a Mon el space whene e ^(X) and E so a e, hence, i and only i X is disc e e and E is a Mon el space (see [5], [16]). We s udy wha happens when X has he ollowing weake p ope y: he compac subse s o X a e G-spaces (see below o de ini ions). We ob ain in Theo em 4.4 ha i £ is a F eche -Mon el space and X has ha p ope y, hen ^( X, E) is a G o hendieck locally con ex space. The key in he p oo is he ollowing ac : e e y coun able equicon inuous subse o ^(X, E)' lies, ia a Radon-Nikodym heo em, in a sui able L τ9 Eβ). As a consequence o a heo em o Mύjica [10], he same esul is ue when E is a s ic induc i e limi o F eche -Mon el spaces. In §3 we s udy he con e se o 4.4. In Co olla y 3.3 i is p o ed ha i X con ains an in ini e compac subse , E is a F eche sepa able space and #( X, E) is a G o hendieck space, hen E is a Mon el space. This p ope y cha ac e izes he Mon el spaces among he F eche sepa able spaces. 345 346 FRANCISCO J. FRENICHE Finally, in §5 we s udy he G o hendieck p ope y in &(Σ, E), he space o Σ- o ally measu able unc ions, by using he esul s o %?{X, E). 2. Gene ali ies. A compac Hausdo opological space K is called a G-space whene e ^(K) is a G o hendieck Banach space, i.e. he weak-s a and weak sequen ial con e gences coincide in %>(Ky [6]. We ex end he e his concep o comple ely egula spaces. 2.1. DEFINITION. X is a G-space i e e y compac subse K o X is a G-space. I XΊs compac , bo h de ini ions coincide [6]. Le us ema k ha he e exis non-compac non-disc e e (/-spaces. Indeed, he opological subspace o he S one-Cech compac i ica ion o a coun able disc e e se ob ained emo ing a clus e poin , is such a space. We in oduce a new de ini ion o G o hendieck locally con ex space, so ha <&{ X) is a G o hendieck space i and only i X is a G-space. 2.2. DEFINITION. E is a G o hendieck space whene e he σ(E', E)- and σ(E E'^-sequen ial con e gences coincide in he equicon inuous subse s o E'. In [17] he ΓG-spaces a e de ined as hose spaces E in which he σ(E', E)- and σ(E Zs'^-sequen ial con e gences coincide. When one deals wi h ^( X) spaces, ou de ini ion seems o be mo e easonable han ha o [17] (see 2.4 and 2.5). The ollowing pe manence p ope ies o he class o G o hendieck locally con ex spaces a e easy o see, hus we s a e hem wi hou p oo . 2.3. PROPOSITION, (a) E is a G o hendieck space i and only i e e y, o some, dense subspace o E so is. (b) Le T: E —> F be a linea con inuous ope a o such ha o e e y bounded subse B o F he e is a bounded subse C o E so ha B is con ained in he closu e o T(C). Then F is a G o hendieck space i E so is. (c) // E is he induc i e limi o he sequence (En) o G o hendieck spaces, and i e e y bounded subse o E is con ained in some En, hen E is a G o hendieck space. 2.4. THEOREM. <g(X, E) is a G o hendieck space i and only i < (K, E) so is o e e y compac subse K o X. In pa icula , X is a G-space i and only i %?(X) is a G o hendieck space. GROTHENDIECK LOCALLY CONVEX SPACES 347 P oo . Le us ecall ha , i K is a compac subse o X, he es ic ion map Γis a con inuous linea ope a o om ^(X, E) in o ^(K, E). I B c %{K, E) is bounded, hen he bounded subse C o <V(X9 E whose elemen s g can be w i en g = Σπ<w /„(-)en wi h/M € ^( X), 0 < /„ < 1, Σn<m n ^ 1? and <?„ e U{A(*:): Λ e 5}, sa is ies Γ(C) z> 5 (see [14,1.5.3]). I «χ X, £) is a G o hendieck space, #( JBΓ, E) so is by 2.3(b). Con e sely, le (g'n) be an equicon inuous and σ(V(X, E) V(X9 £))-null sequence. By [14, III.3 and III.4], he e exis a compac subse K o X and an equicon inuous sequence (h'n) in ^(K, E)' such ha g'n=h'noT o all /ι e N. Since (A'J is σ(V(K9 E)'9 T( (K, £)))-null and equicon inuous, i is also σ(<g(K9 E) <V(K9 E)")-nυΆ i V(K, E) is a G o hendieck space. I ollows ha (g'n) is σ(ί (Z, E) V(X, E)")-mύl. 2.5. REMARK. We use an example o Haydon [4] o show ha , while in he class o ba elled spaces he ΓG-spaces and he G o hendieck spaces do coincide, his is no ue in gene al. Choose, o each in ini e sequence in N, a clus e poin in he S one-Cech compac i ica ion o N, and le X be he opological subspace o ha compac i ica ion, o med by N and hese clus e poin s. Then e e y compac subse o X is ini e, V(X) is in aba elled and e e y /e V(X) is bounded. By Theo em 2.4, X is a G-space. Le /„'(/) = n~ι (n) o all/ e <g(X) and n e N. Then (/;) is a σ(^(X) , V(X))-nuΆ sequence in &(X) ha is no o((S(X) <^{Xyynu because i is no equicon inuous. 3. Necessa y condi ions o Ή(X, E) o be a G o hendieck space. I is well known, and easy o see, ha Φ(X) and E a e opologically isomo phic o complemen ed subspaces o V(X9 E). By 2.3(b), V(X) and E mus be G o hendieck spaces i ΦζX, E) is such a space. Howe e , unless X is pseudo ini e, i.e. hei compac subse s a e ini e (hence ί (-Y, E) is a G o hendieck space i and only i E so is, by Theo em 2.4), E has a s onge p ope y i Φ(X9 E) is a G o hendieck space, as we p o e in he nex heo em. To p o e i we ecall he ollowing esul o [2]: THEOREM A. Le E and F be Hausdo locally con ex spaces, and suppose ha F con ains a subspace opologically isomo phic o he subspace o c0 whose elemen s ha e only ini ely many non-ze o coo dina es. I he in ec i e enso p oduc F <8>εE is a G o hendieck space, hen he σ(E E)- and β(E E)-sequen ial con e gences coincide in he equicon inu- ous subse s o E . 348 FRANCISCO J. FRENICHE As was no ed in [2], i X is no pseudo ini e, hen ^( X) con ains a subspace opologically isomo phic o he abo e men ioned subspace o cQm Mo eo e , he injec i e enso p oduc <g(X)®εE can be linea and opologically iden i ied wi h a dense subspace o ^{X, E), namely, he subspace o all ini e dimensional alued elemen s o ^{X, E). Thus we ob ain om Theo em A and P oposi ion 2.3 (a): 3.1. THEOREM. // ^(X, E) is a G o hendieck space and X con ains an in ini e compac subse , hen he σ(E', E)- and β(E E)-sequen ial con e - gences coincide in he equicon inuous subse s o E'. 3.2. REMARK. By Theo em 2.4, i X is pseudo ini e and £ is a G o hendieck Banach space, ^(X, E) is a G o hendieck space. Howe e , i E is in ini e dimensional, he conclusion o Theo em 3.1 does no hold [11]. Using Theo em 3.1 and [7, 11.6.2], we ob ain he ollowing co olla y, con e se o Theo em 4.4: 3.3. COROLLARY. I E is a F eche sepa able space, X is no pseudo ini e and ^(X, E) is a G o hendieck space, hen E is a Mon el space. 3.4. REMARK. I is unknown o us i Co olla y 3.3 is ue wi hou he sepa abili y assump ion on E. This is ela ed wi h he ollowing ques ion aised in [7, pg. 247]: is a F eche space E al eady a Mon el space i e e y σ(E', ^-con e gen sequence in E' con e ges o β(E', E)Ί 4. Su icien condi ions o V(X9 E) o be a G o hendieck space. We shall need some ac s abou ec o in eg a ion, many o hose can be ound in [1] and [15]. Le (X, Σ, T) be a comple e measu e space wi h τ(X) < 1. We deno e by S?(Σ,E) ( esp. @(Σ,E), L τ, E), L°°(τ, E)) he ec o space o Σ-simple ( esp. Σ- o ally measu able, τ-in eg able, τ-essen ially bounded) E- alued (classes o ) unc ions. Recall ha S {Σ,E) and @(Σ, E) a e endowed wi h he uni o m con e gence opology, and ha he opology o Lι{τ, E) is de ined by he semino ms u -> /p(u(x)) dτ(x), whe e/? uns o e he se o all con inuous semino ms in E (unless con a y speci ica- ion, all in eg als will be ex ended o X). The ollowing Radon-Nikodym heo em is p o ed in [1]: THEOREM B. // E is a quasi-comple e (CM)-space, μ: Σ -> E is a coun ably addi i e ec o measu e, o bounded a ia ion and τ-absolu ely con inuous, hen he e exis s u e Lι(τ, E) such ha μ{A) = jA u(x) dτ(x) o e e y A ^ Σ. GROTHENDIECK LOCALLY CONVEX SPACES 349 Le us ecall ha E is a quasi-comple e (CM)-space, i , o ins ance, i is ei he a F eche -Mon el space o a (DF)-Mon el space [1]. Fi s ly we ex end he classical duali y heo em Lι - L00 o L τ9 Eβ), whe e E is a F eche -Mon el space. The ollowing lemma can be easily p o ed. As usual, pL will deno e he gauge o he absolu ely con ex se L in i s linea span. 4.1. LEMMA. // u e^(Σ, E' namely, u = Σi^mχAe/ i wi h (Ai)i^m disjoin in Σ, hen ί pBo(u(x))dτ{x) < τl U AήsuppBo(e;) o e e y bounded subse B o E. 4.2. THEOREM. Le E be a F eche -Mon el space. The ela ion (1) u'{u)= u{x){υ{x))dτ{x) o all u €= Z/(τ, E'β) de ined o u e L τ> Eβ)' and e L°°(τ, E), is an algeb aic isomo phism be ween L τ, E'β)' andU°(τ9 E P oo . Le υ e L°°(τ, E). The map x -» u{x)(υ(x)) is measu able o e e y u e L 9 Eβ), because is s ongly measu able and he asse ion is clea ly ue when e ^(Σ, E). Fu he mo e, i Z G Σ is a τ-null se such ha B = (S Z) is bounded, hen we ha e (2) u(x)( (x)) <pB0(u(x)) o e e y x ^ X Z. Hence x -> w(jc)(ί;(x)) is τ-in eg able, and we can de ine a linea o m «' on Lι(τ, Eβ) by (1). Mo eo e , i ollows om (2) ha u' is con inuous. Con e sely, ix u' G LX(T, ί^)'. The e exis s a bounded subse B o E such ha (3) ί PB°(U(X)) dτ(x) < 1 implies (w^w)) < 1 o e e y w e L τ, Eβ). We de ine a map μ: Σ -* £"' by (4) μMXeO = «'(χ^0 o e e y A e Σ and e' e £' (i ollows easily om Lemma 4.1 and (3) ha μ(A) e E"). Since £ is e lexi e we can suppose ha μ(A) e £. 350 FRANCISCO J. FRENICHE Clea ly, μ: Σ -> E is a ini ely addi i e ec o measu e. We shall show ha μ is coun ably addi i e: le A be he union o he disjoin sequence (An) in Σ. Gi en an absolu ely con ex ze o-neighbo hood U in E and ε > 0, we chodse λ wi h 0 < λ < oo such ha B c λU, and a0 e N such ha λτ(U >w J( ) < ε o e e y m > m0. Since e'{μ{A)) - £ e'(μ{An)) = u'(xun>mAn e') n<m i ollows om Lemma 4.1 and (3) ha e'(μ(A)) - Σ e'(μ(An)) < e n<m o e e y m > m0 and e' e C/°, as desi ed. Fu he mo e, iΐ A =Un<mAn whe e {An)n<m is disjoin in Σ, and i ε > 0, he e exis s (e'n)n<min U° such ha Σ P (μ(An)) < Σ e'MAH)) + ε = «'( Σ XΛΔ + e. n<m n<m n<m Hence hep^ a ia ion o μ sa is ies he inequali y Vpuμ(A) < λτ(A), om Lemma 4.1 and (3) again. Thus μ is τ-absolu ely con inuous and has bounded a ia ion. By Theo em B, he e exis s e Lx(τ, E) such ha (5) μ(A) = ί υ{x)dτ{x) o e e y A <Ξ Σ. JA We claim ha υ is τ-essen ially bounded and sa is ies (1). Indeed, le {UJ)J be a coun able basis in E o absolu ely con ex ze o-neighbo hoods. Choose, o eachj e N, λy such ha 0 < λy < oo and B c λ^I/,. By Lemma 4.1, (3), (4) and (5), we ha e (6) ( e'( (x)) dτ(x) <λ τ(A) JA o all e' <Ξ U/, A e Σ andy <Ξ N. Le (e'j k)k be a sequence in U such ha pυ(e) = supk e'jk(e) o e e y e & E. By (6), he e exis s Z e Σ wi h τ(Z) = 0 such ha e jΛ{ (x)) < λj o sΛ x ^ X Z and ally, c G N. Hence υ(X Z) is bounded in £ Finally, i ollows om (4) ha (1) is ue o all MGy(Σ, £"), and, by densi y, o e e y u ^ Lι(τ, Eβ). This concludes he p oo . Assume ha X is compac Hausdo and Σ con ains he Bo el subse s o X. Fo each u e L τ, Eβ), deno e by u he ec o measu e o densi y u wi h espec o T. I p is a con inuous semino m in E, he subse GROTHENDIECK LOCALLY CONVEX SPACES 351 F o L τ, Eβ) de ined by he condi ion Vp u{X) < oo, is a linea sub- space. I M G JF hen u has bounded semi a ia ion, hus i de ines a con inuous linea o m on <^(Σ, £), which ex ends by con inui y o he whole space &(Σ, E) [15]. Le ΓwG <g(X, E)' be he es ic ion o V( X, E) o his linea o m, i.e. (7) (Tu)(g) = g(x) d u(x) o e e y g €Ξ <g(X9 E). 4.3. LEMMA. ΓΛe map T: F -> #(*, J?)' έ e/ineJ y (7) is <2 linea con inuous ope a o , when ^{Xy E)' is endowed wi h he s ong opology wi h espec o <g(X9 E). P oo . We ha e, o each u e F, (8) (Tu)(g) = u(x)(g(x)) d (x) o e e y gE ^{X9E). Indeed, he domina ed con e gence heo em and a s anda d densi y a gumen show ha i su ices o see (8) when g belongs o5^(Σ, E), ha is i ially ue. Le H be a bounded subse o ί?( JST, £). Then B = U{g(X): g e i } is a bounded subse o E. Hence, by (8), (Tu)(g) < jpBo(u(x)) dτ(x) and he lemma ollows. We a e now eady o p o e he su icien condi ion: 4.4. THEOREM. Le X be a comple ely egula Hausdo G-space and E a F eche -Mon el space. Then ^(X, E) is a G o hendieck space. P oo . By 2.4 we can suppose, wi hou loss o gene ali y, ha X is compac . Le (g n)n be an equicon inuous sequence in ^(X, E)'. By [14, IΠ.4.5] he e exis s a con inuous semino m p in E such ha Vpμn(X) < 1, o e e y n e N, whe e μn is he ep esen ing measu e o g n [14, III]. Le T = Σn2~nVpμn. T is a coun ably addi i e [0,l]- alued Bo el measu e, by [14, ΠI.2.5]. Le Σ be he comple ed σ- ield o he Bo el ield o X wi h espec o T. We shall deno e also by T and μn he na u al ex ensions o he ea lie measu es o Σ. 352 FRANCISCO J. FRENICHE Since E is a Mon el space, he measu e μn: Σ -> E£ is coun ably addi i e. Clea ly Vpμn < 2", hus μn has bounded a ia ion and is τ-abso- lu ely con inuous (when i is conside ed as an £^- alued measu e). We apply Theo em B, ob aining, o each n G N, a unc ion un G L^T, 2?^) such ha μw is he ec o measu e o densi y un wi h espec o T. Clea ly un G i7 and 7M n = g^, o e e y n G N. Fix g" G <g(X, E)". By Lemma 4.3 and Theo em 4.2, he e exis s υ e L°°(τ, £) such ha g /(g^) = / WΠ(JC)(I;(JC)) dτ(x) o e e y /i G N. Le Z be a se in Σ wi h τ(Z) = 0 and (X Z) bounded. The unc ion λ = χX Z is o ally measu able, because E is Mon el and me izable. Gi en ε > 0, we can choose 2 G y(Σ, E) such ha /?(ί;3(jc)) < ε/2, o e e y x G X9 i u3 = λ — 2. Hence, (9) un(x)( 3(x)) dτ(x) ϋ3(x) dμn{x) <ε/2 o e e y n G N, because F^μ ( X) < 1. On he o he hand, i (g'n) is σ(^(X, £) , ί (Z, £))-nuU, hen (μn(A)(e)) is a null sequence, o e e y e G E and ^EI Indeed, since X is a G-space, o each e G 2S, he weak-s a null sequence (μn( )(^)) in #(Λy, is also weak null, hence (μn(A)(e)) is null o e e y Bo el subse A o X, and so o e e y A ^ Σ. Since u2 is simple, i ollows ha (10) lim [ un(x)( 2(x)) dτ(x) = 0. By (9) and (10), (g/ (g^)) is a nuH sequence, and we ha e shown ha (g'n) is σ(V(X9 E) V(X9 E)")-n M. 4.5. COROLLARY. Le X be a comple ely egula Hausdo G-space and E he induc i e limi o he sequence (En) o F eche -Mon el spaces, such ha e e y bounded subse o E is localized in some En. Then ^(X, E) is a G o hendieck space. P oo . We can again suppose X compac . By [10], he induc i e limi o he sequence (#(Z, En)) is a dense opological subspace o <#(X, E). By P oposi ion 2.3 (a) and (c), and Theo em 4.4, i ollows ha <g(X, E) is a G o hendieck space. GROTHENDIECK LOCALLY CONVEX SPACES 353 4.6. COROLLARY. Le E be a F eche sepa able space. The ollowing condi ions a e equi alen : (a) E is a Mon el space. (b) The e exis s a non-pseudo ini e comple ely egula Hausdo space X such ha Ή(X, E) is a G o hendieck space. (c) Fo e e y comple ely egula Hausdo G-space X, <&(X9 E) is a G o hendieck space. P oo . Use 4.4 and 3.3. 5. Applica ion o spaces o o ally measu able unc ions. Le X be a nonemp y se and Σ a ield o subse s o X. We will say ha a subse B o X is open i o e e y x e B he e is A e Σ wi h x ^ A and A c B. Endowed X wi h his opology, le X* be he Hausdo space associa ed o X, <π X -> X* he quo ien map, and Σ* = {π(A): A e Σ}. The ollowing lemma is easily es ablished: 5.1. LEMMA (a) X* is a comple ely egula Hausdo ze o-dimensional opological space. (b) The map A ^ Σ -> π(A) ^ Σ* is a Boolean isomo phism. (c) The map g e ^(Σ*, £)->g°τ G ^(Σ, £) w α opological iso- mo phism, and i s es ic ion oS (Σ*, E) so is on oS (Σ, E). (d) The map x* e X* -> {5* e Σ*: x*eΰ*}e ^(Σ*) is By using 5.1, when one s udies he linea opological p ope ies o , /?), i can be supposed ha X is a dense subspace o a Hausdo compac ze o-dimensional opological space K (namely, he S one space o he Boolean algeb a Σ), and Σ is he ace in X o he Boolean algeb a o open and closed subse s o K. In his con ex we ha e he ollowing heo em: 5.2. THEOREM. The e exis s a subspace o SS(Σ, E), con aining &'(Σ, E), ha is opologically isomo phic o ^(K, E). P oo . I is easy o check ha he se o es ic ions o X o all elemen s o ^(K, E) is such a subspace. By P oposi ion 2.3 (a), i ollows ha SS(Σ, E) is a G o hendieck space i and only i ^(K, E) so is. Hence we can apply o 38(Σ, E) he esul s o §§3 and 4.