Thus, by Banach™s ¬xed point theorem, we have a unique solution x of x = k(y0 , x)

˜

for every y0 ∈ U and x depends continuously on y0 . As said before, it follows that

t

y := y0 + 0 g(x(s)) ds solves the original di¬erential equation.

Let k(t) := y ’ f (x(t)) be the error we make at time t. For our original di¬erential

equation the error was k(t) = e’ct y, and hence k was the solution of the homo-

geneous linear di¬erential equation k (t) = ’c k(t). In the modi¬ed situation we

have

Claim. If x(t) is a solution, k(t) := y ’ f (x(t)), and g(t) is de¬ned by (f (σt · xt ) ’

f (xt )) x (t) then

k (t) + c σt · k(t) = g(t).

From x (t) = c Ψ(σt (x(t))) · (σt (y ’ f (x(t)))) = c Ψ(σt (x(t))) · (σt k(t)) we conclude

d

that f (σt xt ) · x (t) = c σt kt , and by the chain rule dt f (x(t)) = f (xt ) · x (t) we get

k (t) + cσt k(t) = (f (σt x(t)) ’ f (x(t))) · x (t) =: g(t).

In order to estimate that error k, we now consider a general inhomogeneous linear

di¬erential equation:

51.22

51.22 51. The Nash-Moser inverse function theorem 569

Sublemma. If the solution k of the di¬erential equation k (t) + c σt k(t) = g(t)

exists on [0, T ] then for all p ≥ 0 and 0 < q < c we have

T T

qt

eqt g(t)

dt ¤ C k(0)

e k(t) +C + g(t) dt

p p+q p p+q

0 0

Proof. If k is a solution of the equation above then the coordinates kj are solutions

of the ordinary inhomogeneous linear di¬erential equation

kj (t) + cσ(t ’ j)kj (t) = gj (t).

The solution kj of that equation can be obtained as usual by solving ¬rst the ho-

mogeneous equation via separation of variables and applying then the method of

variation of the constant. For this recall that for a general 1-dimensional inhomo-

geneous linear di¬erential equation k (t) + σ(t)k(t) = g(t) of order 1, one integrates

t

the homogeneous equation dkh(t) = ’σ(t)dt, i.e., log(kh (t)) = C ’ 0 σ(„ )d„ or

(t)

kh

t

kh (t) is a multiple of e’ 0 σ(„ )d„ . For vector valued equations this is no longer

t

a solution, since σ(t) does not commute with 0 σ(„ )d„ in general, and hence

t t

d ’ 0 σ(„ )d„

need not be ’σ(t) · e’ 0 σ(„ )d„ . But in our case, where σ is just

e

dt

a multiplication operator it is still true. For the inhomogeneous equation one

makes the ansatz k(t) := kh (t) C(t), which is a solution of the inhomogeneous

equation g(t) = k (t) + σ(t)k(t) = kh (t) C(t) + kh (t) C (t) + σ(t) kh (t) C(t) =

’σ(t) kh (t) C(t) + kh (t) C (t) + σ(t) kh (t) C(t) = kh (t) C (t) if and only if C (t) =

t

t ρ

kh (t)’1 g(t) = e 0 σ(„ )d„ g(t) or C(t) = C(0) + 0 e 0 σ(„ ) g(t)d„ dρ. So the solution

of the inhomogeneous equation is given by

t

t ρ

’ σ(„ )d„ σ(„ )d„

k(t) = kh (t) C(t) = e C(0) + g(ρ) e dρ .

0 0

0

t

In particular we have k(0) = C(0), and using a(t, s) := e’ s σ(„ ) d„ we get k(t) =

t t

a(0, t)k(0) + 0 a(ρ, t)g(ρ) dρ. We set aj,s,t := exp(’c s σ(„ ’ j)d„ ). Then

t

kj (t) = aj,0,t kj (0) + aj,„,t gj („ )d„.

0

We claim that ect aj,s,t ¤ ec (ecs + ecj ) for 0 ¤ s ¤ t ¤ T .

For t ¤ j + 1 this follows from aj,s,t ¤ 1. Let now t > j + 1. If s ≥ j + 1 then

σ(„ ’ j) = 1 for „ ≥ s, and so aj,s,t = e’c(t’s) and ect aj,s,t = ecs . If otherwise

s ¤ j + 1 then

t

σ(„ ’ j) d„ = e’c(t’j’1) .

aj,s,t ¤ exp ’c

j+1

So ect aj,s,t ¤ ec(j+1) = ec ecj , and the claim holds.

51.22

570 Chapter X. Further Applications 51.22

∞

Next we claim that for 0 < q < c we have s eqt aj,s,t dt ¤ C(eqj + eqs ).

For j ¤ s ≥ t we have aj,s,t ¤ Cec(s’t) by the previous claim and

∞

∞ ∞ (q’c)t

cs e

eqt aj,s,t dt ¤ Cecs (q’c)t

¤ Ceqs

e dt = Ce

q’c

s s t=s

using q < c. For s < j we split the integral into two parts. From aj,s,t ¤ 1 we

j j

conclude that s eqt aj,s,t dt ¤ s eqt dt ¤ Ceqj , and from aj,s,t ¤ Cec(j’t) (by the

∞ ∞

previous claim) we conclude that j eqt aj,s,t dt ¤ Cecj j e(q’c)t dt ¤ Ceqj , which

proves the claim.

Now the main claim. We have

T T

qt

eqt epj kj (t) dt ¤

e k(t) dt =

p

0 0 j

T t

qt pj

¤ ee aj,0,t kj (0) + aj,s,t gj (s) ds dt.

0 0

j

The ¬rst summand is bounded by

T

pj

eqt aj,0,t dt kj (0) ¤ C e(p+q)j kj (0) ¤ C k(0)

e p+q

0

j j

and the second by

T T T

pj qt

epj (eqj + q qs ) gj (s) ds

gj (s) ds ¤

e e aj,s,t dt

0 s 0

j j

T

+ eqs g(s) p ds.

¤C g(s) p+q

0

We will next show that the domain of de¬nition is not ¬nite and that the limit

limt’+∞ x(t) =: x∞ exists and is a solution of f (x∞ ) = y. For this we need the

is su¬ciently small then for n ≥ 2r and

Sublemma. If x is a solution and y 2r

q ≥ 0 we have

T

dt ¤ Cn,q eqT y

¤

x(T ) x (t) n.

n+q n+q

0

Proof. Using the inequality for smoothing operators we get for n ≥ 0 and q ≥ 0

that

= cΨ(σt x(t)) · σt k(t) ¤ C σt kt

x (t) + C σt x(t) σt k(t)

n+q n+q n+q n+q+2r 0

¤ Ceqt kt + x(t) kt .

n n+2r 0

51.22

51.22 51. The Nash-Moser inverse function theorem 571

¤ 1. Later we will show that this is

We will from now on assume that x(t) 2r

automatically satis¬ed. Then

qt

¤ k(t) 0 (1 + x(t)

x (t) 2r )Ce ,

q

¤2

and hence

t t

eqs k(s)

¤ x (s) q ds ¤ C

x(t) ds.

q 0

0 0

Using the estimates for k given by the sublemma, we have to estimate norms of

g(t) = (f (σt x(t)) ’ f (x(t))) · x (t) = ’b(x(t), σt x(t))((1 ’ σt )x(t), x (t)),

1

where b(x0 , x1 )(h0 , h1 ) := 0 f (x0 + t(x1 ’ x0 ))(h0 , h1 ) dt. Obviously, b is smooth,

is bilinear with respect to (h0 , h1 ), and satis¬es a tame estimate of degree 2r with

respect to x0 , x1 and degree r in h0 and h1 with base 0, i.e.

¤C

b(x0 , x1 )(h0 , h1 ) h0 h1 + h0 h1

n n+r r r n+r

+ ( x0 + x1 n+2r ) h0 h1 .

n+2r r r

¤ 1 we obtain

From this and xt 2r

¤C (1 ’ σt )xt (1 ’ σt )xt

g(t) x (t) + x (t)

n n+r r r n+r

¤Ce’rt xt ¤Cert kt ¤Ce’(n+r)t xt ¤Ce(n+r)t kt

0

n+2r n+2r 0

(1 ’ σt )xt

+ ( xt + σt xt n+2r ) x (t)

n+2r r r

¤Ce’rt xt ¤Cert kt

¤ xt n+2r +C xt n+2r 0

2r

¤ C xt kt 0 .

n+2r

In order to estimate x(t) q , we need the following estimate using the sublemma

giving an estimate for the solution k (t) + cσ(t)k(t) = g(t) for c > 2r

T T

qt

eqt

dt ¤ C ¤

e kt k(0) + g(t) + g(t)

p p+q p p+q

0 0

¤ xt ¤ xt

kt kt

p+2r 0 p+q+2r 0

T t t

qt (p+2r)s

e(p+q+2r)s ks

¤C k(0) + eC e ks ds + C ds kt dt

p+q 0 0 0

0 0 0

t

¤Ceqt e(p+2r)s ks ds

0

0

T T

qt

e(p+2r)s ks 0 ds .

¤C dt ·

k(0) + e kt

p+q 0

0 0

This inequality is of recursive nature. In fact, for p = 0 and q = 2r it says

T

KT := 0 e2rt kt 0 dt ¤ C y 2r + CKT and hence KT (1 ’ CKT ) ¤ C y 2r . If

2

KT ¤ 2C then 1’CKT ≥ 1 and hence KT ¤ 2C y 2r . Thus, KT ∈ (2C y 2r , 2C ].

1 1

/

2

51.22

572 Chapter X. Further Applications 51.22

1

Therefore, choosing y 2r < 4C 2 makes this a nonempty interval, and continuity of

T ’ KT and K0 = 0 shows that

T

e2rt kt 0 dt = KT ¤ 2C y ¤ δ.

for all y

2r 2r

0

Let us now show that the requirement x(t) 2r ¤ 1 is automatically satis¬ed.

Suppose not, then there is a minimal t0 > 0 with x(t0 ) 2r ≥ 1 since x(0) = 0.

Thus, for 0 ¤ t < t0 we have x(t) < 1, and hence the above estimates hold on

the interval [0, t0 ]. From x (t) 2r ¤ Ce2rt k(t) 0 we obtain by integration that

T t

x(t) 2r ¤ 0 x (t) 2r ¤ 0 Ce2rt k(t) 0 dt ¤ 2C y 2r . Thus, if y 2r ¤ δ with

Cδ < 1 then x(t0 ) 2r < 1, a contradiction. Note that this shows at the same time

that the sublemma is valid for q = 0 and n = 2r.

Now we proceed to show that

T

eqs ks ds ¤ C y for q ≥ 2r.

0 q

0

In fact, q = 2r and q = 2r + 1 will be su¬cient, and hence we have a common C.

The above estimate for p = 0 gives

T T T

qt qt

e2rs ks

dt ¤ C dt ·

e ks k(0) + e kt ds .

0 q 0 0

0 0 0

y

¤C y 2r ¤C

Thus

T T

qt

eqt ks

(1 ’ c y · dt ¤ C y dt ¤ C y

2r ) e ks and

0 q 0 q

0 0

for q ≥ 2r.Now for q ≥ p + 2r ¤C y ¤C y

q p+2r

T T T

qt qt

e(p+2r)s ks 0 ds

dt ¤ C dt ·

e ks k(0) + e kt

p p+q 0

0 0 0

¤ C( y ·y ¤C y

+y 2r ) p+q .

p+q p+q

¤1

Now we prove the main claim by induction on n = p + 2r. For p = 0 we have shown

it already. Next for n + 1 = p + 1 + 2r: Using the inequality at the very beginning

of the proof of the sublemma for n replaced by p and q by 1 + 2r + q

T T

Ce(1+2r+q)t

dt ¤

x (t) kt + x(t) kt dt

p+1+2r+q p p+2r 0

0 0

T

qT

e(1+2r)t k(t) + e(1+2r)t x(t)

¤ Ce · kt dt

p p+2r 0

0

¤C y p+2r

¤ CeqT C y +C y y 2r+1 +

p+2r+1 p+2r

+y p+2r C y 2r+1

¤ CeqT y +y y .

p+2r+1 p+2r 2r+1

51.22

51.22 51. The Nash-Moser inverse function theorem 573

¤ δ ¤ 1 we get by interpolation that

For y 2r

¤C y ¤C y

y y y p+1+2r ,

p+2r 2r+1 p+2r+1 2r

which eliminates the last summand and completes the induction.

Claim. For su¬ciently small y 2r the solution x exists globally, limt’+∞ x(t) =:

x∞ exists and solves f (x∞ ) = y. Moreover, we have x∞ n ¤ cn y n .

t

Furthermore, x(t) n ¤ 0 x (s) n ds ¤ C y n for n > 2r and y 2r ¤ δ using the

main claim for q = 0.

T

Suppose x exists on [0, ω) with ω chosen maximally. Since 0 x (t) n dt ¤ C y n

ω

with C independent on T < ω we have 0 x (t) n dt ¤ C y n < ∞, and hence

ω s

limt’ω t x („ ) d„ = 0. Thus, we obtain x(s) ’ x(t) n ¤ t x (r) dr ’ 0 for

s, t ’ ω. Hence, x(ω) := limt ω x(t) exists and x(ω) 2r ¤ 1.

Thus, we can extend the solution in a neighborhood of ω, a contradiction to the

maximality of ω. Then by the same argument as before x∞ := limt ∞ x(t) exists

and x∞ 2r ¤ 1 and x∞ n ¤ C y n for all n ≥ 2r. Since x (t) = cΨ(σ(t)x(t)) ·

(σ(t)(y’f (x(t)))) we have that limt’∞ x (t) = cΨ(x∞ )(y’f (x∞ )) exists, and since

∞

x (t) n dt ¤ C y n < ∞ we have that (x∞ )(y ’ f (x∞ )) = limt ∞ x (t) = 0.

0

So we get y ’ f (x∞ ) = 0, hence we have obtained an inverse.

Proof of the inverse function theorem. By what we have shown so far, we

know that f is locally bijective and f ’1 y n ¤ C y n for all n ≥ 2r. Furthermore,

f ’1 y1 ’ f ’1 y0 + ( f ’1 y1 + f ’1 y0

¤C y1 ’ y 0 y 1 ’ y0

n+2r ) ,

n n n+2r 0

which shows continuity and Lipschitzness of the inverse, and the inverse f ’1 is

tame and locally Lipschitz.

We next show that f ’1 is Gˆteaux-di¬erentiable with derivative as expected, i.e.

a

(f ’1 ) (y)(k) = f (f ’1 y) · k.

For this let c(t) := f ’1 (y + t k), x := c(0) = f ’1 (y) and := Ψ(x) := f (f ’1 (y))’1 .

Then c is locally Lipschitz, and we have to show that c(t)’c(0) ’ · k for t ’ 0.

t

Now

c(t) ’ c(0) c(t) ’ c(0)

’ (k) = ( —¦ f (x)) ’ (k)

t t

c(t) ’ c(0) f (c(t)) ’ f (c(0))

’

= f (x)

t t

1

c(t) ’ c(0)

f (x) ’ f (x + s(c(t) ’ x)) ds ·

= .

t

0

=:g(t,s)

Since t ’ c(t) is locally Lipschitz, the map (t, s) ’ f (x) ’ f (x + s(c(t) ’ x)) is

locally Lipschitz, and hence in particular continuous. Therefore, g(t, s) ’ g(0, s)

51.22

574 Chapter X. Further Applications 51.23

1 1 1

for t ’ 0 uniformly on all s ∈ [0, 1]. Thus, 0 g(t, s)ds ’ 0 g(0, s)ds = 0 ds = 0

0

in L(E, F ), and since c(t)’c(0) stays bounded, this proves the claim.

t

Thus, we have for the Gˆteaux-derivative of the inverse function the formula

a

(f ’1 ) = inv —¦f —¦ f ’1 = Ψ —¦ f ’1 .

Since Ψ and f ’1 are tame, so is (f ’1 ) . By induction, using the chain rule for

di¬erentiable and for tame maps we conclude that (f ’1 ) is a tame smooth map,

since Ψ was assumed to be so. Note that in order to apply the chain-rule it is

not enough to have Gˆteaux-di¬erentiability, but because of tameness (or local

a

Lipschitzness) of the derivative we have the appropriate type of di¬erentiability

automatical. In fact, we have to consider d(f ’1 ) := ((f ’1 ) )§ = Ψ§ —¦ (f ’1 — Id)

and apply induction to that.

Let us ¬nally show that it is enough to assume that Ψ is a tame continuous in order

to assure that it is a tame smooth map.

51.23. Lemma. Let ¦ : E ⊇ U ’ GL(F ) be a tame smooth map and Ψ, de¬ned

by Ψ(x) := ¦(x)’1 , be a continuous tame map. Then Ψ is a tame smooth map.

Proof. For smoothness it is enough to show smoothness along continuous curves,

so we may assume that E = R = U , and so ¦ is a curve denoted c. Then

c(t)’1 ’ c(s)’1 c(t) ’ c(s)

= ’ comp(inv(c(t)), , inv(c(s)))

t’s t’s

and hence is locally bounded, i.e., c is locally Lipschitz. Now let s be ¬xed. Then

c(t + s) ’ c(s)

t ’ ’ comp(inv(c(t + s)), , inv(c(s)))

t

1

c(t+s)’c(s)

is locally Lipschitz, since t ’ = c (s + rt)dr is smooth. In particular,

t 0

c(t + s)’1 ’ c(s)’1

1

+ inv(c(s)) —¦ c (s) —¦ inv(c(s))

t t

is locally bounded, and hence inv —¦c is di¬erentiable with derivative

(inv —¦c) (s) = ’ inv(c(s)) —¦ c (s) —¦ inv(c(s)).

Thus, inv —¦c is smooth by induction, and

Ψ (x)(y) = ’Ψ(x) —¦ ¦ (x)(y) —¦ Ψ(x).

Tameness of Ψ§ follows since the di¬erential of Ψ§ is given by

dΨ§ (x, h; y, k) = Ψ (x)(y)(h) + Ψ(x) (h)(k) = Ψ (x)(y)(h) + Ψ(x)(k)

= (’Ψ(x) —¦ ¦ (x)(y) —¦ Ψ(x))(h) + Ψ(x)(k)

= Ψ§ (x, k) ’ Ψ§ (x, ‚1 ¦§ (x, y)(Ψ§ (x, h))).

51.23

575

52. Appendix: Functional Analysis

The aim of this appendix is the following. This book needs prerequisites from

functional analysis, in particular about locally convex spaces, which are beyond

usual knowledge of non-specialists. We have used as unique reference the book

[Jarchow, 1981]. In this appendix we try to sketch these results and to connect

them to more widespread knowledge in functional analysis: for this we decided to

use [Schaefer, 1971].

52.1. Basic concepts. A locally convex space E is a vector space together with

a Hausdor¬ topology such that addition E — E ’ E and scalar multiplication

R — E ’ E (or C — E ’ E) are continuous and 0 has a basis of neighborhoods

consisting of (absolutely) convex sets. Equivalently, the topology on E can be

described by a system P of (continuous) seminorms. A seminorm p : E ’ R

is speci¬ed by the following properties: p(x) ≥ 0, p(x + y) ¤ p(x) + p(y), and

p(»x) = |»|p(x).

A set B in a locally convex space E is called bounded if it is absorbed by each

0-neighborhood, equivalently, if each continuous seminorm is bounded on B. The

family of all bounded subsets is called the bornology of E. The bornologi¬cation of

a locally convex space is the ¬nest locally convex topology with the same bounded

sets, which is treated in detail in (4.2) and (4.4). A locally convex space is called

bornological if it is stable under the bornologi¬cation, see also (4.1). The ultra-

bornologi¬cation of a locally convex space is the ¬nest locally convex topology with

the same bounded absolutely convex sets for which EB is a Banach space.

52.2. Result. [Jarchow, 1981, 6.3.2] & [Schaefer, 1971, I.1.3] The Minkowski

functional qA : x ’ inf{t > 0 : x ∈ t.A} of a convex absorbing set A containing 0

is a convex function.

{rA : r > 0} is the whole space.

A subset A in a vector space is called absorbing if

52.3. Result. [Jarchow, 1981, 6.4.2.(3)] For an absorbing radial set U in a locally

convex space E the closure is given by {x ∈ E : qU (x) ¤ 1}, where qU is the

Minkowski functional.

52.4. Result. [Jarchow, 1981, 3.3.1] Let X be a set and let F be a Banach space.

Then the space ∞ (X, F ) of all bounded mappings X ’ F is itself a Banach space,

supplied with the supremum norm.

52.4

576 52. Appendix: Functional analysis 52.8

52.5. Result. [Jarchow, 1981, 3.5.6, p66] & [Schaefer, 1971, I.3.6] A Hausdor¬

topological vector space E is ¬nite dimensional if and only if it admits a precompact

neighborhood of 0.

A subset K of E is called precompact if ¬nitely many translates of any neighborhood

of 0 cover K.

52.6. Result. [Jarchow, 1981, 6.7.1, p112] & [Schaefer, 1971, II.4.3] The abso-

lutely convex hull of a precompact set is precompact.

A set B in a vector space E is called absolutely convex if »x + µy ∈ B for x, y ∈ B

and |»| + |µ| ¤ 1. By EB we denote the linear span of B in E, equipped with the

Minkowski functional qB . This is a normed space.

52.7. Result. [Jarchow, 1981, 4.1.4] & [Horvath, 1966] A basis of neighborhoods

of 0 of the direct sum C(N) is given by the sets of the form {(zk )k ∈ C(N) : |zk | ¤

µk for all k} where µk > 0.

The direct sum i Ei , also called the coproduct i Ei of locally convex spaces Ei is

the subspace of the cartesian product formed by all points with only ¬nitely many

non-vanishing coordinates supplied with the ¬nest locally convex topology for which

the inclusions Ej ’ i Ei are continuous. It solves the universal problem for a

coproduct: For continuous linear mappings fi : Ei ’ F into a locally convex space

there is a unique continuous linear mapping f : i Ei ’ F with f —¦ inclj = fj

for all j. The bounded sets in i Ei are exactly those which are contained and

bounded in a ¬nite subsum. If all spaces Ei are equal to E and the index set is “,

we write E (“) for the direct sum.

52.8. Result. [Jarchow, 1981, 4.6.1, 4.6.2, 6.6.9] & [Schaefer, 1971, II.6.4 and

II.6.5] Let E be the strict inductive limit of a sequence of locally convex vector spaces

En . Then every En carries the trace topology of E, and every bounded subset of E

is contained in some En , i.e., the inductive limit is regular.

Let E be a functor from a small (index) category into the category of all locally

convex spaces with continuous linear mappings as morphisms. The colimit colim E

of the functor E is the unique (up to isomorphism) locally convex space together

with continuous linear mappings li : E(i) ’ colim E which solves the following

universal problem: Given continuous linear gi : E(i) ’ F into a locally convex

space F with gj —¦E(f ) = gi for each morphism f : i ’ j in the index category. Then

there exists a unique continuous linear mapping g : colim E ’ F with g —¦ li = gi

for all i.

‘‘

ee ‘‘‘

i E(i)

eg ‘‘‘‘‘ gi

e “

‘

li

&w

h &&&&&&& (

g

f E(f )

j

h

colim E F

u hh &

lj

&&

u gj

j E(j)

52.8

52.13 52. Appendix: Functional analysis 577

The colimit is given as the locally convex quotient of the direct sum i E(i)

by the closed linear subspace generated by all elements of the form incli (x) ’

(inclj —¦E(f ))(x) for all x ∈ E(i) and f : i ’ j in the index category. Compare

[Jarchow, 1981, p.82 & p.110], but we force here inductive limits to be Hausdor¬.

A directed set “ is a partially ordered set such that for any two elements there

is another one that is larger that the two. The inductive limit is the colimit of a

functor from a directed set (considered as a small category); one writes limj Ej for

’’

this. A strict inductive limit is the inductive limit of a functor E on the directed

set N such that E(n < n + 1) : E(n) ’ E(n + 1) is the topological embedding of a

closed linear subspace.

The dual notions (with the arrows between locally convex spaces reversed) are

called the limit lim E of the functor E, and the projective limit limj Ej in the case

←’

of a directed set. It can be described as the linear subset of the cartesian product

i E(i) consisting of all (xi )i with E(f )(xi ) = xj for all f : i ’ j in the index

category.

52.9. Result. [Jarchow, 1981, 5.1.4+11.1.6] & [Schaefer, 1971, III.5.1, Cor. 1]

Every separately continuous bilinear mapping on Fr´chet spaces is continuous.

e

A Fr´chet space is a complete locally convex space with a metrizable topology,

e

equivalently, with a countable base of seminorms. See [Jarchow, 1981, 2.8.1] or

[Schaefer, 1971, p.48].

Closed graph and open mapping theorems. These are well known if Banach

spaces or even Fr´chet spaces are involved. We need a wider class of situations

e

where these theorems hold; those involving webbed spaces. Webbed spaces were

introduced for exactly this reason by de Wilde in his thesis, see [de Wilde, 1978]. We

do not give their (quite lengthy) de¬nition here, only the results and the permanence

properties.

52.10. Result. Closed Graph Theorem. [Jarchow, 1981, 5.4.1] Any closed

linear mapping from an inductive limit of Baire locally convex spaces into a webbed

locally convex space is continuous.

52.11. Result. Open Mapping Theorem. [Jarchow, 1981, 5.5.2] Any contin-

uous surjective linear mapping from a webbed locally convex space into an inductive

limit of Baire locally convex spaces vector spaces is open.

52.12. Result. The Fr´chet spaces are exactly the webbed spaces with the Baire

e

property.

This corresponds to [Jarchow, 1981, 5.4.4] by noting that Fr´chet spaces are Baire.

e

52.13. Result. [Jarchow, 1981, 5.3.3] Projective limits and inductive limits of

sequences of webbed spaces are webbed.

52.13

578 52. Appendix: Functional analysis 52.21

52.14. Result. The bornologi¬cation of a webbed space is webbed.

This follows from [Jarchow, 1981, 13.3.3 and 5.3.1.(d)] since the bornologi¬cation

is coarser that the ultrabornologi¬cation, [Jarchow, 1981, 13.3.1].

52.15. De¬nition. [Jarchow, 1981, 6.8] For a zero neighborhood U in a locally

convex vector space E we denote by E(U ) the completed quotient of E with the

Minkowski functional of U as norm.

52.16. Result. Hahn-Banach Theorem. [Jarchow, 1981, 7.3.3] Let E be a

locally convex vector space and let A ‚ E be a convex set, and let x ∈ E be not in

the closure of A. Then there exists a continuous linear functional with (x) not

in the closure of (A).

This is a consequence of the usual Hahn-Banach theorem, [Schaefer, 1971,II.9.2]

52.17. Result. [Jarchow, 1981, 7.2.4] Let x ∈ E be a point in a normed space.

Then there exists a continuous linear functional x ∈ E — of norm 1 with x (x) =

x.

This is another consequence of the usual Hahn-Banach theorem, cf. [Schaefer, 1971,

II.3.2].

52.18. Result. Bipolar Theorem. [Jarchow, 1981, 8.2.2] Let E be a locally

convex vector space and let A ‚ E. Then the bipolar Aoo in E with respect to the

dual pair (E, E — ) is the closed absolutely convex hull of A in E.

between vector spaces E and F and a set A ⊆ E the polar

For a duality ,

of A is Ao := {y ∈ F : | x, y | ¤ 1 for all x ∈ A}. The weak topology σ(E, F ) is

the locally convex topology on E generated by the seminorms x ’ | x, y | for all

y ∈ F.

52.19. Result. [Schaefer, 1971, IV.3.2] A subset of a locally convex vector space

is bounded if and only if every continuous linear functional is bounded on it.

This follows from [Jarchow, 1981, 8.3.4], since the weak topology σ(E, E ) and the

given topology are compatible with the duality, and a subset is bounded for the

weak topology, if and only if every continuous linear functional is bounded on it.

52.20. Result. Alao˜lu-Bourbaki Theorem. [Jarchow, 1981, 8.5.2 & 8.5.1.b]

g

& [Schaefer, 1971, III.4.3 and II.4.5] An equicontinuous subset K of E has compact

closure in the topology of uniform convergence on precompact subsets; On K the

latter topology coincides with the weak topology σ(E , E).

52.21. Result. [Jarchow, 1981, 8.5.3, p157] & [Schaefer, 1971, III.4.7] Let E be

a separable locally convex vector space. Then each equicontinuous subset of E is

metrizable in the weak— topology σ(E , E).

A topological space is called separable if it contains a dense countable subset.

52.21

52.29 52. Appendix: Functional analysis 579

52.22. Result. Banach Dieudonn´ theorem. [Jarchow, 1981, 9.4.3, p182] &

e

[Schaefer, 1971, IV.6.3] On the dual of a metrizable locally convex vector space E

the topology of uniform convergence on precompact subsets of E coincides with the

so-called equicontinuous weak— -topology which is the ¬nal topology induced by the

inclusions of the equicontinuous subsets.

52.23. Result. [Jarchow, 1981, 10.1.4] In metrizable locally convex spaces the

convergent sequences coincide with the Mackey-convergent ones.

For Mackey convergence see (1.6).

52.24. Result. [Jarchow, 1981, 10.4.3, p202] & [Horvath, 1966, p277] In Schwartz

spaces bounded sets are precompact.

A locally convex space E is called Schwartz if each absolutely convex neighborhood

U of 0 in E contains another one V such that the induced mapping E(U ) ’ E(V )

maps U into a precompact set.

52.25. Result. Uniform boundedness principle. [Jarchow, 1981, 11.1.1]

( [Schaefer, 1971, IV.5.2] for F = R) Let E be a barrelled locally convex vector

space and F be a locally convex vector space. Then every pointwise bounded set of

continuous linear mappings from E to F is equicontinuous.

Note that each Fr´chet space is barrelled, see [Jarchow, 1981, 11.1.5].

e

A locally convex space is called barrelled if each closed absorbing absolutely convex

set is a 0-neighborhood.

52.26. Result. [Jarchow, 1981, 11.5.1, 13.4.5] & [Schaefer, 1971, IV.5.5] Montel

spaces are re¬‚exive.

By a Montel space we mean (following [Jarchow, 1981, 11.5]) a locally convex vector

space which is barrelled and in which every bounded set is relatively compact. A

locally convex space E is called re¬‚exive if the canonical embedding of E into the

strong dual of the strong dual of E is a topological isomorphism.

52.27. Result. [Jarchow, 1981, 11.6.2, p231] Fr´chet Montel spaces are separable.

e

52.28. Result. [Jarchow, 1981, 12.5.8, p266] In the strong dual of a Fr´chet

e

Schwartz space every converging sequence is Mackey converging.

The strong dual of a locally convex space E is the dual E — of all continuous linear

functionals equipped with the topology of uniform convergence on bounded subsets

of E.

52.29. Result. Fr´chet Montel spaces have a bornological strong dual.

e

Proof. By (52.26) a Fr´chet Montel space E is re¬‚exive, thus it™s strong dual Eβ

e

is also re¬‚exive by [Jarchow, 1981, 11.4.5.(f)]. So it is barrelled by [Jarchow, 1981,

52.29

580 52. Appendix: Functional analysis 52.34

11.4.2]. By [Jarchow, 1981, 13.4.4] or [Schaefer, 1971, IV.6.6] the strong dual Eβ

of a metrizable locally convex vector space E is bornological if and only if it is

barrelled and the result follows.

52.30. Result. [Jarchow, 1981, 13.5.1] Inductive limits of ultrabornological spaces

are ultrabornological.

Similar to the de¬nition of bornological spaces in (4.1) we de¬ne ultrabornological

spaces, see [Jarchow, 1981, 13.1.1]. A bounded completant set B in a locally convex

vector space E is an absolutely convex bounded set B for which the normed space

(EB , qB ) is complete. A locally convex vector space E is called ultrabornological if

the following equivalent conditions are satis¬ed:

(1) For any locally convex vector space F a linear mapping T : E ’ F is

continuous if it is bounded on each bounded completant set. It is su¬cient

to know this for all Banach spaces F .

(2) A seminorm on E is continuous if it is bounded on each bounded completant

set.

(3) An absolutely convex subset is a 0-neighborhood if it absorbs each bounded

completant set.

52.31. Result. [Jarchow, 1981, 13.1.2] Every ultra-bornological space is an induc-

tive limit of Banach spaces.

In fact, E = limB EB where B runs through all bounded closed absolutely convex

’’

sets in E. Compare with the corresponding result (4.2) for bornological spaces.

52.32. Nuclear Operators. A linear operator T : E ’ F between Banach

spaces is called nuclear or trace class if it can be written in the form

∞

T (x) = »j x, xj yj ,

j=1

1

where xj ∈ E , yj ∈ F with xj ¤ 1, yj ¤ 1, and (»j )j ∈ . The trace of T is

then given by

∞

tr(T ) = »j yj , xj .

j=1

The operator T is called strongly nuclear if (»j )j ∈ s is rapidly decreasing.

52.33. Result. [Jarchow, 1981, 20.2.6] The dual of the Banach space of all trace

class operators on a Hilbert space consists of all bounded operators. The duality is

given by T, B = tr(T B) = tr(BT ).

52.34. Result. [Jarchow, 1981, 21.1.7] Countable inductive limits of strongly nu-

clear spaces are again strongly nuclear. Products and subspaces of strongly nuclear

spaces are strongly nuclear.

A locally convex space E is called nuclear (or strongly nuclear) if each absolutely

convex 0-neighborhood U contains another one V such that the induced mapping

52.34

52.37 52. Appendix: Functional analysis 581

E(V ) ’ E(U ) is a nuclear operator (or strongly nuclear operator). A locally convex

space is (strongly) nuclear if and only if its completion is it, see [Jarchow, 1981,

21.1.2]. Obviously, a nuclear space is a Schwartz space (52.24) since a nuclear op-

erator is compact. Since nuclear operators factor over Hilbert spaces, see [Jarchow,

1981, 19.7.5], each nuclear space admits a basis of seminorms consisting of Hilbert

norms, see [Schaefer, 1971, III.7.3].

52.35. Grothendieck-Pietsch criterion. Consider a directed set P of non-

negative real valued sequences p = (pn ) with the property that for each n ∈ N

there exists a p ∈ P with pn > 0. It de¬nes a complete locally convex space (called

K¨the sequence space)

o

Λ(P) := {x = (xn )n ∈ KN : p(x) := pn |xn | < ∞ for all p ∈ P}

n

with the speci¬ed seminorms.

Result. [Jarchow, 1981, 21.8.2] & [Treves, 1967, p. 530] The space Λ(P) is nuclear

if and only if for each p ∈ P there is a q ∈ P with

pn 1

∈ .

qn n

The space Λ(P) is strongly nuclear if and only if for each p ∈ P there is a q ∈ P

with

pn r

∈ .

qn n r>0

52.36. Result. [Jarchow, 1981, 21.8.3.b] H(Dk , C) is strongly nuclear for all k.

Proof. This is an immediate consequence of the Grothendieck-Pietsch criterion

(52.35) by considering the power series expansions in the polycylinder Dk at 0. The

set P consists of r(n1 , . . . , nk ) := rn1 +···+nk for all 0 < r < 1.

52.37. Silva spaces. A locally convex vector space which is an inductive limit

of a sequence of Banach spaces with compact connecting mappings is called a Silva

space. A Silva space is ultrabornological, webbed, complete, and its strong dual is

a Fr´chet space. The inductive limit describing the Silva space is regular. A Silva

e

space is Baire if and only if it is ¬nite dimensional. The dual space of a nuclear

Silva space is nuclear.

Proof. Let E be a Silva space. That E is ultrabornological and webbed follows

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)

¤ ( P± + Pβ ) x ’ xγ < 2 µ.

If x ∈ E is arbitrary, then P± (x) ∈ P± (E), hence by (1)

Pβ (x) = Pβ (P± (x)) ’ P± (P± (x)) = P± (x) for β ±.

(8) ⇐ (1) & (6) Let µ > 0. Then the set {β : β < ±, Rβ (x) ≥ µ} is ¬nite,

since otherwise there would be an increasing sequence (βn ) such that Rβn (x) ≥ µ

and since P±+1 ’ P± = (1 ’ P± )P±+1 ≥ 1 also (Pβn +1 ’ Pβn )(x) ≥ µ. Let

β∞ := supn βn . Then β∞ ¤ ± is a limit ordinal and Pβ∞ (x) = limβ<β∞ Pβ (x)

according to (6), a contradiction.

(9) ⇐ (6) We prove by trans¬nite induction that P± (x) is in the closure of the

linear span of {Rβ (x) : ω ¤ β < ±} ∪ Pω (x).

53.13

53.13 53. Appendix: Projective resolutions of identity on Banach spaces 589

For ± = ω this is obviously true. Let now ± = β + 1 and assume Pβ (x) is in

the closure of the linear span of {Rγ (x) : ω ¤ γ < β} ∪ Pω (x). Since P± (x) =

Pβ (x) + P± ’ Pβ Rβ (x) we get that P± (x) is in the closure of the linear span of

{Rγ (x) : ω ¤ γ < ±} ∪ Pω (x).

Let now ± be a limit ordinal and let Pβ (x) be in the closure of the linear span of

{Rγ (x) : ω ¤ γ < ±} ∪ Pω (x) for all β < ±. Then by (6) we get that P± (x) =

limβ<± Pβ (x) is in this closure as well.

Proposition. Suppose all complemented subspaces of a Banach space E have PRI

then E has a SPRI.

Proof. We proceed by induction on µ := dens E. For µ = ω nothing is to be

shown. Now let (P± )0¤±¤µ be a PRI of E. For every ± < µ we have ± + 1 < µ and

so µ± := dens((P±+1 ’ P± )(E)) ¤ dens(P±+1 (E)) ¤ ± < µ, hence there is a SPRI

±

(Pβ )0¤β<µ± of (P±+1 ’ P± )(E). Now consider

± ±

P±,β := P± + Pβ (P±+1 ’ P± ) = (P± + Pβ (1 ’ P± ))P±+1

for ω ¤ ± < µ and ω ¤ β ¤ µ± with the lexicographical ordering. This is a

well-ordering and since the cardinality of µ2 is µ and µ± < µ it corresponds to the

ordinal segment [ω, µ). In fact for any limit ordinal ± > ω we have

|[ω, µ± )| ¤ |[ω, ±)|2 ¤ |[ω, ±)|.

|[ω, ±)| = 1¤

ω¤β<± ω¤β<±

Obviously the P±,β are projections that satisfy (1) and (3).

±

(1) For P±,β with the same ± this follows from (1) for Pβ : R± (E) ’ R± (E): In

fact

± ±

P±,β P±,β := P± + Pβ (P±+1 ’ P± ) P± + Pβ (P±+1 ’ P± )

2 ± ±

= P± + Pβ (P±+1 ’ P± )P± + P± Pβ (P±+1 ’ P± )

± ±

+ Pβ (P±+1 ’ P± )Pβ (P±+1 ’ P± )

2 ±

= P± + 0 + 0 + Pmin{β,β } (P±+1 ’ P± )

For di¬erent ± this follows, since P±1 ,β E ⊆ P±1 +1 E ⊆ P±2 and

P± E ⊆ P±,β ⊆ P±+1

±

ker P± ⊇ ker P±,β = ker(P± + Pβ (1 ’ P± ))P±+1 ⊇ ker P±+1

(3) The density of P±,β E is less or equal to ± + 1.

±

And clearly they satisfy (7) as well, since R±,β = Rβ (P±+1 ’ P± ).

β

(9) Since this is true for the P± and the P± it follows for P±,β as well.

In fact P±,β (x) belongs to the closure of the linear span of P± (x) and the R±,β =

± ±

Rβ (P±+1 ’P± )(x) for β < β by the property of the Pβ . Furthermore P± (x) belongs

53.13

590 53. Appendix: Projective resolutions of identity on Banach spaces 53.14

to the closure of the linear span of R± (x) for ± < ± and Pω (x) by the property of

±

the P± and R± (x) belongs to the closure of the linear span of all Rβ (R± x) for all

β < dens R± E.

(8) For x in the linear span of all R±,β E we obviously have that (R±,β (x))±,β ∈ c0 .

n i i

In fact for x := i=1 » R±i ,βi (xi ), we have that R±i ,βi (x) = » R±i ,βi (xi ) and

R±,β (x) = 0 for all (±, β) ∈ {(±1 , β1 ), . . . , (±n , βn )}.

/

R± Rβ = (P±+1 ’ P± )(Pβ+1 ’ Pβ ) = (1 ’ P± )P±+1 Pβ+1 (1 ’ Pβ ) = 0,

if ± + 1 ¤ β or β + 1 ¤ ±, since the factors commute. For general x we ¬nd by (9)

a point x in the linear span of the R±,β x with x ’ x < µ. Then

˜ ˜

{(±, β) : R±,β (x) ≥ µ} ⊆ {(±1 , β1 ), . . . , (±n , βn )}.

Note however that we don™t have P±,β = 1.

53.14. Theorem. Let E be a Banach space with projective generator •. Then E

admits a PRI (P± )± , where each P± is based on a norming pair (A± , B± ) with

(1) |A± | ¤ ±, |B± | ¤ ± for all ±;

(2) Aβ ⊆ A± and Bβ ⊆ B± for all β ¤ ±;

(3) ω¤β<± Aβ = A± for all limit ordinals ±;

(4) ω¤β<± Bβ = B± for all limit ordinals ±;

Proof. Choose a dense subset {x± : ± < dens E}. We construct by trans¬nite

recursion for every ordinal ± ¤ dens E a norming pair (A± , B± ) with

A± ⊇ {xβ : β < ±}, |A± | ¤ ±, |B± | ¤ ±, •(B± ) ⊆ A±

Aβ ⊆ A± and Bβ ⊆ A± for β ¤ ±.

For the ordinal ω let A0 := {x± : ± < ω} and let B0 be a countable subset of E

such that

x = sup{| x, x— : x— ∈ B0 } for all x ∈ A0 .

By (53.7) there is a norming pair (Aω , Bω ) with |Aω |, |Bω | ¤ ω, Aω ⊇ A0 , Bω ⊇ B0

and •(Bω ) ⊆ Aω .

If ± is a successor ordinal, i.e. ± = β + 1, then let A0 := Aβ ∪ {xβ } and B0 := Bβ .

Again by (53.7) we get a norming pair (A± , B± ), such that

A0 ⊆ A± , B 0 ⊆ B± ⊆ E , |A± | ¤ ±, |B± | ¤ ±, •(B± ) ⊆ A±

If ± is a limit ordinal, we set

A± := Aβ

β<±

Bβ ⊆ E .

B± :=

β<±

53.14

53.17 53. Appendix: Projective resolutions of identity on Banach spaces 591

Then obviously (A± , B± ) is a norming pair with •(B± ) ⊆ A± .

Now using the property of the projective generator • we have that there are norm-1

projections P± ∈ L(E) with P± (E) = A± and ker P± = (B± )o = (B± )o . Hence

(53.13.1) P± Pβ = Pβ = Pβ P± for β ¤ ±

—

dens P± E ¤ ±, dens P± (E )σ ¤ ±,

(53.13.3)

(53.13.5) P± (E) = A± = Pβ E

β<±

(53.13.4) P± = 1

and since {x± : ± < dens E} is dense in E we also have (53.13.2). Furthermore we

have that B± is weak— -dense in P± E .

—

53.15. Corollary. WCD and duals of Asplund spaces have SPRI.

53.16. De¬nition. A compact set K is called Valdivia compact if there exists

some set “ with K ⊆ R“ and {x ∈ K : carr(x) is countable} being dense in K.

53.17. Lemma. For a Valdivia compact set K ⊆ R“ we consider the set E :=

{x ∈ R“ : carr(x) is countable}. Let µ be the density number of K © E. Then there

exists an increasing long sequence of subsets “± ⊆ “ for ω ¤ ± ¤ µ satisfying:

(i) |“± | ¤ ±;

(ii) β<± “β = “± for limit ordinals ±;

(iii) “µ = cup{carr(x) : x ∈ K};

and such that K± := Q“± (K) ⊆ K, where Q“ : R“ ’ R“ ’ R“ , i.e.

for γ ∈ “

xγ

Q“ (x)γ := .

for γ ∈ “ \ “

0 /

Thus K± ⊆ K is a retract via Q“± .

Note that for any Valdivia compact set K ⊆ R“ we may always replace “ by

{carr(x) : x ∈ K} = {carr(x) : x ∈ K © E}, and then (iii) says “µ = “.

Proof. The proof is based on the following claim: Let ∆ ⊆ “ be a in¬nite subset.

˜ ˜ ˜

Then there exists some subset ∆ with ∆ ⊆ ∆ ⊆ “ and |∆| = |∆| and Q∆ (K) ⊆ K.

˜

By induction we construct a sequence ∆ =: ∆0 ⊆ ∆1 ⊆ · · · ⊆ ∆k ⊆ · · · ⊆ “ with

|∆k | = |∆0 | and Q∆k ({x ∈ K © E : carr(x) ⊆ ∆k+1 }) being dense in Q∆k (K):

(k+1) Since K © E is dense in K, we have that Q∆k (K © E) is dense in Q∆k (K) ⊆

R∆k — {0} ⊆ R“ . And since the topology of R∆k has a basis of cardinality |∆k |,

there is a subset D ⊆ K © E with |D| ¤ |∆k | and Q∆k (D) dense in Q∆k (K). Let

∆k+1 := ∆k ∪ x∈D carr(x) then ∆k+1 ⊇ ∆k and |∆k+1 | = |∆k |. Furthermore

Q∆k ({x ∈ K © E : carr(x) ⊆ ∆k+1 }) ⊇ Q∆k (D) is dense in Q∆k (K).

˜