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