<<

. 26
( 27)



>>

Now ∆ := k ∆k is the required set. In order to show that Q∆ (K) ⊆ K let x ∈ K
˜
be arbitrary. Since Q∆k (x) is contained in the closure of Q∆k ({xk ∈ K © E :

53.17
592 53. Appendix: Projective resolutions of identity on Banach spaces 53.19

˜
carr(xk ) ⊆ ∆k+1 }) and hence in the closed set Q∆k ({xk ∈ K : carr(xk ) ⊆ “}).
˜
Thus there is an xk ∈ K with carr(xk ) ⊆ “ and such that x agrees with xk on ∆k .
˜
Thus K xk ’ Q∆ (x), since every ¬nite subset of ∆ is contained in some ∆k and
˜
˜
outside ∆ all xk and Q∆ (x) are zero. Since K is closed we get Q∆ (x) ∈ K.
˜ ˜

Without loss of generality we may assume that µ > ω. Let {x± : ω ¤ ± < µ} be a
dense subset of K © E. Let “ω := carr(xω ). By trans¬nite induction we de¬ne

(“β ∪ carr(xβ ))∼ for ± = β + 1,
“± :=
“β for limit ordinals ±.
β<±

Then the “± satisfy all the requirements.

53.18. Corollary. Let K be Valdivia compact. Then C(K) has a PRI.

Proof. We choose “± as in (53.17) and set K± := Q“± (K). Let Q± := Q“± |K .
Then Q± is a continuous retraction.

wK

! IdK±
g
e
g
ee


±

 eeeQ

incl ±

K



uU
R
U
Id
R    u

C(K± ) C(K± )
R
incl R  Q
£
 


±

C(K) u |E ±

We have dens(C(R“± )) = |±|, since we have a base of the topology of this space of
that cardinality. Hence dens(C(K± )) ¤ |±|. Let E± := (Q± )— (C(K± )). Then E± is
a closed subspace of C(K) and (53.13.3) holds. Furthermore P± := Q± —¦ incl— ± is
K
a norm-1 projection from C(K) to E± . The inclusion “± ⊆ “β for ± ¤ β implies
(53.13.1). To see (53.13.6) and (53.13.5) let µ > 0 and choose a ¬nite covering of
K± by sets
Uj := {x ∈ R“± : |xγ ’ xj | < δj for all γ ∈ ∆j },
γ

where xj ∈ R“± , δj > 0 and ∆j ⊆ “± is ¬nite and such that for x , x ∈ Uj © K
we have |f (x ) ’ f (x )| < µ. Now choose ±0 < ± such that “±0 ⊇ ∆j for all of the
¬nitely many j. Since the Uj cover K± , we have x ∈ K± © Uj for some j and hence
Qβ (x) ∈ K± © Uj for all ±0 ¤ β < ±. Hence |f (x) ’ f (Qβ (x))| < µ for all x ∈ K±
and so P± (f ) ’ Pβ (f ) = (1 ’ Pβ )P± (f ) ¤ µ. Thus we have shown that E has
a PRI (P± )± , with all E± ∼ C(K± ) and dens(K± ) ¤ |“± | ¤ ±.
=

53.19. Remark. The space C([0, ±]) has a PRI given by

for µ ¤ β
f (µ)
Pβ (f )(µ) := .
for µ ≥ β
f (β)

53.19
53.20 53. Appendix: Projective resolutions of identity on Banach spaces 593

However, there is no PRI on the hyperplane E := {f ∈ C([0, ω1 ]) : f (ω1 ) = 0} of
the space C[0, ω1 ]. And, in particular, C[0, ω1 ] is not WCD.

Proof. Assume {P± : ω ¤ ± ¤ ω1 } is a PRI on E. Put ±0 := ω0 . We may ¬nd
β0 < ω1 with
P±0 E ⊆ Eβ0 := {f ∈ E : f (±) = 0 for ± > β0 },
because for each f in dense countable subset D ⊆ P±0 E we ¬nd a βf with f (±) = 0
for ± ≥ βf . Since Eβ0 is separable, there is an ±0 < ±1 < ω1 such that

Eβ0 ⊆ P±1 E,

in fact D ⊆ Eβ0 is dense and hence for each f ∈ D and n ∈ N there exists an
˜ ˜
±f,n < ω1 and f ∈ P±f,n E such that f ’ f ¤ 1/n. Then ±1 := sup{±f,n : n ∈
N, f ∈ D} ful¬lls the requirements.
Now we proceed by induction. Let ±∞ := supn ±n and β∞ := supn βn . Then

P±n E = Fβ∞ := {f ∈ E : f (±) = 0 for ± ≥ β∞ }.
P±∞ E =
n

But Fβ∞ is not the image of a norm-1 projection: Suppose P were a norm-1 pro-
jection on Fβ∞ . Let π : E ’ C(X) be the restriction map, where X := [0, β∞ ].
˜ ˜
It is left inverse to the inclusion ι given by f ’ f with f (γ) = 0 for γ ≥ β∞ .
˜ ˜
Let P := π —¦ P —¦ ι ∈ L(C(X)). Then P is a norm-1 projection with image
˜
Cβ∞ (X) := {f ∈ C[0, β∞ ] : f (β∞ ) = 0}. Then C(X) = ker(P ) • Cβ∞ (X).
˜ /˜
We pick 0 = f0 ∈ ker(P ). Since f0 ∈ P (C(X)) = Cβ∞ (X) = ker(evβ∞ ), we
have f0 (β∞ ) = 0, and without loss of generality we may assume that f0 (β∞ ) = 1.
˜ ˜
For f ∈ C(X) we have that f ’ P (f ) ∈ ker P and hence there is a »f ∈ R
˜
with f ’ P (f ) = »f f0 . In fact evaluating at β∞ gives f (β∞ ) ’ 0 = »f 1, hence
˜
P (f ) = f ’ f (β∞ ) f0 . Since β∞ is a limit point, there is for each µ > 0 a xµ < β∞
with f0 (xµ ) > 1 ’ µ. Now choose fµ ∈ C(X) with fµ = 1 = ’fµ (β∞ ) = fµ (xµ ).
Then

= fµ ’ fµ (β∞ ) f0
P fµ ∞ ∞

≥ |fµ (xµ ) ’ fµ (β∞ ) f0 (xµ )|
≥ 1 + 1(1 ’ µ) = 2 ’ µ.

˜
Hence P ≥ 2, a contradiction.
Note however that every separable subspace is contained in a 1-complemented sep-
arable subspace.

53.20. Theorem. [Bistr¨m, 1993, 3.16] If E is a realcompact (i.e. non-measurab-
o
le) Banach space admitting a SPRI, then there is a non-measurable set “ and a
injective continuous linear operator T : E ’ c0 (“).

Proof. We proof by trans¬nite induction that for every ordinal ± with ± ¤ µ :=
dens(E) there is a non-measurable set “± and an injective linear operator T± :

53.20
594 53. Appendix: Projective resolutions of identity on Banach spaces 53.21

E± := P± (E) ’ c0 (“± ) with T± ¤ 1.
Note that if E is separable, then there are x— ∈ E with x— ¤ 1, and which are
n n
σ(E , E) dense in the unit-ball of E . Then T : E ’ c0 (N), de¬ned by T (x)n :=
1—
n xn (x), satis¬es the requirements: It is obviously a continuous linear mapping into
c0 , and it remains to show that it is injective. So let x = 0. By Hahn-Banach
there is a x— ∈ E with x— (x) = x and x— ¤ 1. Hence there is some n with
|(x— ’ x— )(x)| < x and hence x— (x) = 0.
n n
In particular we have Tω0 : Eω0 ’ c0 (“ω0 ).
For successor ordinals ± + 1 we have E±+1 ∼ E± — (E±+1 /E± ) = E± — (P±+1 ’
=
P± )(E). Let R± := (P±+1 ’ P± )/ P±+1 ’ P± , let F := (P±+1 ’ P± )(E) and let
T : F ’ c0 be the continuous injection for the, by (53.13.7), separable space F
with T ¤ 1. Then we de¬ne “±+1 := “± N and T±+1 : E±+1 ’ c0 (“±+1 ) by

T± ( P± (x) )γ for γ ∈ G±

T±+1 (x)γ := .
for γ ∈ N
T (R± (x))γ
Now let ± be a limit ordinal. We set

“± := “ω “β+1 ,
ω¤β<±

and de¬ne T± : E± := P± (E) ’ c0 (“± ) by

Tω ( Pω (x) ) for γ ∈ “ω

T± (x)γ :=
for γ ∈ “β+1
Tβ+1 (Rβ (x))γ
We show ¬rst that T± (x) ∈ c0 (“± ) for all x ∈ E. So let µ > 0. Then the set
{β : Rβ (x) ≥ µ, β < ±} is ¬nite by (53.13.8).
Obviously T± is linear and T± ¤ 1. It is also injective: In fact let T± (x) = 0
for some x ∈ E± . Then Rβ (x) = 0 for all β < ± and Pω (x) = 0, hence by
x = P± (x) = 0.
As card(E) is non-measurable, also the smaller cardinal dens(E) is non-measur-
able. Thus the union “± of non-measurable sets over a non-measurable index set
is non-measurable.

53.21. Corollary. The WCD Banach spaces and the duals of Asplund spaces
continuously and linearly inject into some c0 (“). The same is true for C(K), where
K is Valdivia compact.

For WCG spaces this is due to [Amir, Lindenstrauss, 1968] and for C(K) with K
Valdivia compact it is due to [Argyros, Mercourakis, Negrepontis, 1988.]

Proof. For WCD and duals of Asplund spaces this follows using (53.15). For
Valdivia compact spaces K one proceeds by induction on dens(K) and uses the
PRI constructed in (53.18). The continuous linear injection C(K) ’ c0 (“) is then
given as in (53.20) for ± := dens(K), where Tβ exists for β < ±, since Eβ ∼ C(Kβ )
=
with Kβ Valdivia compact and dens(Kβ ) ¤ β < ±.


53.21
53.22 53. Appendix: Projective resolutions of identity on Banach spaces 595

53.22. Theorem. [Bartle, Graves, 1952] Let T : E ’ F be a bounded linear
surjective mapping between Banach spaces. Then there exists a continuous mapping
S : F ’ E with T —¦ S = Id.

Proof. By the open mapping theorem there is a constant M0 > 0 such that for all
y ¤ 1 there exists an x ∈ T ’1 (y) with y ¤ M0 . In fact there is an M0 such
that B1/M0 ⊆ T (B1 ) or equivalently B1 ⊆ T (BM0 ). Let (fγ )γ∈“ be a continuous
partition of unity on oF := {y ∈ F : y ¤ 1} with diam(supp(fγ )) ¤ 1/2. Choose
xγ ∈ T ’1 (carr(fγ )) with xγ ¤ M0 and for y ¤ 1 set

S0 y := fγ (y)xγ and recursively
γ∈“
1
Sn (an (y ’ T Sn y)),
Sn+1 y := Sn y +
an
n
where an := 22 .
By induction we show that the continuous mappings Sn : {y : y ¤ 1} ’ E satisfy
n’1
y ’ T Sn y ¤ 1/an and Sn y ¤ Mn := M0 · k=0 (1 + 1/ak ).
(n = 0) Obviously S0 y ¤ fγ (y) xγ ¤ M0 and
γ

1
y ’ T S0 y = fγ (y)(y ’ T xγ ) ¤ fγ (y) y ’ T xγ ¤ = a0 ,
2
γ γ∈“y


where “y := {γ ∈ “ : fγ (y) = 0}.
(n + 1) For y ¤ 1 and yn := an (y ’ T Sn y) we have yn ¤ 1 by induction
hypothesis. Then

1 1
Sn+1 y ¤ Sn y + S n y n ¤ Mn + Mn = Mn+1 .
an an

Furthermore
1
y ’ T Sn+1 y = y ’ T Sn y ’ T Sn (an (y ’ T Sn y))
an
1 1 1 1
¤ y n ’ T S n yn ¤ · = .
an an an an+1

Now (Sn ) is Cauchy with respect to uniform convergence on {y : y ¤ 1}. In fact

1 Mn M∞
Sn+1 y ’ Sn y ¤ Sn (an (y ’ T Sn y)) ¤ ¤ ,
an an an

where M∞ := limn Mn . Thus S := limn Sn is continuous and y’T Sy = limn y’
T Sn y = 0, i.e. T Sy = y. Now S : F ’ E de¬ned by S(y) := y S( y ) and
y
S(0) := 0 is the claimed continuous section.



53.22
596




53.22
597

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Index 611

Index

1, 1-norm, 137 “, real analytic, 264
∞ , ∞-norm, 139 “, smooth, 264
„µ0 , ¬rst countable cardinal, 46 “, stereographic, 512
, diagonal mapping, 59 “, vector bundle, 287
augmented local convenient C ∞ -algebra, 316
—β bornological tensor product, 55
β , bornological tensor algebra, 57 automorphism, inner, 373
A<k> , set of points in Ak+1 with pairwise
, bornological symmetric algebra, 57
distinct coordinates, 227
, bornological exterior algebra, 57
1-form, kinematic, 337
B
“, modular, 337
Bµ (x), µ-ball centered at x, 156
“, operational, 337
barrelled locally convex space, 579
1-isolating, 203
base of a vector bundle, 287
A “ space of a ¬ber bundle, 376
basis of a ¬ber bundle, 376
absolutely convex, 576
Bezoutiant matrix, 537
“ convex Lipschitz bound, 17
Bianchi identity, 377
absorbing, 575
biholomorphic mappings, 264
absorbs, 34
bilinear concomitants, natural, 367
addition, local, 441
bipolar U oo , 16
adherence of order ±, Mackey, 49
bornivorous, 34
“, Mackey, 48, 51
“ set, 35
“, sequential, 41
adjoint mapping — , 8 bornological approximation property, 70, 280
“ embedding, 48
“ representation, 373
“ isomorphism, 8
algebra, bounded, 63
“ locally convex space, 575
“, commutative, 57
“ mapping, 19
“, De Rham cohomology, 354
“ tensor product —β , 55
“, exterior, 57
“ vector space, 34
“, formally real commutative, 305
bornologically compact set, 62, 88
“, graded-commutative, 57
“ compact subset, 41
“, symmetric, 57
“ real analytic curve, 99
“, tensor, 57
bornologi¬cation, 35
“, Weil, 306
“ of a locally convex space, 575
algebraic bracket of vector valued di¬erential
forms, 359 bornology of a locally convex space, 8, 575
“ derivation, 358 “ on a set, 21
almost complex structure, 368 bounded set, 575
“ continuous function, 87 “ algebra, 63
alternating tensor, 57 “ completant set, 580
alternator, alt, 57 “ linear mapping, 8
analytic subsets, 241 “ mapping, 19
anti-derivative, 20 “ modules, 63
approximation of unity, 27 bounding set, 19
“ property, bornological, 70, 280 bump function, 153
“ property, Mackey, 70
C
arc-generated vector space, 39
c∞ -approximation property, 70
Asplund space, 135
c∞ -complete space, 20
“ space, weakly, 136
c∞ -completion, 47
associated bundle, 382
c∞ -open set, 19
atlas, 264
c∞ -topology, 19
“, equivalent, 264
C ∞ , smooth, 30
“, holomorphic, 264
C ∞ (R, E), space of smooth curves, 28
“, principal bundle, 380
612 Index

C ∞ (U, F ), space of smooth mappings, 30 complete space, Mackey, 15
∞ “ space, locally, 20
Cb , space of smooth functions with bounded
derivatives, 159 completely regular space, 46
∞ -algebra, augmented local convenient, 316 completion of a locally convex space, 16
C
C ∞ -algebras, Chart description of functors complex di¬erentiable mapping, 81
complexi¬cation MC of manifold M , 20
induced by, 316
∞ -structure, 264 composition, smoothness of, 444
C
C∞ (M, N ), manifold of smooth mappings, 439 “, truncated, 431
C k -topology, compact-open, 436 conjugate pair, dual, 586
C k -topology, Whitney, 436 “ pair, 583
Cb , space of C k -functions with k-th derivative
k conjugation, 373
connection, 366
bounded, 159
k , space of C k -functions with k-th derivative “ form, Lie algebra valued, 387
CB
“ on a ¬ber bundle, 376
bounded by B, 159
ω -manifold structure of C ω (M, N ), 442 “, classifying, 485
C
C ω -manifold structure on C∞ (M, N ), 443 “, induced, 394, 394
“, linear, 396, 397
c0 -ext, a class of locally convex spaces, 212
“, principal, 387
c0 (“), space of 0-sequences, 142
connections, space of, 479
c0 (X), 50
connector, 397
Cc (X), 50
contact di¬eomorphisms, 467
canonical ¬‚ip, 293
“ distribution, 467
“ involution, 293
“ form, 467
carrier of a mapping, 153
“ graph of a di¬eomorphism, 470
Cartan developing, 427
“ manifold, 467
Cartesian closedness, 30
ˇ “ structure, exact, 467
Cech cohomology set, 288
continuous derivation over eva , 276
chain rule, 33
convenient co-algebras, 246
characteristic vector ¬eld, 467
“ C ∞ -algebra, augmented local, 316
chart changing mapping, 264
“ description of functors induced by C ∞ - “ vector space, 2, 7, 20
convex function, dual of a, 131
algebras, 316
convolution, 27
“ description of Weil functors, 307
coproduct, 576
“ of a foliation, distinguished, 273
cotangent bundle, kinematic, 337
“, vector bundle, 287
“ bundle, operational, 337
“, submanifold, 268
CO-topology, 434
Christo¬el forms, 377
covariant derivative, 397
classifying connection, 485
“ exterior derivative, 392, 399
“ space, 485, 487
“ Lie derivative, 399
closed di¬erential forms, 353
covering space, universal, 271
co-algebras, convenient, 246
COk -topology, 436
co-commutative, 246
curvature, 366, 398
cocurvature of a connection, 366
curve, bornologically real analytic, 99
cocycle condition, 288, 376, 414
“, di¬erentiable, 8
“ of transition functions, 288, 376
“, holomorphic, 81
cohomologous transition functions, 288, 380
“, locally Lipschitzian, 9
cohomology algebra, De Rham, 354
“, smooth, 9
“ classes of transition functions, 288
“, topologically real analytic, 99
co-idempotent, 247
colimit, 576
D
commensurable groups, 510
δ, natural embedding into the bidual, 16
commutative algebra, 57
d, di¬erentiation operator, 33
comp, the composition mapping, 31
dn , iterated directional derivative, 26
compact-open C k -topology, 436 v
(d) d [j]
Da E := j=1 D , space of operational
“ topology, 434
tangent vectors of order ¤ d, 278
compatible vector bundle charts, 287
(D(k) ) M , operational cotangent bundle, 337
completant set, bounded, 580
Index 613

[d]
Da E, space of operational tangent vectors of Dunford-Pettis property, 200
homogeneous order d, 278
E
De Rham cohomology algebra, 354
E , space of bounded linear functionals, 8
de¬nite integral, 16
E — , dual space of continuous linear
density of subset of N, 22
functionals, 8
“ number densX of a topological space, 152
EB , linear space generated by B ⊆ E, 11, 576
dentable subset, 135
Der(C ∞ (M, R)), space of operational vector Eborn , bornologi¬cation of E, 35
ˆ
¬elds, 322 E, completion of E, 16
derivation over eva , continuous, 276 embedding of manifolds, 269
“, algebraic, 358 “, bornological, 48
“, order of a, 277 equicontinuous sets, 15
“, graded, 358 equidistant di¬erence quotient, 119
derivative of a curve, 8 equivalent atlas, 264
“, covariant, 397 evolution, right, 410
“, covariant exterior, 392, 399 exact contact structure, 467
“, covariant Lie, 399 “ di¬erential forms, 353
“, directional, 128 expansion at x, 311

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