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)