# Borel sets and σ-fragmentability of a Banach space

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A b stra ct

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### C on ten ts

A b s t r a c t ... 2

A ck now ledgm ents... 5

I n tr o d u c tio n ... 6

1 cr-fragm entability 10 1.1 Some definitions and rem arks ... 10

1.2 cr-fragmentability of a m etric space ... 13

1.3 C ountable cover by special sets of small local diam eter . . . . 15

1.4 Borel s e t s ... 20

1.5 D ecom position of discrete f a m i l i e s ... 27

2 O n e-to -o n e m aps 31 2.1 In tr o d u c tio n ...31

2.2 d. cr-d. m aps ... 32

2.3 Inverse m a p p i n g s ... 39

2.4 Some topological invariants ... 41

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2.5 T he Co-sum of some Banach s p a c e s ...42

3 Spaces w ith a countable cover by sets o f sm all local d iam eter 48

3.1 T he space c o ( r ) ... 48 3.2 P rojectional Resolutions of I d e n t i t y ...50 3.2.1 d. a-d. m aps into co(T) using P R I ... 53 3.2.2 W CD spaces, duals of Asplund spaces and C (K ) spaces 64 3.3 M arkushevich basis... 67

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A cknow ledgm ents

Sincere th an k s to m y supervisor Professor J.E . Jayne whose ideas and sug­ gestions I tried to develop in this thesis. All the problem s we solved were suggested by him .

T he research was fully funded by a grant from CAM, C aja de Ahorros del M editerraneo, under th eir joint program w ith th e B ritish Council, “Becas p ara am pliacion de estudios en el Reino Unido de G ran B retan a e Irlanda del N orte” .

I would also like to th ank Dr. B. Cascales and Dr. J. O rihuela for th eir support and discussions from which I greatly benefited.

The encouragem ent I received from m y family throughout these two years, and especially from my parents, has played a very im p o rtan t p art in the re­ alisation of this thesis. Imm ense thanks to my girl-friend, Ju an a Inès, whose constant support, from th e very beginning, m ade it possible.

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Introduction

T hroughout this thesis two notions will be constantly used. B oth of th e m were introduced by Jayne, Namioka and Rogers in [12, 13],

Let (X , r ) be a Hausdorff space and let p be a m etric on X not necessarily related to th e topology on X .

T he space X is said to be a-fragmented by th e m etric g if, for each e > 0, it is possible to w rite

OO

X = \ J X i , i=l

where each set X{ has the property th a t each non-em pty subset of X{ has a non-em pty relatively r-o p en subset of ^-diam eter less th a n e.

We shall say th a t (X , r ) has a countable cover by sets of small local

g-diam eter if, for each c > 0, X can be expressed as a union

OO

### (0.1)

1 = 1

each non-em pty X{ having th e property th a t each of its points belongs to some relatively non-em pty r-o p en subset of g-diam eter less th a n e.

A lthough our aim is to find results on Banach spaces we give th e statem en ts, when possible, in term s of topological spaces and m etrics defined on them .

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equivalent concepts when th e topology on th e space is generated by a m etric. We also prove th a t if a topological space { X , t ) has a countable cover by sets of small local diam eter and th e m etric involved is lower sem icontinuous for th e topology T, th en th e sets in equation (0.1) can be tak en to be differences of r

-closed sets. W hen applied to a B anach space we obtain th a t if it has a countable cover by sets of sm all local diam eter, th en it has a countable cover by differences of weakly closed sets of small local diam eter. This condition was shown to be sufficient, by th e authors above, to have B o re l{X j\\ • ||) = B o r e l[ X ,w e a k ) ,

see [9], p. 215 and [11], Lem m a 2.1. We prove som ething stronger, we show th a t if a B anach space X has a countable cover by sets of small local diam eter, then b o th X and any || • ||-closed subset of X are Borel subsets of (X**,iu*), and this im plies th e coincidence of th e Borel subsets of X . We finish this C hapter characterizing both th e property of having a countable cover by sets of small local diam eter and th e cr-fragmentability in term s of decom positions of II • ||-discrete families into a countable num ber of relatively weakly discrete families.

C h ap ter 2 is concerned w ith one-to-one m aps betw een B anach spaces (or ra th e r betw een topological spaces w ith m etrics defined on th em ). l i T : X Y

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cr-decom posable families in {Y, || • ||y). (Such a m ap will be said to be d. c7-d.). Moreover, if we denote by T||.||y th e m etric given by

^ ||- ||y K ^ ) = 11^"^ - T v\\y ,u,v G X ,

we show th a t (X , T||.||^) has a countable cover by sets of additive class a (for th e topology 7]|.||y) of small local || • ||%-diameter if and only if T~^ is of Borel class a and T is d. cr-d.

We also show th a t (i) cr-fragmentability (first proved in [19]), (ii) to have a countable cover by sets of small local diam eter, and (iii) th e coincidence of th e weak and norm Borel sets on a Banach space are each topological invariants for th e weak topology. We prove too th a t th e Co-sum of B anach spaces which have countable covers by sets of small local diam eter (resp. cr-fragm entability) has a countable cover by sets of small local diam eter (resp. cr-fragm entability), (th e (T-fragmentability case is Theorem 6.1 in [9]), and apply this to th e case of C { K ) spaces w ith th eir weak topologies w ith K = U{ür„ : n G N } , w here

K , K ij K2, ... are com pact spaces, when each {C (K n ),w e a k ) has either one of th e properties, [15].

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p articu lar cases: W CD spaces, duals of A splund spaces and C { K ) spaces w ith

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### S o m e d efin itio n s and rem arks

We s ta rt by giving two definitions introduced by Jayne, Namioka and Rogers which can be found in [9].

Let (X , r ) be a Hausdorff space and let p be a m etric on X not necessarily related to th e topology on X .

We define th e local g-diameter of a non-em pty subset A of X to be th e infim um of th e positive num bers e w ith th e property th a t each point x oi A

belongs to a non-em pty relatively open subset of A of g-diam eter less th an e.

D e fin itio n 1.1 .1 The space X is said to be a-fragmented by the metric g if, fo r each c > 0, it is possible to write

X = u t=i

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Chapter 1: a-{ragm entability 11

where each set X i has the property that each non-empty subset o f X{ has a

non-empty relatively open subset o f g-diameter less than e.

D e fin itio n 1. 1.2 We shall say that the fam ily B is a cover o f X by sets o f small local g-diameter i f each point o f X belongs to sets o f B having arbitrarily

small local g-diameter.

We shall usually say th a t X has a countable cover by sets of small local diam eter when b o th th e original topology and th e m etric on X are clearly distinguished and it does not create any confusion.

N ote th a t X has a countable cover by sets of small local diam eter, if and only if, for each g > 0, % can be expressed as a union

OO ^ = u

1=1

each non-em pty X i having th e property th a t each of its points belongs to some non-em pty relatively open subset of g-diam eter less th an e.

W hen X is a B anach space and the m etric is the norm m etric we shall talk of th e local diam eter of a set rath er th an its local norm -diam eter.

We shall give some more definitions which will be of interest later on in this chapter. T he definitions as well as some properties concerning th em can be found in [5, 6].

D efin ition 1.1.3

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C hapter 1: a -fragm entability 12

called e-discrete (or m etrically discrete w ith separating distance e) if

d { y i,y2) > £ whenever yi G A s , y2 G At and s ^ t .

ii) A fam ily A = of subsets of a topological space X is called discrete

if for each æ G X there exists an open neighbourhood of x which intersects at m ost one elem ent of th e family A .

iii) A fam ily A = of subsets of a topological space X is called a-discrete if it can be w ritten as a countable union of families each of which is discrete.

iv) A fam ily A = of subsets of a topological space X is called dis­ cretely a-decomposable (d. cr-d., for short) if for each m £ N there exists a discrete fam ily such th a t

A . = y] S t” ), for all a e 5.

m=l

R em ark 1.1.4

1) If A — is a discrete fam ily of subsets of a B anach space X , we see th a t it is cr-decomp os able into a countable set of m etrically discrete families as follows. Define

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C hapter 1: a-fragm entability 13

for all n > 1 and s G S. Then

### = Û

for all Æ 6 5.

n=l

And for each n G AT th e fam ily : 5 G 5 } is (^)-d iscrete in X.

2) We will m ake use of th e fact th a t in a m etric space if a fam ily is discrete in its union, then it is d. a-à. in the whole space.

3) It is also well known th a t in a m etric space any open cover of the space has a <7-discrete open refinem ent (see for exam ple [16], p. 234).

### cr-fragm entability o f a m etric sp a ce

Our first result shows th a t having a countable cover by sets of small local diam eter, although apparently stronger th an a-fragm entability, tu rn s out to be equivalent to th e la tte r when th e topology on the space is given by a m etric. The ideas in th e proof are from [9], Theorem 2.4. It reads as follows:

P r o p o s iti o n 1 . 2.1 Let ( X , d ) be a metric space and g he another metric de­ fined on X . The following conditions are equivalent:

i) ( X , d ) is a-fragmented by g;

ii) ( X , d ) has a countable cover by sets of small local g-diameter.

When the sets in i) can be taken to be differences o f d-closed sets (or more

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C hapter 1: a -fragm entability 14

P roof. ii)=^ i) Is clear by definition.

i)=^ ii) Given e > 0 there is a decom position of X given by th e c-fragm enta- bility of th e space

X = [ J C i.

1 = 1

Fix 2 Ç i\T. Because of th e cr-fragmentability, there exists a fam ily of d-open sets, : 0 < a < /x}, covering C, such th a t

Ci n [/• \ U and e-diam (Ci n % \ U U'^) < ^

0</9<a 0</3<a

for 0 < a < /X. Define

= {x e X : d{x, X \ U i ) > - } and = ( Q H \ U «7* ).

^ 0</3<a

It is clear th a t ^ -d ia m (if” J < e. Now ior (3 th e sets i f ” ,- and iZ^^, when non-em pty, are separated by d-distance at least i . So for each n G TV the fam ily { if ” j : 0 < a < /x} is discrete in (%, d).

Set i f ” = U { if” ,- : 0 < a < /x}. We have

### =

n=l n=l 0 < o t< n n=l 0<a</i 0</3<a

### = U (C,n % \ U I7*) = C',,

0<ot<fj. 0< j3 < a

and therefore

### X = u U

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C hapter 1: a-fragm entability 15

Let us see th a t for each n , i G N th e set i f " has local g-diam eter less th a n

e. Take x G if " . We have th a t for some a , x G if^^. Since th e fam ily

{ H ^ i : 0 < O' < ;/} is discrete in (X , d) th ere m ust be a d-open neighbourhood y of X such th a t

V n if ^ j = 0 for P ^ CL.

So

g -d iam (y fl i f f ) = g -d iam (y fl i f f J < e.

Now if th e Ci are differences of d-closed sets, and since th e are d- closed, we have th a t th e i f f are differences of d-closed sets. So each i f " is th e discrete union of sets which are differences of d-closed sets. Since d-open sets are iV -sets, it follows th a t th e i f f ’s are also f^-sets. ■

### lo c a l d ia m eter

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C hapter 1: a -fragm entability 16

T h e o r e m 1 .3 .1 Let { X , r ) he a topological space and g he a lower semicon­ tinuous metric on X . I f [ X , r ) has a countable cover hy sets o f small local

g-diameter, then { X , r ) has a countable cover hy differences of r-closed sets o f

small local g-diameter. Moreover, i f the g-topology is stronger than r , then the

sets can he taken to he g-closed.

P r o o f . Let e > 0 be any positive num ber. We shall show th a t X has a countable cover by differences of r-closed sets of local diam eter less th a n e.

Let { [/” : a E A , n N } be a cr-discrete refinement of a cover of X by balls of diam eter less th a n e. We can suppose th a t each {Cf” : a G A } is m etrically discrete w ith separating distances 6n > 0.

For n Ç: N let { C ^ : m G N } be a countable cover of X w ith local diam eter of each (7” < T hen for each n , m N th e fam ily {Î7” H C ^clç^a is discrete in (Each point in (7^ has a non-em pty relatively open neighbourhood w ith diam eter less th a n 8n- This open subset of C ” can intersect at m ost one of th e U2, n (7” ’s.)

For each œ G Î7” fl (7^ th ere exists a r-o p en neighbourhood of x, say such th a t

n ( U : nc z ) f 0 ,

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C hapter 1: a -fragm entability 17

Set

### o r = u

xg£/jnC“ is a r-o p en set w ith

### n C")

and

n ( a ; n c : ) = 0 for

Set = ( % n % ) n GS'” - It is clear th a t

^ = U U ( U K '" *

)-77% n a e A

We notice th a t, since g is r-low er semicontinuous and

diam(C7'” fl C ” ) < e,

we have

d ia m (% H % "") < £.

We now show th a t

U ( 1% n c : ] " n G D = U ( % n % ) ^ n U © r •

aG^ a 6^ otE^

Since fl (C/jg fl C ” ) = 0 for /5 ^ a , it is clear th a t

U [ ( % n G : ) " n G : - ] = U ( % n G : ) " n U G’c

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C hapter 1: a-fragm entability 18

Now, we always have

### " c U (% n %) '

aeA aeA

and so we m ust check th a t

### U (% n%)"n[U O

-aeA aeA aeA aeA

Set Ba = n and Ga = (j”’”' and suppose th a t th ere exists

such th a t

aeA aeA

aeA aeA

T hen, since

### X e U Ga,

aeA

th ere m ust be C/a;, a r-o p en neighbourhood of x, such th a t Ux C Gao for some

« 0 G A and

### % n ( U B .) # 0.

aeA

Thus Ux n Bao ^ 0, since Gao H = 0 for /? ao- Suppose now th a t there exists a r-o p en neighbourhood Vx of x such th a t

### =

0-Take 0 7^ C/x H 14 C Gao • Since U xC V x is a r-o p en neighbourhood of x and

### .

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Chapter 1: a-fragm entability 19

we have

### 0,

which implies th a t Ui

Vi

### n

B a ^ ^ 0. This is a contradiction. Now we show th a t for each n , m Ç: N th e set

### n(U G.)

cxE A O.Ç.A

has local diam eter less th a n e. Consider

® 6 U n ( U G„) = U ( ^ a " n Ga).

aeA aeA oc

Then th ere exists ao such th a t x G Gaa and

diam (

H

### ( (J

Ga) fl Gao) = diam (

### |J

{Ba fl Ga) fl Gao) =

aeA aeA aeA

= diam (5ao H Gao) < diam(Bao ’”) < 6.

So we have

^ = U U ( U K '™ ),

neN meN aeA

where for each n , m Ç: N th e set

U K ' ”'

aeA

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C hapter 1: a-fragm entability 20

As a corollary of th e previous result we have:

T h e o r e m 1 .3 .2 Let X he a Banach space. I f [X, w ea k) has a countable cover by sets o f small local norm-diameter, then { X ,w e a k ) has a countable cover by

differences o f weakly closed sets o f small local norm-diameter and also by norm

closed sets o f small local norm-diameter.

P r o o f . Note th a t by Theorem 1,3.1, for each e we have

OO

X = \ J { F i r \ G i ) ,

»=1

where th e Fi are closed and th e Gi are open in the weak topology, and therefore in th e norm topology. Now each Gi is a 1^-set, say Gi — : n G N } ,

w ith th e Hi^nS being closed sets. So we have

i=l n=l

and clearly for each i , n G N th e set Fi fl Hi^n has local diam eter less th a n e.l

### B o r e l se ts

Let ( X , II • II) be a Banach space. The norm || • || is called a Kadec norm if th e weak and norm topologies agree on {æ G % : ||cc|| = 1}. It was shown by E dgar th a t if a Banach space X adm its an equivalent K adec norm then:

i) B o r e l{ X , || • ||) = B o r e l{ X ,w e a k ), ([3], Theorem 1.1);

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C hapter 1: a-fragm entability 21

On th e other hand, Jayne, Namioka and Rogers proved, ([9], Theorem 2,3), th a t a B anach space th a t adm its an equivalent Kadec norm has a countable cover by sets th a t are differences of weakly closed sets of sm all local diam eter.

We improve E d g ar’s result by showing th a t a Banach space w ith th e JN R p ro p erty verifies ii) above. Moreover, any norm-closed subset of th e space is also a Borel subset of its bidual when considered w ith th e weak* topology, from which we get i) above as a corollary.

R e m a r k 1 .4 .0 It was proved in [9], p. 215, and [11] th a t a topological space (X , r ) which has a countable cover by differences of r-closed sets of sm all local p-diam eter, for some m etric g, any g-open set can be w ritten as a countable union of differences of r-closed sets and therefore th e Borel stru ctures for b o th the r-topology and the p-topology agree.

T h e o r e m 1 .4 .1 Let X be a Banach space and suppose that (^X^weak) has a countable cover by sets of small local diameter. Then X is the countable

intersection o f countable unions of differences o f w*-closed sets in X** and

therefore X G Borel(X**,w*). Moreover, any closed subset o f X is o f the same type.

P r o o f . Let n ^ o: G G N } be a cr-discrete refinem ent of a cover of

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C hapter 1: a-fragm en tab ility 22

For n , p G N let : m G N } be a countable cover of X w ith the local diam eter of each < 6n,p. Then for each n , m , p G N th e fam ily

{ ^ a , n ^ n , m } c c e A IS Weakly discrete in

U K » n C ^ , m ) ,

a e A

in fact it is discrete in (7^^. So for each x G H there exists a iy*-open neighbourhood of x in X**, say such th a t

### # 0,

n ( % . n = 0, for # a .

Set

_ I I T T n,m ,p

a,n U ^x,a

is clearly a u;*-open set with

### D ([/;,. n c ;,„ )

and

G” ? n n C ^ J = 0, for /3 5^ a .

Set = (tfS,„ n c i m ) n We show th a t

### ^a,Tn,n,p}’

p m n aeA

N otice th a t, since diam(C/J„ fl < -, we have

diam(Z7a,n fl Cn,m ) < - .

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C hapter 1: a-fragm entability 23

Let

x - e X - n f l U U U

p m n o te A

For each p Ç: N there exist n , m E N and a G A such th a t

SO th e re exists Xp E D such th a t ||cc** — ccp|| < Thus we have

• 11- lim x„ = X

' p -* o o p ~

and therefore x** G X .

We now show th a t

### u

( ( % " n c i m ) n =

### U

n CK.,n) n (J

ct£A ot£A a £ A

We have

U ( ( % " n CS,„) n G ^,j) = U (GJ,„ n CS,„) n U GT,?.

cxE A a £ A a £ A

since H „ D = 0, for P ^ a. So we m ust check th a t

U ( % " n CS,„) n [ U G” ^] c U ( % . n CS,„) n [ U

olÇA ot^A ol^A

Set B a — LI Gnjn and suppose th a t th ere exists

such th a t

U n [ U G„] O.Ç.A a £ A

X ^ u ” n [ U G„]

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C hapter 1: a -fragm entability 24

T h ere m ust be a it;*-open neighbourhood Ux of cc, since

X G [J Ga, aeA

such th a t Ux C Gaa for some œq G A and

aeA

T h us Ux n BaQ 7^ 0, since Ga^ fl 5 / 3 = 0 for /3 7^

ao-Suppose now th a t there exists a w*-open neighbourhood Vx oi x such th a t

Vx n Bao =

0-Choose Ux w ith 0 7^ C/x D C Gao ■ Since UxHVx is a tu*-open neighbourhood of X and

we have

### >

aeA

( % n i / ) n ( U a . ) f 0,

aeA

which im plies again th a t t/x fl 14 fl Bao 7^ 0- This is a contradiction. If 5 is a norm closed subset of X , consider the families

{ F n n c ^ J a e A

and follow th e proof. In this case th e vectors Xp^s belong to F and, since F is

closed, th e lim it also belongs to F.

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C hapter 1: a-fragm entability 25

C o r o lla r y 1 .4 .1 Let ( X, || • ||) 6e a Banach space with a countable cover by sets o f sm all local diameter, then B o re l{X , || • ||) = B o r e l[ X ,w e a k ).

P r o o f . Let A be a norm-closed subset of X . By T heorem 1,4.1, A G

B orel{X **,w eak*), i.e. there exists B G X**, B G B orel(X **,w eak*) and

A = B n X . B ut th e weak topology on X coincides w ith th e restriction to X

of th e weak* topology on X** and therefore A is a weak-Borel subset of X . ■ A t th is point we need to give another definition which can be found in [16], pages 345-346, as well as some basic properties.

D e f in itio n 1 .4 .1 Let Fq be th e fam ily of closed sets of a m etric space. Suppose th a t for an ordinal num ber a , we have defined the families for ^ < a. So th e sets of th e fam ily Fa are countable intersections or unions of sets belonging to F^ w ith ^ < a according to w hether a is even or odd (th e lim it ordinals are u nderstood to be even). It is known th a t the transfinite union of this families gives us th e fam ily of all Borel sets.

We can also do th e same using open sets. Set Go to be th e fam ily of open sets. Suppose th a t for an ordinal num ber a, we have defined th e families for ( < CK. So th e sets of th e fam ily Ga are countable unions or intersections of sets belonging to G^ w ith ( < according to w hether a is even or odd.

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C hapter 1: a-fragm en ta bility 26

a fam ily will be said to be of multiplicative class a. Similarly, th e families Fa

w ith odd indices as well as th e families Ga w ith even indices are additive and form th e additive class a.

O ur last result of this section is a generalization of the one m entioned above from [11]. T he ideas for th e proof also come from there. It will be used later on in C h apter 2.

L e m m a 1 .4 .1 Let ( X ,t) be a topological space such that any open set is an Ff^-set. Let g be a m etric on X and suppose that X has a countable cover by

sets o f additive class a o f sm all local g-diameter, then each g-open subset o f

X is o f additive class a..

P r o o f . Let G be a g-open subset of X . Let m > 1, be a countable cover of X by sets of additive class a of small local diam eter.

For n > 1 set

M (n ) = {m Ç: N \ m > l and Dm having local diam eter less th a n —}.

T hen for each n > 1 th e fam ily {Dm • G M ( n ) } covers X .

W rite

Gn = { x ^ X : {y e X : g{y, x) C G }}.

For each n > 1 and each m > 1 in M (n ), we consider th e points cc G Gn H

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Chapter 1: a-fragm entability 27

T hus X G U{x) C G. Hence, for m > 1 and m G M ( n ) th e set

Un,m = U{C/(œ) : x e GnH Dm}

is a relatively open subset of Dm- So for every n ,m ^ N there exist Vn,m,

T-open subsets of X , such that Un,m = Dm H V^,m- Now we have

OO

### = u

i = l

where th e sets are r-closed in X . Hence

OO

1 = 1

and therefore th e sets Un,m are of additive class a. They contain (7„ fl Dm and are contained in G. Hence

U{C/n,m : 71 > 1, 771 G M {ti)}

is a set of additive class a th a t coincides w ith G.

### D e c o m p o sitio n o f d isc r e te fa m ilies

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C hapter 1: a-fra gm enta b ility 28

P r o p o s i t i o n 1 .5 .1 Let { X , r ) be a topological space and g he a m etric on X . Then the follow ing conditions are equivalent:

i) { X ^t) has a countable cover by sets o f sm all local g-diameter;

ii) There exists a decomposition o f X ,

OO

X = \ J C i

i = l

such that i f A = is a g-discrete fa m ily o f subsets o f X , then fo r

each s Ç: S there is a decomposition

A . = \ j

1=1

with A \ C Ci such that each fam ily is discrete in

P r o o f . i)=> ii) For each n Ç: N , consider OO X = u c r ,

1 = 1

w here th e sets C ^ have local g-diam eter less th a n K

Let A = {A,}ag5 be a discrete fam ily in (X , g). Then there exist ^ -d iscrete fam ilies such th a t

A ,

### = IJ

AT", for s e S.

771 = 1

W rite {A*’^ } = A ^ H C ^ , for m , i E N and s Ç: S and fix z ,m G N . Take

X G Cf^. T h ere exists a T-open neighbourhood of x, say U, such th a t

d ia m fy n C f ) < — .

(30)

C hapter 1: a-fragm entability 29

Therefore U m eets at m ost one elem ent of th e family So is discrete in r ) and we have

### = U U^mncr =

U4'"-m=l m=l t=l t,m

Set = Fn and So we have

OO

A , = \ J B : , B : C Fn,

n=l

and is discrete in ( F n ,r ) .

ii)=> i) Given c > 0, let { [/” : a G F} be a a-discrete open refinement of an open cover of X by balls of radius less th an | . Let {Cm}m=:i a countable cover of X such th a t for n , m Ç: N ,

y n ^

### 0

^n,n. m=l

and is discrete in {Cm^T)- W rite

a er Obviously,

### % = U c .

n , m

We now show th a t for each G W, th e set has local diam eter less th a n e.

Take x £ F ^ . T hen x G B^'J^ and therefore there exists a T-open neigh­ bourhood U OÎ X such th a t

(31)

«o-Chapter 1: a-fragm entability 30

So diam(C/' D F ^ ) = diam(C/' fl < diam (j5”’”^) < e. ■ We give an analogous statem en t for th e case of the space being cr-frag-m ented. We ocr-frag-m it the proof because it follows th e sacr-frag-me line as th e previous one.

P r o p o s iti o n 1 .5 .2 Let ( X ^ r ) be a topological space and g be a m etric on X . Then the following conditions are equivalent:

i) (%, r ) is a-fragm ented by g;

ii) There exists a decomposition o f X ,

OO X = \ J C i ,

1 = 1

such that i f A = is a g-discrete fa m ily o f subsets o f X , then fo r

each s Ç: S there is a decomposition

### Ci

1= 1

such that each fam ily satisfies the following condition:

(32)

### In tr o d u c tio n

Suppose we have two B anach spaces X and Y , and a one-to-one m ap T :

X — > Y . If we assume th a t Y has a countable cover by sets of small local diam eter, w hat kind of condition do we have to im pose on T for X to have such a cover as well? W hat can we say about T ” ^? We shall give some answers to these questions in this chapter, b u t first of all we have to fix some notation.

If Ti and T2 are two topologies on a topological space üT, we shall say th a t Ti is stronger th a n Tg, denoted by Tg T%, if any T2-open subset of H is also Ti-open.

Let (X ,T i) and T2) be topological spaces and pi , p 2 be m etrics defined on X and y , respectively. If T : X — > Y is a one-to-one m ap, we define: Tr^

(33)

C hapter 2: O ne-to-one m aps 32

to be the topology on X given by the family {T~^( U) : U Tg-open} and to

be the topology on X associated to the metric T o Q2, i.e, the family

{ T~^{U) : U p2-open }. Note that if T is t i -T2 continuous (respectively gi -Q2

continuous), then (resp. Tg^ ■;< g i ) .

D e f in itio n 2 . 1.1 We shall say that a map T , as above, is discretely a-decom ­ posable, d. a-d. fo r short, fo r the pair (p i, P2) ^ 2^ is one-to-one and it m aps gi-discrete fam ilies o f subsets o f X into g2~d. a-d. fam ilies o f subsets o f Y .

These m aps were used by Hansell, see [6], where we refer for fu rth er proper­ ties, w ith o u t im posing on th e m ap th e condition of being one-to-one. It is easy to see from R em arks 1.1.4 th a t p2-d. cr-d. families in T { X ) are also ^2-d. a-d. i n Y .

### m ap s

We now prove a lem m a which will be used in Theorem 2.2.2 below.

L e m m a 2 . 2.1 L et { X, t i ) and (Y, T2) be topological spaces and gi, g2 be m etrics defined on X and Y , respectively. Suppose that there exists a d. a-d. map

T : X — y Y fo r the pair (g i, ^2)- (Y, T2) has a countable cover by sets o f sm all local g2-diam eter, then there exists a decomposition o f X ,

X = \ j E i ,

1 = 1

(34)

Chapter 2: One-to-one m aps 33

then there is a decomposition

OO

=

A i ,

### 4

C Ei,

t=l

such that each fa m ily ^ discrete in the space

(^Ei^Tr^)-P roof. Consider th e m ap T : X — > Y. T hen th e fam ily T A is d. cr-d. Therefore th ere exit discrete families p G such th a t

GO T A , = U

p=l

By Proposition 1.5.1 there exists a decom position of Y ,

Y = l ) C „ ,

n=l

and discrete families {B^'^}aç.s in th e space (Cn,T2), w ith

OO

•Bf = U BI"'", and B "'’’ C C„ for p e iV, a e S.

n=l

Since T is trivially T^^-Tg continuous it follows th a t th e family

T - ' ( { B n . e s )

is discrete in th e space ( T “ ^(7n, TT2). It is clear th a t

00 00 00

^ = U U = u

p=l n=l 1 = 1

(35)

Chapter 2: One-to-one m aps 34

We now give th e analogous lem m a for <j-fragmentability.

L e m m a 2 .2 .2 Let (X ,T i) and (y ,T 2) be topological spaces and let gi,Q2 be

m etrics defined on X and Y , respectively. Suppose that there exists a d. a-d.

map T : X — > Y fo r the pair (p i, ^2)- / / (Y, T2) is a-fragm ented by Q2, then

there exists a decomposition o f X,

00

X = \ J E , ,

i = l

such that i f A = is a gi-discrete fam ily o f subsets o f X , then fo r each

s E S , there is a decomposition

A s = \ J A ) , A \ C Ei , 1 = 1

such that each fa m ily satisfies the following condition:

fo r any i E N , i f : 5 € 5} ^ 0, then fo r every non-em pty subset A o f U{A* : 5 G 5 } there exists a r-open subset U o f X such that U f) A C A], fo r exactly one s E S .

T h e o r e m 2 .2 .1 Let (X ,T i) and (Y,T2) be topological spaces and let pi , P2 be m etrics defined on X and Y , respectively. Suppose that there exists a

one-to-one map T : X — > Y. Then the following conditions are equivalent:

i) T is d. a-d. fo r the pair (pi, ^2)/

(36)

C hapter 2: One-to-one m aps 35

iii) (X, Tg^) is a-fragm ented hy gi.

P roof. ii) Given e > 0, let {C/” : a G F} be a cr-discrete open refinem ent of an open cover of X by balls of pi-radius less th a n | .

For every n G N th e family { T (f/” )}aer is d. a-d. in Y . T hus we have OO

### = u

m=l

w ith {B2'”*}e<gr being gj-discrete.

Define E J ’"* = C U^. It is clear th a t

n,m a

We show th a t for any n , m Ç: N th e set

### K’”'

»er has local pi-diam eter less th an e.

So take x G Then Tcc G and so th ere exists a ^2-open set, say V , such th a t V fl = 0 for a œq. Nowtake G = T~^{V)j which is

Tgj-open, and we have

gi-diam (G fl ( \J = gi-diam {G fl E^ f ^ ) < ^i-diam(C/^’"") < e. aer

ii)<=> iii) Proposition 1.2.1.

ii)=> i) Because of Proposition 1.5.1, any pi-discrete family, A =

can be decomposed in th e following way: OO

A , = \ J

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C hapter 2: One-to-one m aps 36

w ith th e fam ily {A*}aG5 being T^j-discrete in its union. Since Tg^ is a m etric on X , it follows th a t these families are Tg^-d. cr-d, in X and so A is also

Tg^-d. cr-d. Hence th e m ap T is d. cr-d. for th e pair (pi, ^2)- ■

T h eorem 2 .2 .2 Let (X ,T i) and (Y,T2) he topological spaces and let gi,Q2 he

m etrics defined on X and Y , respectively, with T2 d 9 2- Suppose that there

exists a one-to-one map T : X> Y and that (Y, T2) has a countable cover hy sets o f sm all local Q2-diameter. Then the following conditions are equivalent:

i) T is d. cr-d. fo r the pair (gi, P2)/

ii) ( X, Tg^) has a countable cover hy sets o f small local gi-diam eter;

iii) (X, Tg^) is a-fragm ented hy gi;

iv) [ XjTt^) has a countable cover hy sets o f small local gi-diam eter;

v) (X , TV2 ) ^5 a-fragm ented hy g i.

P roof. i)=>iv) For each e > 0 and p Ç: N , let Ap = be discrete families in X such th a t U{Ap : p G N } is a refinement of a cover of X by balls of Pi-radius less th a n | .

By Lem m a 2.2.1 th ere exists a num ber of sets C j , w ith

### ^ = U

i=i and decompositions

A l = \ _ } A l ' ’ , a € S , p € N , A ^-^C lC j

(38)

C hapter 2: One-to-one m aps 37

such th a t each fam ily is discrete in {Cj^Tr^)’

Now define

B? = U aes It is clear th a t

OO OO

### X = u u

J=1 p-1

We show th a t th e fam ily is a countable cover of X by sets of sm all local gi-diam eter. Let x G Since the family is discrete in (C j’jTVj) th ere exists a Tr^ open neighbourhood y of æ such th a t

( y n C , ) n A r f 0,

for some sq ^ S and

(V n Cj) n = 0, for s ^ 5q.

If V n Cj n = 0 for every 5 G 5', we would have

y n c,- n ( U = 0, aes

and this would im ply th a t x ^ B j , which is a contradiction. So th ere exists 5q G 5 such th a t

V n C j H A ÿ ^ 0 and y

Cj

### n

A ^ / = 0, for 5 ^ So.

Now, since A^'J is contained in a ball of radius less th a n | , pi-diam(Af^^) < e.

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C hapter 2: One-to-one m aps 38

g i-d ia m (y fl B j ) = p i-d ia m (y fl ( | J = g i-d ia m (y D A ^ f ) < e. ses

iv)=>v) Obvious.

v)=4>iii) Obvious. ■

T h eo rem 2.2.3 Let (X, Ti) and (T,T2) he topological spaces and let g i , g2 he

m etrics defined on X and Y , respectively, with T2 p2- Suppose that there

exists a one-to-one map T : X> Y and that (Y, T2) is a-fragm ented hy p2-

Then the following conditions are equivalent:

i) T is d. a-d. fo r the pair (gi, ^2);

ii) (X , Tgj) has a countable cover hy sets of small local gi-diameter;

iii) (X , Tpj) is a-fragm ented hy gi;

iv) (X , Tr2 ) is a-fragm ented hy g i.

P ro o f. i)=^iv) The same proof as above bu t using Lem m a 2.2.2.

iv)=>iii) Obvious. ■

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Chapter 2: One-to-one m aps 39

### I n v e r s e m a p p in g s

We sta rt by recalling a definition from [16].

D efin ition 2.3.1 A mapping f : X — > Y is said to be o f Borel class a if, fo r every closed subset F g Y , the set f ~ ^ [ F ) is Borel o f multiplicative class c l.

(Equivalently, f ~^ { G) is o f additive class a, fo r every open set G).

O ur next result improves p a rt of a result in [14], Corollary 7.

T h eorem 2.3.1 Let X , Y be two Banach spaces and f : X — > Y be a con­ tinuous linear injection. Define ip = f~ ^ : /( % ) — > X . Then the following conditions are equivalent:

i) p is o f Borel class a and f is d. a-d.;

ii) (X ,/||.||y ) has a countable cover by sets o f additive class ex. (for the topol-ogy fw-Wy) o f small local || • \\x~diameter.

P roof. ii)=> i) By Theorem 2.2.2 we have th a t / is d. <7-d.

Now let G he a norm open subset of X . Since th e topology /||.||y in X

verifies th a t any open set is an F^ set, we apply Lem m a 1.4.1 and ob tain th a t

G is of additive class a in (X , /||.||y)- Hence the set p~^(G) is of additive class

a in ( / ( X ) , II • ||y ) and therefore p is of Borel class a.

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C hapter 2: One-to-one m aps 40

discrete families {B ^^}aeA such th a t OO

### b ; = u

Bir-t=i

N ote th a t since th e sets C/” are open in X and ip is of Borel class a, th e sets B 2 = are of additive class a.

Now fix 2, n G N . T he fam ily H B2}aeA is discrete in f { X ) and its sets are of additive class a.

Set

### B2).

olE A

T h en is a discrete union of sets of additive class a and therefore is itself of additive class a. (For a proof of this fact see [16], page 358, T heorem 1). It is obvious th a t

f ( X ) = U H .> . i , n

Define The sets Ci^^s are of additive class a and they form a countable cover of (%, /||.||y).

Now tak e x G and f { x ) = y E So y E Ba^ fl B^^ for only one 0=0 G A. Thus th ere exists an open neighbourhood U oi y in Y such th a t

U n { B T n 5 ” ) = 0 for a ^ qq.

W rite V = f ~^{U). V is a /||.||y-open neighbourhood of x and

d ia m (y n Ci,n) = d i a m ( / - '( B fl H .>)) = d ia m (/- ^ (B n n <

(42)

Chapter 2: One-to-one m aps 41

Therefore Ci^n has local diam eter less th an e.

### S o m e to p o lo g ic a l in v a r ia n ts

The following lem m a provides us w ith a useful tool to check when some m aps are d. cr-d. For a proof of it see [17].

L em m a 2.4.1 Let X and Y be Banach spaces with norm s || • \\x cind || • ||y,

respectively. Let T : X — > Y he a one-to-one map such that fo r every hounded

sequence in X converging to some point x in the T||.||y topology we have that X n converges weakly (or pointwise in the case o f X being a C (K ) space)

to X . Then T is d. a-d.

T h eorem 2.4.1 Let (%, || • \\x) and (Y", || • ||y) he Banach spaces. Suppose that there exists an hom eom orphism (f) : [ X , w e a k ) — ^ (Y^weak). Then the following results hold.

i) ( X j We a k ) has a countable cover hy sets o f small local || • \\x~diameter i f and only i f [Y, weak) has a countable cover hy sets o f small local || • ||y -

diameter.

ii) (X , lyeaA;) is a-fragm ented hy || • \\x i f and only i f (Y, weak) is a-fragm en­ ted hy II | | y .

iii) B o r e l { X , w e a k ) = B o r e l(X ,\\ • ||x ) i f and only i f B orel{Y , weak ) =

(43)

C hapter 2: One-to-one m aps 42

P r o o f . In this case it is clear th a t th e conditions in Lem m a 2.4.1 are fullfiled and so our m ap (f) and its inverse are d. a-d. Moreover, since there is weak to weak continuity, by R em ark 2.2.1 and Theorem s 2.2.2 and 2.2.3, i) and ii) hold. To prove iii) assume th a t B orel(Y ^w eak) = B orel{Y , || • ||y).

D enote by R(x,||.||;^) th e closed un it ball of {X , || • ||%). B(^x,\\-\\x) is a

w-closed subset of X and, since ÿ is a hom eom orphism , ÿ(jB(_y,||.||^)) is a iw-closed subset of Y . Thus th e norm || • ||% is lower semi continuous on (X , ^||.||y). Since

(j) is d. a-d ., (X , ^||.||y) has a countable cover by differences of ÿ||.||y-closed sets of sm all local || • ||%-diameter. So if G is a || • ||%-open subset of X , then, see R em ark 1.4.0,

### G = U C"

1 = 1

where Ci is th e difference of two <ÿ||.||y-closed sets for every % E X .

Set Bi = <!>{Ci). T hen th e B {S are differences of || • ||y-closed sets and therefore th ey are w-Borel sets. Thus ^~^[Bi) = Ci are ly-Borel in X . We conclude th a t G is a countable union of weak-Bore\ sets and therefore is itself

a w eak-B orel sub et of X . ■

### T h e Co-sum o f so m e B a n a c h s p a c e s

N o ta tio n : Let A he a set. We shall denote hy \A\ the cardinality o f the set A .

(44)

C hapter 2: One-to-one m aps 43

norm s {|| - Un : G N } . We denote hy co{X„ : n G N } the Banach space o f all sequences x = with Xn G X n , fo r n £ N , and such that given c > 0

there exists m E N such that ||aJ„||„ < e fo r n > m . The norm in this space is

In order to give our next result we need th e next proposition. The proof of p a rt i) can be found in [9], Theorem 6.1.

T h eorem 2.5.1 Let {Xn, || • ||n )^ i a sequence o f Banach spaces. Set X =

co{Xn : n G N } . Then the following results hold.

i) I f fo r each n E N the space X n is a-fragm ented hy its norm, then the space X is a-fragm ented hy its norm.

ii) I f fo r each n E N the space X n has a countable cover hy sets o f small local

### II

' \\n~diameter, then X has a countable cover hy sets o f sm all local

### \\oo-

diameter.

P roof. For each n G iV, we can find a sequence Bn = {^m)m=i subsets of X n such th a t each point of belongs to sets of Bn having arb itrarily small local II • 11 n-diam eter. For r , k E N define

C* = { x = {xn)Z=i G X : \{xn : ||a;„||n > ^ } | = r}.

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C hapter 2: One-to-one m aps 44

For r > 1, A: > 1 define

^ n n ... n

for m.}< — ^771 j j ... j TTij*^ j dj. — ^cîj j ... ^ dj- ^ &nd

■^o,k — G X ; ||x|[oo ^ ^}*

I t ’s clear th a t

OO OO OO

### 4T“'U(U^0,.).

k=l r=l d^e{l r}^mrE{ji, ^=1

So we have a countable collection of sets covering X and now we will show th a t given x = G X and e > 0 th ere exist r, k, d^, n ir such th a t

X e

and th e local || • ||oo-diam(i4™j’’*’') < e.

Take k E: N such th a t ^ < f- T hen either ||x||oo < ^ or th ere exists

r £ N such th a t x ^ In th e first case, x G Ao^k and obviously th e local II • 11 OO-diameter of Ao^k is less or equal th a n e.

So suppose th a t there exists r E N such th a t x G (7*, i.e., ||cci||,- > \ for % — d\ ,..., dj- « Let

r • r ll®»l|t ~ k ^ J J 1

0 = r m n { — - , - : z = c ti,..., dr}.

Now for i G {d%,..., dr} we have th a t X{ G X{. Thus we can find m*- G N and

(46)

C hapter 2: One-to-one m aps 45

open neighbourhood of X{ in X{, say Vi, such th a t

II • ||i-diam(K- n < 5.

Set

P r \ V i ) .

i = l

y is a weak open neighbourhood of x in X . We show th a t

II • ||oo-diam (y PI < e.

Take y Cl For i = d i , d r ,

l|y%||: > llk :||i - ll^i -2 /i||i| > p

SO for i 0 {di, ...,d r}, ||i/i||i < Therefore

||®t- - 2/i||i < < I for z = d i , dr.

and

1 1 2 e

ll^i - 2/*||i = - otherwise.

Hence || • ||oo-diam (y fl < e as required. ■

Following th e notatio n in Theorem 2.5.2 below, in [15], Kenderov and Moors showed th a t if th e spaces {C{Kn) , pt wi se) are or-fragmentable, th en

{ C { K ) , p t w i s e ) is also cr-fragmentable. We prove it for th e weak topology and also in th e case of countable covers by sets of small local diam eter.

(47)

Chapter 2: One-to-one m aps 46

i) I f fo r each n ^ N the space { C{ Kn) , we a k ) is a-fragmentable, then the space ( C { K ) , w e a k ) is a-fragmentable.

ii) I f fo r each n ^ N the space [C{Kn)j Weak) has a countable cover by sets o f sm all local diameter, then the space {C i^K \w ea k') has a countable

cover by sets o f small local diameter.

P r o o f . Define the m ap

T : C { K ) — co(C (ir„), II • H o c )

by th e form ula

### T(/) =

1.

By Theorem 2.5.1 and R em ark 2.2.1, since T is clearly weak to weak con­ tinuous, we only have to show th a t T is d. cj-d.

So take { f m ) m = u f ^ C'(R') and suppose th a t {T{f m) ) converges to T { f )

in th e norm of cq, i.e., given e > 0 there exists m E N such th a t for all Â: > m we have

m f k ) - T ( / ) |U < e,

I.e..

\\—f k \ K n f\Kn\\oo ^ £ for all n E N .

n n

Now, \{ X E K , th ere exists n E N such th a t x E K n , and so

\—f k M ~ ~ / ( ^ ) l — ^ for all k > m .

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C hapter 2: One-to-one m aps 47

Thus fjn converges to / in th e pointwise topology and by L em m a 2.4.1 we

(49)

### by sets o f sm all local diam eter

We shall give some exam ples of spaces w ith countable covers by sets of sm all local diam eter as well as some w ith well behaved injections into Co(r) for some set

### T h e s p a c e co(r)

D e fin itio n 3 .1 .1 Let V be a set. We define co(F ) to he the set

c o (r) = { x Ç: BF : fo r a// £ > 0 |{i G F : |æ(i)| > s}| < oo}.

When endowed with the swpremum norm, co(F ) is a Banach space.

O ur next proposition is based on L em m a 3.2 from [22]. It also follows from

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C hapter 3: Spaces w ith a countable cover by sets o f sm all local diam eter 49

[9], Theorem 2.1 b), since Co(T) has an equivalent locally uniform ly convex norm and on bounded sets in co(T) th e weak and pointwise topologies coincide. P r o p o s iti o n 3 .1 .2 {cq{T), p oin tw ise) has a countable cover hy differences o f pointwise closed sets o f small local diameter.

P r o o f . Given e > 0, for n G iV w ith ^ < | and k Ç: N we pu t

= {y ^ Co(r) : |{t E r : \yt\ > - } | = k}. n

We show th a t is the intersection of an open and a closed set in th e pointwise topology. Set

. . . , CLm) = {2: G Co(r) : > - , 2 = 1, . . . ,m } , a{ G T.

and

6

### r»"

is open and clearly

We now show th a t th e sets B ^ have local diam eter less or equal th a n e. Let y' G and let {t G F : \y[\ > ^} = { t i , . . . ,tk}- T hen there exists 0 < (^ < I such th a t \y’f..| — <^ > F ,% = 1, . . . , A;. The set

2

### \yt^ ~ Vth\

is an open neighbourhood of y ' in Bk in th e pointwise topology. Take x G U. T h en for 2 = 1, . . . , A;

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Chapter 3: Spaces w ith a countable cover by sets o f sm all local diam eter 50

A nd since x G and \xn\ > = 1, . . . , A:, we have \xt\ < ^ i o r t ^ ti.

So take x , y £ U. L et’s show th a t ||a; — y He» < For z G { 1 , . . . , A;},

\^u — Vti\ < \^ti ~ 2 / t J + \Vti “ 2/ t i l ^ S -h S = 26 < e.

A nd for t G F w ith t ^ ti,

1 1 2

\xt - yt\ < \xt\ + jytj < - + - = - < £ .

n n n

### P r o je c tio n a l R e s o lu tio n s o f I d e n tity

D e fin itio n 3 .2 .1 Let X be a Banach space. The density character o f X,

denoted by dens(X), is th e smallest cardinal num ber of a dense subset of X .

D e fin itio n 3 .2 .2 Let X be a Banach Space. We denote by p the smallest ordinal such th a t its cardinality \p\ = d en s(X ). A projectional resolution o f identity, P R Iio r short, on X is a collection {P« : ujq < a < p \ oi projections

from X into X th a t satisfy, for every a w ith (jUq < a < p, th e following conditions:

i) l|Fa|| = l;

ii) PaoP/3 = P/goPa = FL if Wo < a < < ft;

(52)

Chapter 3: Spaces w ith a countable cover by sets o f sm all local dhmet(er 51

iv) U{-f/3+ i(-^) : CÜQ < /3 < a } is norm dense in

v) = I d x .

T he following lem m a lists some properties of th e P R F s tha; we will need later on.

L e m m a 3 .2 .1 Let X be a Banach space and {Pa : ujq < a < p- h a P R f on X . We pu t Pa+i — Pa = Ta> foT ljq < CL < p. Then the followinj r^.sults hold.

i) For every x G X , i f a is a lim it ordinal, ojq < a < p, we havi

Pa{x) = II • II — lim f^ (æ ).

/ 3 < a

ii) For every x £ X , {||Th(a:)|| • ol G [ujq, p )} belongs to cq{[ljo,p)).

P r o o f , i) Let a be a lim it ordinal, < cl < p, and let x £ ï . l î

^ e U

say X £ P ^ q ( X ) , th en Pfi{x) = x i f j3 > ^ q . Thus, in this case, wehcve

II • II — lim Pj3{x) = X = Pa(aj). (1)

If X G P a { X ) , by (iv) in Definition 3.2.2, given e > 0 there exsti

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C hapter 3: Spaces w ith a countable cover b y sets o f sm all local diam eter 52

such th a t 11 a; — y \\ < | . For this c > 0 there m ust he f3o < a such th a t for any

l3, /3o < /3 < a , we have

Thus

I l f . W - P;3(x)|| < ||P „(x ) - f .W I I + ||P „(ÿ ) - P0(y)\\ + ||P^(ÿ) - Pp{x)\\ <

< ll-P a lllk -y ll + - f . ( y ) || + ||-P^III|œ -ÿ|| < E,

whenever (3 > po.

Finally for x £ X and P < a we have

P^{x) - Pa(x) = Pfi{Pcc{x)) - Pcc{x)

and thus (1) holds for every x Ç: X .

ii) If th e assertion is false, there exists æo G X , £ > 0, and

Wo < 0=1 < #2 < . . . <

such th a t for every i > 1, ||7hX^o)|| > Set

a = sup{«i : i > 1}.

It is clear th a t

lim Pfi{xo) ^ Poc{xo)y

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Chapter 3: Spaces w ith a countable cover by sets o f sm all local diam eter 53

### d. cr-d. m aps into co(F) using P R I

The next theorem is a straightforw ard ad ap atatio n to th e case of P R F s from th a t in [22].

T h e o r e m 3 .2 . 1. 1 Let p be a class o f Banach spaces such that the following results hold:

i) fo r every X G p there exists a P R I on X , {Pa : ujq < a < p }.

ii) (P„+i - P .)(% ) 6 p.

Then fo r every X G p there exists a one-to-one hounded linear map

T : X ^ co(r),

fo r some set T, such that fo r every discrete fam ily A = o f subsets o f X , the fa m ily T A = { T Aa]aç.s is d. a-d. in co(r).

P r o o f . Let X G p. We proceed by induction on densi^X).

W hen X is separable, we have th a t is m etrizable and separable. So let { fn : n > 1} be a dense subset of {Bx*)W*) and define

T : X ^ c o ( i V ) b y r ( x ) = . / n = l

T is clearly a linear m ap, and it is one-to-one because {fn)^=i is dense in

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C hapter 3: Spaces with a countable cover by sets o f sm all local diam eter 54

Let A = be a discrete family in X and let be an open base for th e topology on X . For each 5 G *?, pick a, G Ag. T hen th ere exists

Ug^ open neighbourhood of such th a t U g r\A t — 0, for t ^ s. Now for each 5 G 5 th ere exists G AT such th a t Ug G Bn^ C Ug. Hence |6'| < |JV| and therefore S m ust be countable.

So let A = {An}neiv^ be a discrete family and define

£(n) ^ and g W = 0, for n 5^ m

Then for every m & N th e fam ily is discrete and oo

T{A n) = I J for every n e N .

771 = 1

So th e fam ily T A is d. <7-d.

Let X he an uncountable cardinal and X G p such th a t d e n s (X ) = %. Suppose th a t th e result is tru e for every V £ p w ith d e n s (Y ) < %. Let p be th e sm allest ordinal w ith cardinality \p\ = %.

Let {Pa : LJo < a < p } he a P R I on X . For any ol^ Uq < a < p, we set

X a = {Bct+i — P a){X ). T hen X a G p and dens{X a) < |o:| < d e n s{X ). Thus there exist sets Fq and one-to-one continuous linear m aps

Jcc'-Xa ----^ Co(Fa)

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Chapter 3: Spaces with a countable cover b y sets o f sm all local diam eter 55

Since is separable, there is also Jq : Puq[ X ) — > Cq[N ) sharing these properties. Set

v = N c U

and define T : X — > ^oo(r) by

T {x ){n ) = Jo (fL o (^ ))W for n ^ N , and

T (x )(7 ) = ij« (P « + i(a j) - fL (æ ))(7 ) for 7 E

T is clearly a linear m ap and is continuous. We have

||r x || = sup ||ij„ (P « + i(œ ) - <

UQ<OL<H ^

< \ sup ||J a ||( ||P a + l(a ;)- P a (a j)||) < ^ SUp ( ||Pa+i || || CC || + || P^ || || æ || ) < ^ UQ<ot<^x ^ wo<a<fi

< ^ sup (||æ|| + ||æ||) = ||æ||.

We show th a t T { X ) C co(r). By Lem m a 3.2.1, ii), given x E X th e set

{||^a(a:)|| : « G [wo, /^)}

belongs to Co([wo,//)), where Ta = Pa+i —

Pa-So, given any e > 0, there exist a i , . . . , a „ G [wo, p) such th a t

||Tai(a;)|| > e and ||7]g(æ)|| < e for /5 0!i,z = 1, . . . , n.

For each i E {1, . . . ,n } , J a X ^ X ^ ) ) ^ co(FaJ. Hence th ere exist 71, . . . , 7m, G

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C hapter 3: Spaces w ith a countable cover b y sets o f sm all local diam eter 56

Now we prove th a t T is an injection. Take x Ç: X and suppose th a t T(æ) = 0. We show th a t P^{x) = 0 for loq < ( < p.

For ( = Wo, we have Jo{P^^q{x)) = 0 and, since Jq is an injection, we

conclude th a t P^oi^) = 0.

Fix ljq < ( < p, and suppose th a t = 0 for Wo < a < (. If ( is a

non-lim it ordinal, then ^ = a + 1 for some a < (. Now

- Pcc{x)) = 0 = > P^{x) = 0,

because Pa{x) = 0 and is an injection. If ^ is a lim it ordinal, we have

P^(x) = lim Pot(a3) = 0.

For C — Pi since p is a lim it ordinal, we have

P^i{x) = X = lim P a (x ) = 0,

hence x = 0 and so T is an injection.

Now we introduce some notation. Set Ya = Pa{X) for ujq < a < p and

Zcx = {y ^ co(F) :yt = 0{oT t e | J F J .

a<C<H

Define ha : co(F) — > Za C co(F) by

0 for t G U{F( : a < ^ < p}, K { y ) = y' w ith y[ = <

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