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from the permanence properties of ultrabornological spaces (52.30) and of webbed
spaces (52.13). The inductive limit describing E is regular and E is complete by
[Floret, 1971, 7.4 and 7.5]. The dual E is a Fr´chet space since E has a countable
e
base of bounded sets as a regular inductive limit of Banach spaces. If E is nuclear
then the dual is also nuclear by [Jarchow, 1981, 21.5.3].
If E has the Baire property, then it is metrizable by (52.12). But a metrizable Silva
space is ¬nite dimensional by [Floret, 1971, 7.7].


52.37
582




53. Appendix: Projective Resolutions
of Identity on Banach spaces


One of the main tools for getting results for non-separable Banach spaces is that of
projective resolutions of identity. The aim is to construct trans¬nite sequences of
complemented subspaces with separable increment and ¬nally reaching the whole
space. This works for Banach spaces with enough projections onto closed subspaces.
We will give an account on this, following [Orihuela, Valdivia, 1989]. The results in
this appendix are used for the construction of smooth partitions of unity in theorem
(16.18) and for obtaining smooth realcompactness in example (19.7)

53.1. De¬nition. Let E be a Banach space, A ⊆ E and B ⊆ E Q-linear sub-
spaces. Then (A, B) is called norming pair if the following two conditions are
satis¬ed:

∀x ∈ A : x = sup{| x, x— | : x— ∈ B, x— ¤ 1}
∀x— ∈ B : x— = sup{| x, x— | : x ∈ A, x ¤ 1}.


53.2. Proposition. Let (A, B) be a norming pair on a Banach space E. Then
¯¯
(1) (A, B) is a norming pair.
(2) Let A0 ⊆ A, B0 ⊆ B, ω ¤ |A0 | ¤ », and ω ¤ |B0 | ¤ » for some cardinal
number ».
˜˜ ˜ ˜
Then there exists a norming pair (A, B) with A0 ⊆ A ⊆ A, B0 ⊆ B ⊆ B,
˜ ˜
|A| ¤ » and |B| ¤ ».
(3)

x ∈ A, y ∈ B o ’ x ¤ x + y , in particular A © B o = {0}
x— ∈ Ao , y — ∈ B ’ y — ¤ y — + x— , in particular Ao © B = {0}.

¯
Proof. (1) Let x ∈ A and µ > 0. Thus there is some a ∈ A with x ’ a ¤ µ and
we get

x ¤ x ’ a + a ¤ µ + sup{| a, x— | : x— ∈ B, x— ¤ 1}
¤ µ + sup{| a ’ x, x— | : x— ∈ B, x— ¤ 1}
+ sup{| x, x— | : x— ∈ B, x— ¤ 1}
¤ µ + a ’ x + sup{| x, x— | : x— ∈ B, x— ¤ 1}
¤ 2µ + x ,

53.2
53.3 53. Appendix: Projective resolutions of identity on Banach spaces 583

and for µ ’ 0 we get the ¬rst condition of a norming pair. The second one is shown
analogously.
(2) For every x ∈ A and y — ∈ B choose a countable sets ψ(x) ⊆ B and •(y — ) ⊆ A
such that

x = sup{| x, x— | : x— ∈ ψ(x)} and y — = sup{| y, y — : y ∈ •(y — )}

By recursion on n we construct subsets An ⊆ A and Bn ⊆ B with |An | ¤ » and
|Bn | ¤ »:

∪ {ψ(x) : x ∈ An Q}
Bn+1 := Bn Q

∪ {•(x— ) : x— ∈ Bn Q }.
An+1 := An Q


˜ ˜ ˜˜
Finally let A := n∈N An and B := n∈N Bn . Then (A, B) is the required norming
pair. In fact for x ∈ An we have that

x = sup{| x, x— | : x ∈ ψ(x)} ¤ sup{| x, x— | : x ∈ Bn+1 } ¤ x

˜ ˜
•(b) ⊆ A.
Note that •(B) := ˜
b∈B
(3) We have

x = sup{| x, x— | : x— ∈ B, x— ¤ 1}
= sup{| x + y, x— | : x— ∈ B, x— ¤ 1}
¤ sup{| x + y, x— | : x— ¤ 1} = x + y

and analogously for the second inequality.

53.3. Proposition. Let (A, B) be a norming pair on a Banach space E consisting
of closed subspaces. It is called conjugate pair if one of the following equivalent
conditions is satis¬ed.
(1) There is a projection P : E ’ E with image A, kernel B o and P = 1;
(2) E = A + B o ;
σ(E ,E)
(3) {0} = Ao © B ;
(4) The canonical mapping A ’ E ∼ (E , σ(E , E)) ’ (B, σ(B, E)) is onto.
=

Proof. We have the following commuting diagram:

u 99
G
Bo
9Aδ
9
y
ker

se Eu SS ww (E , σ(E , E))
v
{0}e u
E
eg y ST
e T
S y
u
δ

Av w (B, σ(B, E)) B
δ| A
53.3
584 53. Appendix: Projective resolutions of identity on Banach spaces 53.7

(1)’(2) is obvious.
(2)”(3) follows immediately from duality.
(2)’(4) Let z ∈ (B, σ(B, E)) . By Hahn-Banach there is some x ∈ E with x|B = z.
Let x = a + b with a ∈ A and b ∈ B o . Then a|B = x|B = z.
(4)’(1) By (4) the mapping δ : A ’ E ∼ (E , σ(E , E)) ’ (B, σ(B, E)) is
=
bijective, since A © B o = {0}, and hence we may de¬ne P (x) := δ ’1 (x|B ). Then
P is the required norm 1 projection, since δ : x ’ x|B has norm ¤ 1 and δA has
norm 1 since (A, B) is norming.

53.4. Corollary. Let E be a re¬‚exive Banach space. Then any norming pair
(A, B) of closed subspaces is a conjugate pair.

Proof. In fact we then have
σ(E ,E)
Ao © B = Ao © B = Ao © B = {0},
since the dual of (E , σ(E , E)) is E and equals E the dual of (E , ). By
[Jarchow, 1981, 8.2.5] convex subsets as B have the same closure in these two
topologies.

53.5. De¬nition. A projective generator • for a Banach space E is a mapping
• : E ’ 2E for which
(1) •(x— ) is a countable subset of {x ∈ E : x ¤ 1} for all x— ∈ E ;
(2) x— = sup{| x, x— | : x ∈ •(x— )};
¯¯
(3) If (A, B) is norming, with •(B) := b∈B •(b) ⊆ A, then (A, B) is a conju-
gate pair.
Note that the ¬rst two conditions can be always obtained.
¯¯
We say that the projection P de¬ned by (53.3) for (A, B) is based on the norming
¯ ¯
pair (A, B), i.e. P (E) = A and ker(P ) = B o = B o .

53.6. Corollary. Every re¬‚exive Banach space has a projective generator •.

Proof. Just choose any • satisfying (53.5.1) and (53.5.2). Then (53.5.3) is by
(53.2.1) and (53.4) automatically satis¬ed.

53.7. Theorem. Let • be a projective generator for a Banach space E. Let
A0 ⊆ E and B0 ⊆ E be in¬nite sets of cardinality at most ».
Then there exists a norm 1 projection P based on a norming pair (A, B) with
A0 ⊆ A, B0 ⊆ B, |A| ¤ », |B| ¤ » and •(B) ⊆ A.

Proof. By (53.2.3) there is a norming pair (A, B) with
A0 ⊆ A, B0 ⊆ B, |A| ¤ », |B| ¤ ».
Note that in the proof of (53.2.3) we used some map •, and we may take the
projective generator for it. Thus we have also •(B) ⊆ A. By condition (53.5.3) of
the projective generator we thus get that the projection based on (A, B) has the
required properties.


53.7
53.8 53. Appendix: Projective resolutions of identity on Banach spaces 585

53.8. Proposition. Every WCD Banach space has a projective generator.

A Banach space E is called WCD, weakly countably determined , if and only if there
exists a sequence Kn of weak— -compact subsets of E such that for every

∀x ∈ E ∀y ∈ E \ E ∃n : x ∈ Kn and y ∈ Kn .
/

Every WCG Banach space is WCD:
In fact let K be weakly compact (and absolutely convex) such that n∈N K is
dense in E. Note that (E, σ(E, E )) embeds canonically into (E , σ(E , E )). Let
Kn,m := n K + m {x ∈ E : x ¤ 1}. Then Kn,m is weak— -compact, and for
1

any x ∈ E and y ∈ E \ E there exists an m > 1/ dist(y, E) and an n with
1
dist(x, n K) < m . Hence x ∈ Kn,m and y ∈ E + 1/m {x ∈ E : x ¤ 1} ⊇ Kn,m .
/
The most important advantage of WCD over WCG Banach spaces are, that they
are hereditary with respect to subspaces.
For any ¬nite sequence n = (n1 , . . . , nk ) let
σ(E ,E )
Cn1 ,...,nk := E © Kn1 © · · · © Knk .

Then these sets are weak— -compact (since they are contained in Knk ) and if E is
not re¬‚exive, then for every x ∈ E there is a sequence n : N ’ N such that

x∈ Cn1 ,...,nk ⊆ E.
k=1

In fact choose a surjective sequence n : N ’ {k : x ∈ Kk }. Then x ∈ Cn1 ,...,nk

for all k, hence x ∈ k=1 Cn1 ,...,nk . If y ∈ E \ E, then there is some k, such that
y ∈ Knk and hence y ∈ Cn1 ,...,nk ⊆ Knk .
/ /

Proof of (53.8). Because of (53.6) we may assume that E is not re¬‚exive. For
every x— ∈ E we choose a countable set •(x— ) ⊆ {x ∈ E : x ¤ 1} such that

x— = sup{| x, x— : x ∈ •(x— )} and
sup{| x, x— | : x ∈ Cn1 ,...,nk } = sup{| x, x— | : x ∈ Cn1 ,...,nk © •(x— ) }

for all ¬nite sequences (n1 , . . . , nk ). We claim that • is a projective generator:
¯¯
Let (A, B) be a norming pair with •(B) ⊆ A. We use (53.3.3) to show that (A, B)
σ(E ,E)
is norming. Assume there is some 0 = y — ∈ Ao © B . Thus we can choose
x0 ∈ E with |y — (x0 )| = 1 and a net (yi )i in B that converges to y — in the Mackey


topology µ(E , E) (of uniform convergence on weakly compact subsets of E). In fact
this topology on E has the same dual E as σ(E , E) by the Mackey-Arens theorem
[Jarchow, 1981, 8.5.5], and hence the same closure of convex sets by [Jarchow, 1981,
8.2.5]. As before we choose a surjective mapping n : N ’ {k : x0 ∈ Kk }. Then

x0 ∈ C := Cn1 ,...,nk ⊆ E.
k=1

53.8
586 53. Appendix: Projective resolutions of identity on Banach spaces 53.10

and C is weakly compact, hence we ¬nd an i0 such that

sup{|yi0 (x) ’ y — (x)| : x ∈ C} <
— 1
2

and in particular we have

|yi0 (x0 )| ≥ |y — (x0 )| ’ |yi0 (x0 ) ’ y — (x0 )| > 1 ’
— — 1
= 1.
2 2

Since the sets forming the intersection are decreasing, Cn1 is σ(E , E )-compact
and
W := {x—— ∈ E : |x—— (yi0 ’ y — )| < 1 }

2

is a σ(E , E )-open neighborhood of C there is some k ∈ N such that Cn1 ,...,nk ⊆ W ,
i.e.
sup{|yi0 (x) ’ y — (x)| : x ∈ Cn1 ,...,nk } ¤ 1 .

2
— —
By the de¬nition of • there is some y0 ∈ Cn1 ,...,nk © •(yi0 ) with |yi0 (y0 )| > 1 ’ 1 ,
2
thus
|y — (y0 )| ≥ |yi0 (y0 )| ’ |yi0 (y0 ) ’ y — (y0 )| > 1 ’ 1 = 0.
— —
2 2

Thus y — (y0 ) = 0 and y0 ∈ •(B) ⊆ A, a contradiction.

Note that if P ∈ L(E) is a norm-1 projection with closed image A and kernel B o ,
then P — ∈ L(E ) is a norm-1 projection with image P — (E) = ker P o = B oo = B
and kernel ker P — = P (E)o = Ao . However not all norm-1 projections onto B can
be obtained in this way. Hence we consider the dual of proposition (53.3):

53.9. Proposition. Let (A, B) be a norming pair on a Banach space E consisting
of closed subspaces. It is called dual conjugate pair if one of the following equivalent
conditions is satis¬ed.
(1) There is a norm-1 projection P : E ’ E with image B, kernel Ao ;
(2) E = B • Ao ;
σ(E ,E )
(3) {0} = B o © A ;
( )|A
(4) The canonical mapping B ’ E ’ ’ ’ A is onto.
’’

Proof. This follows by applying (53.3) to the norming pair (B, A) ⊆ (E , E ).

The dual of de¬nition (53.5) is

53.10. De¬nition. A dual projective generator ψ for a Banach space E is a
mapping ψ : E ’ 2E for which
(1) ψ(x) is a countable subset of {x— ∈ E : x— ¤ 1} for all x ∈ E;
(2) x = sup{| x, x— | : x— ∈ ψ(x)};
¯¯
(3) If (A, B) is norming, with ψ(A) := a∈A ψ(a) ⊆ B, then (A, B) satis¬es
the condition of (53.9).
Note that the ¬rst two conditions can be always obtained.

From (53.7) we get:

53.10
53.13 53. Appendix: Projective resolutions of identity on Banach spaces 587

53.11. Theorem. Let ψ be a dual projective generator for a Banach space E. Let
A0 ⊆ E and B0 ⊆ E be in¬nite sets of cardinality at most ».
Then there exists a norm 1 projection P in E with A0 ⊆ P — (E ), B0 ⊆ P (E ),
|P — (E )| ¤ », |P (E )| ¤ ».

53.12. Proposition. A Banach space E is Asplund if and only if there exists a
dual projective generator on E.

Note that if P is a norm-1 projection, then so is P — . But not all norm-1 projections
on the dual are of this form.

Proof. (⇐) Let ψ be a dual projective generator for E. Let A0 be a separable
subspace of E. By (53.11) there is a separable subspace A of E and a norm-1
projection P of E such that A0 ⊇ A, P (E ) is separable and isomorphic with A
via the restriction map. Hence A is separable and also A0 . By [Stegall, 1975] E is
Asplund.

: x— ¤1}
-weak— upper semi-continuous mapping φ : X ’ 2{x
(’) Consider the
given by
φ(x) := {x— ∈ E : x— ¤ 1, x, x— = x }.

By the Jayne-Rogers selection theorem [Jayne, Rogers, 1985], see also [Deville,
Godefroy, Zizler, 1993, section I.4] there is a map f : E ’ {x— ∈ E : x— ¤ 1}
with f (x) ∈ φ(x) for all x ∈ E and continuous fn : E ’ {x— : x— ¤ 1} ⊆ E with
fn (x) ’ f (x) in E for each x ∈ E. One then shows that

ψ(x) := {f (x), f1 (x), . . . }

de¬nes a dual projective generator, see [Orihuela, Valdivia, 1989].

53.13. De¬nition. Projective Resolution of Identity. Let a “long sequence”
of continuous projections P± ∈ L(E, E) on a Banach space E for all ordinal numbers
ω ¤ ± ¤ dens E be given. Recall that dens(E) is the density of E (a cardinal
number, which we identify with the smallest ordinal of same cardinality). Let
E± := P± (E) and let R± := (P±+1 ’ P± )/( P±+1 ’ P± ) or 0, if P±+1 = P± . Then
we consider the following properties:
P± Pβ = Pβ = Pβ P± for all β ¤ ±.
(1)
(2) Pdens E = IdE .
dens P± E ¤ ± for all ±.
(3)
(4) P± = 1 for all ±.
(5) β<± Pβ+1 E = P± E, or equivalently β<± Eβ = E± for every limit ordinal
± ¤ dens E.
For every limit ordinal ± ¤ dens E we have P± (x) = limβ>± Pβ (x), i.e.
(6)
± ’ P± (x) is continuous.
E±+1 /E± is separable for all ω ¤ ± < dens E.
(7)
(R± (x))± ∈ c0 ([ω, dens E]) for all x ∈ E.
(8)
P± (x) ∈ Pω (x) ∪ {Rβ (x) : ω ¤ β < ±} .
(9)

53.13
588 53. Appendix: Projective resolutions of identity on Banach spaces 53.13

The family (P± )± is called projective resolution of identity (PRI ) if it satis¬es (1),
(2), (3), (4) and (5).
It is called separable projective resolution of identity (SPRI ) if it satis¬es (1), (2),
(3), (7), (8) and (9). These are the only properties used in (53.20) and they follow
for WCD Banach spaces and for duals of Asplund spaces by (53.15). For C(K)
with Valdivia compact K this is not clear, see (53.18) and (53.19). However, we
still have (53.21) and in (16.18) we don™t use (7), but only (8) and (9) which hold
also for PRI, see below.
2
Remark. Note that from (1) we obtain that P± = P± and hence P± ≥ 1, and
E± := P± (E) is the closed subspace {x : P± (x) = x}.
2
Moreover, P± Pβ = Pβ = Pβ P± for β ¤ ± is equivalent to P± = P± , Pβ (E) ⊆ P± (E)
and ker Pβ ⊇ ker P± .
(’) Pβ x = P± Pβ x ∈ P± (E) and P± x = 0 implies that Pβ x = Pβ P± x.
(⇐) For x ∈ E there is some y ∈ E with Pβ x = P± y, hence P± Pβ x = P± P± y =
P± y = Pβ x. And Pβ (1 ’ P± )x = 0, since (1 ’ P± )x ∈ ker P± ⊆ ker Pβ .
Note that E±+1 /E± ∼ (P±+1 ’ P± )(E), since E± ’ E±+1 has P± |E±+1 as right
=
inverse, and so E±+1 /E± ∼ ker(P± |E ) = (1 ’ P± )P±+1 (E) = (P±+1 ’ P± )(E).
= ±+1


(5) ⇐ (9), since for x ∈ E± we have x = P± (x) and Eω ∪ {Rβ (x) : β < ±} ⊆ E±
for all ±.
(3) ⇐ (5) & (7) By trans¬nite induction we get that for successor ordinals ± =
β + 1 we have dens(E± ) = dens(Eβ ) + dens(E± /Eβ ) = dens(Eβ ) ¤ β ¤ ±,
since dens(E± /Eβ ) ¤ ω. For limit ordinals it follows from (5), since dens(E± ) =
dens( β<± Eβ ) = sup{dens(Eβ ) : β < ±} ¤ sup{β : β < ±} = ±.
(6) ⇐ (4) & (1) & (5) For every limit ordinal 0 < ± ¤ dens E and for all x ∈ E the
net (Pβ (x))β<± converges to P± (x).
Let ¬rst x ∈ P± (E) and µ > 0. By (5) there exists a γ < ± and an xγ ∈ Pγ (E) with
x ’ xγ < µ. Hence for γ ¤ β < ± we have by (1) that Pβ (xγ ) = P± (xγ ) and so

P± (x) ’ Pβ (x) = P± (x ’ xγ )| + P± (xγ ) ’ Pβ (xγ )| ’ Pβ (xγ ’ x)

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