Some properties and applications of weakly equicompact sets

Some p ope ies and applica ions o weakly equicompac se s E. Se ano, C. Pi˜ nei o, and J. M. Delgado Abs ac . Le Xand Ybe Banach spaces. A se M⊂W(X, Y ) ( he space o all weakly compac ope a o s om Xin o Y)isweakly equicompac i , o e e y bounded sequence (xn)inX, he e exis s a subsequence (xk(n)) so ha (Tx k(n)) is uni o mly weakly con e gen o T∈M. In his pape , he no ion o weakly equicompac se is used o ob ain cha ac e iza ions o spaces X such ha X← 1, o spaces Xsuch ha BX∗is weak∗sequen ially compac and also o ob ain se e al esul s conce ning o he weak ope a o and he s ong ope a o opologies. As ano he applica ion o weak equicompac ness, we conclude a cha ac e iza ion o ela i ely compac se s in L(X, Y ) when his space is endowed wi h he opology o uni o m con e gence on he class o all weakly null sequences. Finally, we show ha simila a gumen s can be applied o he s udy o uni o mly comple ely con inuous se s. Ma hema ics Subjec Classifica ion (2000). 47B07. Keywo ds. Weakly compac ope a o s, weakly equicompac se , collec i ely weakly compac se , p ecompac se , uni o m spaces, uni o mly comple ely con inuous. 1. In oduc ion. Le us conside ( eal o complex) Banach spaces Xand Y.As usual L(X,Y ), K(X,Y ), V(X,Y ), W(X,Y ) and CW(X,Y ) will deno e, espec- i ely, he ec o space o all bounded, compac , comple ely con inuous, weakly compac o condi ionally weakly compac linea ope a o s om Xin o Yen- dowed wi h he ope a o no m. In [10], he au ho s defined weakly equicompac ( espec i ely condi ionally weakly equicompac ) se s as hose subse s Mo W(X, Y ) ( espec i ely o CW(X,Y )) sa is ying ha , o e e y bounded sequence (xn)inX, he e exis s a subsequence (xk(n)) so ha (Txk(n)) is uni o mly weakly con e - gen ( espec i ely uni o mly weakly Cauchy) o T∈M. They ha e p o ed he ollowing cha ac e iza ion: Vol. 89 (2007) Some p ope ies and applica ions o weakly equicompac se s 267 P oposi ion A ([10, P oposi ion 2.2]). Le Mbe a subse o W(X,Y )( espec i ely o CW(X,Y )). The ollowing s a emen s a e equi alen : (a) Mis weakly equicompac ( espec i ely condi ionally weakly equicompac ). (b) Msa isfies he ollowing p ope ies: (i) Fo e e y bounded sequence (xn)in X, he e is a subsequence (xk(n)) so ha (Txk(n))is weakly con e gen ( espec i ely weakly Cauchy) o e e y T∈M. (ii) M∗y∗is ela i ely compac in X∗ o e e y y∗∈Y∗. Rema k 1.1. I M⊂K(X,Y ) and Mis weakly equicompac , hen o e e y bounded sequence (xn)inX, he e is a subsequence (xk(n)) so ha (Txk(n))is con e gen o e e y T∈Mbu his con e gence is no necessa ily uni o m o T∈M. The e a e many si ua ions in which condi ion (b)-ii implies i sel ha a se Mis weakly equicompac (see, o example, [10, co olla y 2.3]). In his pape , we gi e an example o a se Msuch ha M∗y∗is ela i ely compac o e e y y∗∈Y∗bu M is no weakly equicompac ; in ac , i BX∗is no weak∗sequen ially compac , he e exis always a Banach space Yand a se o weak∗–weakly con inuous ope a o s in W(X∗,Y) as in he example (Theo em 2.6). I is also p o ed ha a Banach space Xdoes no con ain a copy o 1i and only i he poin wise ela i e compac ness o M∗implies he weak equicompac ness o M, ega dless o he Banach space Y and he se M⊂W(X,Y ). Sec ion 3 is de o ed o ob ain se e al esul s on compac ness in he weak ope a o and s ong ope a o opologies (in sho WOT and SOT, espec i ely) abou weakly equicompac se s. Among o he esul s, we p o e ha , i a se M⊂CW(X,Y ) is condi ionally weakly equicompac and Mo M∗is sequen- ially WOT-compac , hen M∗is sequen ially SOT-compac and Mis condi ion- ally collec i ely weakly compac ( ecall ha a se M⊂W(X,Y ) ( espec i ely M⊂CW(X,Y )) is called collec i ely weakly compac ( espec i ely condi ionally collec i ely weakly compac ) iff he se T∈MT(BX) is ela i ely weakly compac ( espec i ely condi ionally weakly compac ) in Y). In sec ion 4, we conside he locally con ex opologies Tw0and Twc on L(X,Y ) o he uni o m con e gence, espec i ely, on he class w0o all weakly null sequences and he class wc o all weakly Cauchy sequences in X( he space Y endowed wi h he weak opology). To con inue wi h he wo k ha has been done p e iously in [10, Sec ion 3], we p o e ha he ela i ely compac se s a e, in bo h opologies, he weakly w0−equicompac and poin wise weakly compac se s (Theo em 4.6). Finally, in sec ion 5, we apply ou echniques o cha ac e ize ela i ely compac se s in V(X,Y ) (again, V(X,Y ) endowed wi h he opology o uni o m con e gence on he classes w0o wc and he space Ywi h he opology induced by i s no m) in e ms o uni o mly comple ely con inuous se s. 268 E. Se ano, C. Pi˜ nei o, and J. M. Delgado A ch. Ma h. Ou no a ion is s anda d. I Xis a Banach space, BXwill deno e i s closed uni ball and X∗will be he opological dual o X. Fo an a bi a y se I,we will w i e 1(I,X) ( espec i ely ∞(I,X)) o he Banach space o all absolu ely summable ( espec i ely bounded) X- alued unc ions defined on I, endowed wi h he no m ξ=i∈Iξi( espec i ely ξ= sup {ξi:i∈I}) o each ξ= (ξi)i∈I∈1(I,X) ( espec i ely ξ=(ξi)i∈I∈∞(I,X)). As usual, we will w i e 1(I) ( espec i ely ∞(I)) ins ead o 1(I,IR ) ( espec i ely ∞(I,IR )) and ejwill be (δj i)i∈I o each j∈I. 2. Weakly equicompac se s and poin wise compac ness. To s a , we a e going o show some si ua ions in which condi ion (b)-ii in P oposi ion A implies i sel he weak equicompac ness o a se M⊂W(X, Y ). Fo example, his s a emen is ue when he Banach space Xdoes no con ain a copy o 1, as an applica ion o he Rosen hal 1– heo em [8]. Fu he mo e, in iew o P oposi ion A, we can also ob ain he same conclusion i BX∗is weak∗sequen ially compac and he ope a o s in M⊂W(X∗,Y) a e weak∗–weakly con inuous (in ac , hese a e he only possible examples i we o ce he implica ion ega dless o he Banach space Y). Finally, we will p o e ha he abo e implica ion occu s as well i he Banach space Y∗is sepa able (Co olla y 2.3). I M⊂L(X, Y ) is bounded, define Vy∗:x∈X−→ (Tx,y∗)T∈M∈∞(M) o e e y y∗∈Y∗and pu  M={Vy∗:y∗∈BY∗}. P oposi ion 2.1. Le Mbe a bounded subse o W(X,Y ). The ollowing s a emen s a e equi alen : (a) Mis weakly equicompac . (b) Fo e e y bounded sequence (xn)in X, he e exis s a subsequence (xk(n))so ha (Vy∗xk(n))con e ges o all y∗∈Y∗. (c) Msa isfies he ollowing p ope ies: (i) Fo e e y bounded sequence (xn)in X, he e is a subsequence (xk(n)) so ha (Vy∗xk(n))is weak∗con e gen o all y∗∈Y∗. (ii)  Mis a subse o K(X,∞(M)). P oo . (a)⇒(b) is a di ec consequence o he weak equicompac ness o Mand (b)⇒(c) is ob ious. Thus, only (c)⇒(a) needs o be p o ed. Le ’s see ha Mis weakly equicompac ia P oposi ion A. I (xn) is a bounded sequence in X,i admi s a subsequence (xk(n)) such ha (Vy∗xk(n))isweak ∗con e gen o e e y y∗∈Y∗. Since he weak∗con e gence in ∞(M) coincides wi h he con e gence coo dina ewise, his means ha (Txk(n)) is weakly Cauchy o all T∈M.As he ope a o s in Ma e weakly compac , (Txk(n)) is weakly con e gen o all T∈M. Now, define he ope a o Uy∗:(ξT)T∈M∈1(M)−→ T∈MξTT∗y∗∈X∗ o each y∗∈Y∗. No ice ha Uy∗is he es ic ion o (Vy∗)∗ o 1(M). Then, he se M∗y∗⊂Uy∗(B1(M)) is ela i ely compac .  Vol. 89 (2007) Some p ope ies and applica ions o weakly equicompac se s 269 Rema k 2.2. No ice ha , gi en a bounded se M⊂L(X,Y ),  Msa isfies he condi ion (c)-ii in he abo e p oposi ion i and only i M∗y∗is ela i ely compac o e e y y∗∈Y∗. Co olla y 2.3. Le M⊂W(X,Y )be a bounded se . I Y∗is sepa able and M∗y∗ is ela i ely compac o e e y y∗∈Y∗, hen Mis weakly equicompac . P oo . Acco ding o Rema k 2.2, he ope a o φ:y∗∈Y∗−→ Vy∗∈K(X, ∞(M)) is well defined. I (y∗ n) is a sequence in Y∗so ha BY∗⊂ {y∗ n:n∈IN }, hen  M⊂{φ(y∗ n): n∈IN }is sepa able. A s anda d a gumen o diagonaliza ion yields he weak equicompac ness o M om P oposi ion 2.1.  Now, we gi e an example o a se M ailing o be weakly equicompac bu such ha M∗y∗is ela i ely compac o e e y y∗∈Y∗. Example 2.4. Fo e e y s∈IR , conside he ope a o Ts:∞(IR )−→ c0(IR ) defined by Tsx=xses o all x=(x ) ∈IR and pu M={Ts:s∈IR }. Ob iously, Mis a subse o K(∞(IR ),c 0(IR )). Fi s o all, we a e going o p o e ha Mis no weakly equicompac . By con adic ion, e e y sequence (xn)inB∞(IR)(xn= (xn ) ∈IR o e e y n∈IN ) admi s a subsequence (xk(n)) so ha (Tsxk(n))nis con e gen o all s∈IR (Rema k 1.1). This means ha (xk(n) s)ncon e ges o all s∈IR and, he e o e, (xk(n))isweak ∗con e gen in ∞(IR ). This is a con adic ion because B∞(IR)is no weak∗sequen ially compac [3, p. 226]. On he o he hand, no ice ha Vy∗x=(x ·y ) ∈IR whe e y∗=(y ) ∈IR ∈ 1(IR ). Each ope a o Vy∗is compac so, acco ding o Rema k 2.2, M∗y∗is ela- i ely compac . Rema k 2.5. In he p e ious example, we ha e made use o he weak∗sequen ially noncompac ness o B∞(IR); i is also significan o men ion ha he ope a o s in Ma e weak∗–weakly con inuous. The ollowing esul shows his is no acciden al. In he ollowing esul , W∗(Y∗,Z) will deno e he subspace o W(Y∗,Z)o all weak∗–weakly con inuous ope a o s om Y∗in o Z. Theo em 2.6. Le Ybe a Banach space. The ollowing s a emen s a e equi alen : (a) Fo e e y Banach space Zand e e y N⊂W ∗(Y∗,Z),Nis weakly equicom- pac whene e N∗z∗is ela i ely compac o all z∗∈Z∗. (b) Fo e e y Banach space Xand e e y M⊂K(X, Y ),M∗is weakly equicom- pac whene e M∗∗x∗∗ is ela i ely compac o all x∗∗ ∈X∗∗. (c) BY∗is weak∗sequen ially compac . P oo . Only (b)⇒(c) needs o be p o ed. Fo e e y β=(βy)y∈BY∈2(BY), we define he ope a o Tβ:(αy)y∈BY∈2(BY)−→ y∈BY(βyαy)y∈Yand pu M=Tβ:β∈B2(BY). I is easy o check ha M⊂K(2(BY),Y). Since M=M∗∗ and Mβ =Tβ(B2(BY)) o all β∈B2(BY), i ollows ha M∗is weakly 270 E. Se ano, C. Pi˜ nei o, and J. M. Delgado A ch. Ma h. equicompac . Now, gi en a bounded sequence (y∗ n)⊂Y∗, he e exis s (y∗ k(n)) such ha (T∗ ezy∗ k(n))=(z,y∗ k(n)) con e ges o all z∈BY. Theo em 2.7. Le Xbe a Banach space. The ollowing s a emen s a e equi alen : (a) Fo e e y Banach space Yand e e y M⊂W(X,Y ),Mis weakly equicompac whene e M∗y∗is ela i ely compac o all y∗∈Y∗. (b) Xdoes no con ain a copy o 1. P oo . Only (a)⇒(b) needs o be p o ed. Fo e e y x∗∈X∗, we define he ope a o Tx∗=x∗⊗ex∗and pu M={Tx∗:x∗∈BX∗}⊂K(X,c0(BX∗)). In a simila way as in example 2.4, i can be p o ed ha M∗y∗is ela i ely compac o all y∗∈c0(BX∗)∗. Hence, Mis weakly equicompac and, he e o e, e e y bounded sequence (xn)inXhas a subsequence (xk(n)) such ha (Tx∗xk(n)) is con e gen o all x∗∈BX∗. Since Tx∗xk(n)−Tx∗xk(m)=|xk(n)−xk(m),x ∗|, i ollows ha (xk(n)) is weakly Cauchy.  3. Some esul s on compac ness o weakly equicompac se s. P oposi ion 3.1. Le Mbe a condi ionally weakly equicompac subse o CW(X,Y ). I Mo M∗is ela i ely WOT sequen ially compac , hen M∗is ela i ely SOT sequen ially compac . P oo . Le (Tn) be a sequence in M.I Mis ela i ely WOT sequen ially compac , he e exis a subsequence (Tk(n)) and T∈L(X,Y ) such ha (Tk(n))WOT →T, ha is o say, (T∗ k(n)y∗)isweak ∗con e gen o T∗y∗ o e e y y∗∈Y∗.So(T∗ k(n)y∗−T∗y∗) is a weak∗null sequence in he ela i ely compac se M∗y∗−T∗y∗ o e e y y∗∈Y∗(P oposi ion A). F om his, i can be deduced ha (T∗ k(n)y∗−T∗y∗)isa no m null sequence and, hus, (T∗ k(n)) is SOT con e gen . The same a gumen is alid i M∗is ela i ely WOT sequen ially compac .  Defini ion 3.2. Le (Tn) be a sequence in L(X,Y ). We say ha (Tn) has he weak c oss limi p ope y (in sho , wcl p ope y) i e e y bounded sequence (xn)inX admi s a subsequence (xk(n)) such ha weak-limn,m→∞ Tnxk(m)−Tmxk(n)=0. Rema k 3.3. No ice ha i (Tn) is a sequence in L(X,Y ) wi h he wcl p ope y hen (Tn) is, in pa icula , a WOT Cauchy sequence and e e y subsequence o (Tn) has he wcl p ope y. P oposi ion 3.4. Le Mbe a condi ionally weakly equicompac subse o CW(X,Y ) and (Tn)a sequence in M. The ollowing s a emen s a e equi alen : (a) (Tn)has he wcl p ope y. (b) (T∗ n)is SOT con e gen . Vol. 89 (2007) Some p ope ies and applica ions o weakly equicompac se s 271 P oo . I (Tn) has he wcl p ope y and (T∗ n) is no SOT Cauchy, hen he e exis y∗∈Y∗,ε0>0 and inc easing maps h, k:IN −→ IN so ha   (T∗ k(n)−T∗ h(n))y∗  > ε0 o all n∈IN . Gi en n∈IN , choose xnin BXsuch ha xn,(T∗ k(n)−T∗ h(n))y∗>ε 0 (1) Wi hou loss o gene ali y we can assume ha (Tmxn)nis uni o mly weakly Cauchy o m∈IN . Now, ake a subsequence (xl(n))o (xn) sa is ying weak-lim n,m→∞ Tnxl(m)−Tmxl(n)=0. I we pu p(n)=k(l(n)) and q(n)=h(l(n)) o all n∈IN ,weha e: xl(n),(T∗ p(n)−T∗ q(n))y∗≤Tp(n)xl(n)−Tnxl(p(n)),y∗ +Tnxl(p(n)) −Tnxl(q(n)),y∗ +Tnxl(q(n)) −Tq(n)xl(n),y∗. Ha ing in mind ha (Tn) is a condi ionally weakly equicompac se wi h he wcl p ope y, we deduce ha lim n→∞ xl(n),(T∗ k(l(n)) −T∗ h(l(n)))y∗=0, a con adic ion wi h (1). Con e sely, gi en a sequence (xn)inBX, he e exis s a subsequence (xk(n)) such ha (Tmxk(n))nis uni o mly weakly Cauchy o m∈IN . Gi en ε>0 and y∗∈Y∗, he e is a na u al numbe n0so ha  (T∗ p−T∗ q)y∗ <ε/2 and Tmxk(p)−Tmxk(q),y∗<ε/2 o all p, q ≥n0and all m∈IN . Then, aking p, q ≥n0,weha e: Tpxk(q)−Tqxk(p),y∗≤Tpxk(q)−Tpxk(q),y∗+xk(p),(T∗ p−T∗ q)y∗<ε  Theo em 3.5. Le M⊂CW(X,Y )be condi ionally weakly equicompac . I Mo M∗is ela i ely WOT sequen ially compac hen Mis condi ionally collec i ely weakly compac . P oo . Acco ding o P oposi ion 3.1, M∗is ela i ely SOT sequen ially compac . Gi en a sequence (Tnxn) wi h xn∈BXand Tn∈M o all n∈IN , we can assume ha (T∗ n) is SOT con e gen and (Txn) is uni o mly weakly Cauchy o T∈M. By P oposi ion 3.4, he sequence (Tn) has he wcl p ope y, so he e is a subsequence (xk(n)) such ha weak-lim n,m→∞ Tnxk(m)−Tmxk(n)=0.(2) 272 E. Se ano, C. Pi˜ nei o, and J. M. Delgado A ch. Ma h. Fo a fixed y∗in Y∗,weha e Tk(n)xk(n)−Tk(m)xk(m),y∗≤Tk(n)xk(n)−Tnxk(n),y∗ (3) +Tnxk(n)−Tnxk(m),y∗ (4) +Tnxk(m)−Tmxk(n),y∗ (5) +Tmxk(n)−Tk(m)xk(n),y∗ (6) +Tk(m)xk(n)−Tk(m)xk(m),y∗ (7) The SOT con e gence o (T∗ n) in case o (3) and (6), he condi ionally weak equicompac ness applied o (4) and (7) and, finally, (2) allows o s a e ha he sequence (Tk(n)xk(n)) is weakly Cauchy.  Example 3.6. Se Tβ:(αn)∈c0−→ (βnαn)∈c0(β=(βn)∈c0). I we define M={Tβ:β∈Bc0}, i is easy o deduce ha M⊂K(c0,c 0) is weakly equicompac and Mand M∗a e ela i ely WOT sequen ially compac . Ne e heless, Mis no collec i ely weakly compac since β∈Bc0Tβ(Bc0)=Bc0. This shows ha he conclusion in Theo em 3.5 canno be imp o ed, in gene al. 4. Rela i ely compac s se s in Lwc(X,Yw).In [10, Sec ion 2], he au ho s gene - alize he concep o weakly equicompac se . Gi en a class Go bounded sequences in X, we say ha a subse Mo L(X, Y ) ( he se o linea maps om Xin o Y)iscondi ionally weakly G-equicompac ( espec i ely weakly G-equicompac )i e e y sequence (xn)∈Ghas a subsequence (xk(n))∈Gso ha (Txk(n)) is weakly Cauchy ( espec i ely weakly con e gen ) uni o mly o T∈M. We will deno e by w0 he class o all weakly null sequences in Xand by wc he class o all weakly Cauchy sequences. We will occasionally in oke he ollowing esul : Lemma 4.1 ([10, Lemma 2.6]). I M⊂L(X,Y ), he ollowing s a emen s a e equi alen : (a) I (xn)∈w0, hen (Txn)w →0uni o mly o T∈M. (b) I (xn)∈wc, hen (Txn)is weakly Cauchy uni o mly o T∈M. (c) Mis weakly w0-equicompac . (d) Mis condi ionally weakly wc-equicompac . Now, we ecall he defini ion o he opology o uni o m con e gence on a class Go bounded se s. Le Xand Ybe Hausdo ff locally con ex spaces, F(X, Y ) he se o all maps om Xin o Yand Gaco e o X o med by bounded sub- se s o X. Fo each A∈G, we define U(A, W )={ ∈F(X,Y ): (A)⊂W}, whe e W uns o e a 0-neighbo hood basis Bo Y. I is well-known ha he se {U(A, W):A∈G,W∈B}is a 0-neighbo hood subbasis o a opology, deno ed by TG, compa ible wi h he commu a i e g oup s uc u e o F(X, Y ) and he induced opology on L(X,Y ) is locally con ex. We will w i e LG(X,Y ) o his locally con ex space o LG(X, Yτ) i we need o emphasize he opology τo Y(see [2, TVS III.13] and [5, Chap e 8]). Vol. 89 (2007) Some p ope ies and applica ions o weakly equicompac se s 273 A Hausdo ff uni o m space Eis said o be p ecompac i i s comple ion, deno ed by  E, is compac . Wi hin he amewo k o opological ec o spaces his means ha , o a subse Ao a opological ec o space E, he ollowing a e equi alen : (a) Ais p ecompac , (b)Ais ela i ely compac in  Eand (c) o each 0-neighbo hood Vin E he e exis s a fini e subse Fso ha A⊂F+V[4, Theo em 3.5.1]. Ac ually, he opology o uni o m con e gence is ea ed in he amewo k o uni o m spaces (see [1, Chap e X]). We will use he ollowing e sion o Ascoli’s classical heo em Ascoli’s Theo em ([1, Theo em X.17.2]). Conside wo uni o m spaces Xand Y, a co e Go X o med by p ecompac subse s and a se H⊂F(X,Y )such ha he es ic ion o each h∈H o each A∈G,h|A, is uni o mly con inuous. Then His p ecompac in he opology o uni o m con e gence on membe s o Gi and only i i sa isfies he ollowing wo condi ions: (I) His poin wise p ecompac , i.e., H(x)={h(x):h∈H}is p ecompac in Y, o each x∈X. (II) Fo each A∈G, he se H|A={h|A:h∈H}is uni o mly equicon inuous. Fo simplici y, we will deno e by Yw he space (Y,σ(Y,Y ∗)) and by Y· he space (Y,·); we will also deno e by w0( espec i ely wc) he class o med by he anges o all weakly null sequences ( espec i ely weakly Cauchy sequences). We e e o Tw w0and Tw wc as he opologies o uni o m con e gence when we conside , espec i ely, he classes w0and wc in Xand he weak opology in Y;Lw0(X, Yw) and Lwc(X,Yw) deno e L(X, Y ) endowed wi h he opologies Tw w0and Tw wc espec- i ely ( ecall ha L(Xw,Y w)=L(X·,Y ·)). Ob iously WOT Tw w0Tw wc. Mo eo e : he opologies Tw w0and Tw wc only coincide iff Xis weakly sequen ially comple e. Howe e , he au ho s ob ained he ollowing esul in [10]: Theo em B ([10, Theo em 3.2]). Le M⊂L(X, Y )be a bounded se . The ollow- ing s a emen s a e equi alen : (a) Mis p ecompac in Lwc(X,Yw). (b) Mis p ecompac in Lw0(X, Yw). (c) Mis weakly w0-equicompac . The aim o his sec ion is o deduce a cha ac e iza ion o compac se s in he spaces Lwc(X,Yw) and Lw0(X, Yw) ia Theo em B. We design by Ls(X,Y ) he space L(X,Y ) endowed wi h he opology o simple con e gence, ha is o say, he opology TS, whe e Sis he amily o he fini e se s in X(no ice ha he opologies WOT and SOT a e pa icula cases). We will need he ollowing p ope ies, oo: 1) Ls(X,Y ) is closed in Fs(X,Y ) [2, P oposi ion III.16.4]. 2)  Ls(X,  Y)=L s(X,  Y) [5, 8−§39.6(7)]. 274 E. Se ano, C. Pi˜ nei o, and J. M. Delgado A ch. Ma h. No ice ha he na u al inclusion i:Lw0(X,Yw)−→ Ls(X,  Yw) is con inuous and, hus, he e exis s a unique con inuous ex ension i:  Lw0(X,Yw)−→ Ls(X,  Yw). In pa icula , his yields ha i M⊂L(X,Y ), hen M  Lw0(X,Yw)⊆MLs(X,  Yw). A simila a gumen shows ha M  Lwc(X,Yw)⊆MLs(X,  Yw). Lemma 4.2. I M⊂L(X,Y )is bounded and Mx is ela i ely weakly compac o all x∈X, hen MLs(X,  Yw)⊂L(X,Y ). P oo . Since  Yw ∗=Y∗[4, 3.4.4], he weak opology in  Ywis p ecisely σ( Yw,Y∗) and {W(0; y∗,ε):y∗∈Y∗,ε>0}is a 0-neighbo hood subbasis o his opology. Now, ake a map Qin MLs(X,  Yw). Gi en x∈X,y∗∈Y∗and ε>0, he e exis s an ope a o Tε∈Msuch ha |Qx −Tεx, y∗| <ε; in o he wo ds, Qx ∈Mxσ(  Yw,Y ∗). By hypo hesis, Mxσ(Y,Y ∗)is σ( Yw,Y∗)-closed. So we ha e Mxσ(  Yw,Y ∗)⊆Mxσ(Y,Y ∗)σ(  Yw,Y ∗) =Mxσ(Y,Y ∗) om which Qx ∈Mxσ(Y,Y ∗). This yields ha Q∈L(X,Y ). Now, a s aigh o wa d a gumen shows ha he ope a o Qbelongs o L(X,Y ) and Q≤supT∈MT.  Co olla y 4.3. I M⊂L(X,Y )is bounded and Mx is ela i ely weakly compac o all x∈X, hen M  Lwc(X,Yw),M  Lw0(X,Yw)⊂L(X, Y ). The ollowing lemma can be p o ed using a s anda d a gumen : Lemma 4.4. Le Gan a bi a y class o bounded sequences in X.I M⊂L(X,Y ) is condi ionally weakly G-equicompac , hen MLs(X,  Yw)is condi ionally weakly G-equicompac . Co olla y 4.5. Le Gan a bi a y class o bounded sequences in X.I M⊂L(X, Y ) is condi ionally weakly G-equicompac , hen M  Lwc(X,Yw)and M  Lw0(X,Yw)a e con- di ionally weakly G-equicompac . Now, we a e able o s a e one o ou main esul s: Theo em 4.6. Le M⊂L(X,Y )be a bounded se . The ollowing s a emen s a e equi alen : (a) Mis ela i ely compac in Lwc(X,Yw). (b) Mis ela i ely compac in Lw0(X,Yw). (c) Msa isfies he ollowing p ope ies: (i) Mis weakly w0-equicompac . (ii) Mx is ela i ely weakly compac o all x∈X.