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.