Proposition. The space C ∞ (M ← E) of sections of the vector bundle (E, p, M )

with this structure satis¬es the uniform boundedness principle with respect to the

point evaluations evx : C ∞ (M ← E) ’ Ex for all x ∈ M .

If M is a separable manifold modeled on duals of nuclear Fr´chet spaces, and if

e

each ¬ber Ex is a nuclear Fr´chet space then C ∞ (M ← E) is a nuclear Fr´chet

e e

space and thus smoothly paracompact.

Proof. By de¬nition of the structure on C ∞ (M ← E) the uniform boundedness

principle follows from (5.26) via (5.25).

For the statement about nuclearity note that by (6.1) the spaces C ∞ (U± , V ) are

nuclear since we may assume that the U± form a countable cover of M by charts

which are di¬eomorphic to c∞ -open subsets of duals of nuclear Fr´chet spaces, and

e

closed subspaces of countable products of nuclear Fr´chet spaces are again nuclear

e

Fr´chet. By (16.10) nuclear Fr´chet spaces are smoothly paracompact.

e e

30.2. Lemma. Let M be a smooth manifold and let f : M ’ L(E, F ) be smooth,

where E and F are convenient vector spaces.

Then f— (h)(x) := f (x)(h(x)) is a linear bounded C ∞ (M, E) ’ C ∞ (M, F ) with the

natural structure of convenient vector spaces described in (27.17). The correspond-

ing statements in the real analytic and holomorphic cases are also true.

Proof. This follows from the uniform boundedness principles and the exponential

laws of (27.17).

30.2

30.3 30. Spaces of sections of vector bundles 295

30.3. Lemma. Under additional assumptions we have alternative descriptions of

the convenient structure on the vector space of sections C ∞ (M ← E):

(1) If M is smoothly regular, choose a smooth closed embedding E ’ M —F into

a trivial vector bundle with ¬ber a convenient vector space F by (29.8). Then

C ∞ (M ← E) can be considered as a closed linear subspace of C ∞ (M, F ),

with the natural structure from (27.17).

(2) If there exists a smooth linear covariant derivative with unique parallel

transport on p : E ’ M , see , then we equip C ∞ (M ← E) with the initial

structure with respect to the cone:

)—

Pt(c,

C (M ← E) ’ ’ ’ ’ C ∞ (R, Ec(0) ),

∞

’’’

s ’ (t ’ Pt(c, t)’1 s(c(t))),

where c ∈ C ∞ (R, M ) and Pt denotes the parallel transport.

The space C ∞ (M ← E) of sections of the vector bundle p : E ’ M with this struc-

ture satis¬es the uniform boundedness principle with respect to the point evaluations

evx : C ∞ (M ← E) ’ Ex for all x ∈ M .

If M is a separable manifold modeled on duals of nuclear Fr´chet spaces, and if

e

each ¬ber Ex is a nuclear Fr´chet space then C ∞ (M ← E) is a nuclear Fr´chet

e e

space and thus smoothly paracompact.

If in (1) M is even smoothly paracompact we may choose a ˜complementary™ smooth

vector bundle p : E ’ M such that the Whitney sum is trivial E •M E ∼ M — F ,

=

see also (29.8).

: X(M ) — C ∞ (M ← E) ’ C ∞ (M ← E) with

For a linear covariant derivative

unique parallel transport we require that the parallel transport Pt(c, t)v ∈ Ec(t)

along each smooth curve c : R ’ M for all v ∈ Ec(0) and t ∈ R is the unique

solution of the di¬erential equation ‚t Pt(c, t)v = 0. See (32.12) till (32.16).

Proof. This structure is independent of the choice of the vector bundle atlas,

because C ∞ (U± , V ) ’ β C ∞ (U±β , V ) is a closed linear embedding for any other

atlas (Uβ ).

The structures from (30.1) and (1) give even the same locally convex topology if

we equip C ∞ (M, F ) with the initial topology given by the following diagram.

wC

inj—

C ∞ (M ← E) ∞

(M, F )

u u

(pr2 —¦ψ± )—

wC

C ∞ (U± , V ) ∞

(U± , F )

where the bottom arrow is a push forward with the vector bundle embedding h :

U± ’ L(V, F ) of trivial bundles, given by h§ := pr2 —¦ inj —¦ψ± : U± — V ’ F , which

’1

is bounded by (30.2).

30.3

296 Chapter VI. In¬nite dimensional manifolds 30.4

We now show that the identity from description (2) to description (30.1) is bounded.

The restriction mapping C ∞ (M ← E) ’ C ∞ (U± ← E|U± ) is obviously bounded

for description (2) on both sides. Hence, it su¬ces to check for a trivial bundle E =

M — V , that the identity from description (2) to description (30.1) is bounded. For

the constant parallel transport Ptconst the result follows from proposition (27.17).

The change to an arbitrary parallel transport is done as follows: For each C ∞ -curve

c : R ’ M the diagram

wC

)—

Pt (c,

∞ ∞

C (M, V ) (R, V )

¢

u

h—

Ptconst (c, )— = c—

C ∞ (R, V )

commutes, where h : R ’ GL(V ) is given by h(t)(v) = Pt(c, t)v with inverse

h’1 (t)(w) = Pt(c, t)’1 w = Pt(c( +t), ’t)w, and its push forward is bibounded

by (30.2).

Finally, we show that the identity from description (30.1) to description (2) is

bounded. The structure on C ∞ (R, Ec(0) ) is initial with respect to the restriction

maps to a covering by intervals I which is subordinated to the cover c’1 (U± ) of

)—

Pt(c,

R. Thus, it su¬ces to show that the map C (M ← E) ’ ’ ’ ’ C ∞ (I, Ec(0) ) is

∞

’’’

bounded for the structure (30.1) on C ∞ (M ← E). This map factors as

wC

(pr2 —¦ψ± )—

C ∞ (M ← E) ∞

(U± , V )

)—

u

Pt(c,

(c|I)—

C ∞ (R, Ec(0) )

u u

u h—

C ∞ (I, Ec(0) ) C ∞ (I, V )

where

h(t)(v) := Pt(c, t)’1 (ψ± (c(t), v)) = Pt(c(

’1 ’1

+t), ’t)(ψ± (c(t), v))

is again a smooth map I ’ L(V, Ec(0) )).

30.4. Spaces of smooth sections with compact supports. For a smooth

vector bundle p : E ’ M with ¬nite dimensional second countable base M and

∞

standard ¬ber V we denote by Cc (M ← E) the vector space of all smooth sections

with compact supports in M .

∞

Lemma. The following structures of a convenient vector space on Cc (M ← E)

are all equivalent:

∞

(1) Let CK (M ← E) be the space of all smooth sections of E ’ M with supports

contained in the ¬xed compact subset K ‚ M , a closed linear subspace of

30.4

30.4 30. Spaces of sections of vector bundles 297

C ∞ (M ← E). Consider the ¬nal convenient vector space structure on

∞

Cc (M ← E) induced by the cone

∞ ∞

CK (M ← E) ’ Cc (M ← E)

∞

where K runs through a basis for the compact subsets of M . Then Cc (M ←

∞

E) is even the strict and regular inductive limit of spaces CK (M ← E)

where K runs through a countable base of compact sets.

(2) Choose a second smooth vector bundle q : E ’ M such that the Whitney

sum is trivial (29.8): E•E ∼ M —F . Then Cc (M ← E) can be considered

∞

=

∞

as a closed direct summand of Cc (M, F ).

∞

The space Cc (M ← E) satis¬es the uniform boundedness principle with respect to

the point evaluations. Moreover, if the standard ¬ber V is a nuclear Fr´chet space

e

∞

and the base M is in addition separable then Cc (M ← E) is smoothly paracompact.

Proof. Since CK (M ← E) is closed in C ∞ (M ← E) the inductive limit CK (M ←

∞ ∞

∞ ∞

E) ’ Cc (M ← E) is strict. So the limit is regular (52.8) and hence Cc (M ← E)

∞

is convenient with the structure in (1). The direct sum property CK (M ← E) ‚

∞

CK (M, F ) from (30.3.1) passes through the direct limits, so the equivalence of

statements (1) and (2) follows.

∞

We now show that Cc (M ← E) satis¬es the uniform boundedness principle for the

point evaluations. Using description (2) and (5.25) for a direct sum we may assume

∞

that the bundle is trivial, hence we only have to consider Cc (M, V ) for a convenient

∞

vector space V . Now let F be a Banach space, and let f : F ’ Cc (M, V ) be a

linear mapping, such that evx —¦f : F ’ V is bounded for each x ∈ M . Then by

the uniform boundedness principle (27.17) it is bounded into C ∞ (M, V ). We claim

∞

that f has values even in CK (M, V ) for some K, so it is bounded therein, and

∞

hence in Cc (M, V ), as required.

If not we can recursively construct the following data: a discrete sequence (xn ) in

M , a bounded sequence (yn ) in the Banach space F , and linear functionals n ∈ V

such that ±

= 0 if n < k,

| k (f (yn )(xk ))| = 1 if n = k,

< 1 if n > k.

Namely, we choose yn ∈ F and xn ∈ M such that f (yn )(xn ) = 0 in V , and xn has

distance 1 to m<n supp(f (ym )) (in a complete Riemannian metric, where closed

bounded subsets are compact). By shrinking yn we may get | m (f (yn )(xm ))| < 1

for m < n. Then we choose n ∈ V such that n (f (yn )(xn )) = 1.

Then y := n 21 yn ∈ F , and f (y)(xk ) = 0 for all k since | k (f (y)(xk ))| > 0. So

n

∞

f (y) ∈ Cc (M, V ).

/

For the last assertion, if the standard ¬ber V is a nuclear Fr´chet space and the

e

base M is separable then C ∞ (M ← E) is a nuclear Fr´chet space by the propo-

e

∞

sition in (30.1), so each closed linear subspace CK (M ← E) is a nuclear Fr´chet

e

∞

space, and by (16.10) the countable strict inductive limit Cc (M ← E) is smoothly

paracompact.

30.4

298 Chapter VI. In¬nite dimensional manifolds 30.6

30.5. Spaces of holomorphic sections. Let q : F ’ N be a holomorphic

vector bundle over a complex (i.e., holomorphic) manifold N with standard ¬ber

V , a complex convenient vector space. We denote by H(N ← F ) the vector space

of all holomorphic sections s : N ’ F , equipped with the topology which is initial

with respect to the cone

(pr —¦ψ± )—

’ 2 ’’

H(N ← F ) ’ H(U± ← F |U± ) ’ ’ ’ ’ H(U± , V )

’

where the convenient structure on the right hand side is described in (27.17), see

also (7.21).

By (5.25) and (8.10) the space H(N ← F ) of sections satis¬es the uniform bound-

edness principle for the point evaluations.

For a ¬nite dimensional holomorphic vector bundle the topology on H(N ← F )

turns out to be nuclear and Fr´chet by (8.2), so by (16.10) H(N ← F ) is smoothly

e

paracompact.

30.6. Spaces of real analytic sections. Let p : E ’ M be a real analytic

vector bundle with standard ¬ber V . We denote by C ω (M ← E) the vector space

of all real analytic sections. We will equip it with one of the equivalent structures

of a convenient vector space described in the next lemma.

Lemma. The following structures of a convenient vector space on the space of

sections C ω (M ← E) are all equivalent:

(1) Choose a vector bundle atlas (U± , ψ± ), and consider the initial structure

with respect to the cone

(pr —¦ψ± )—

C ω (M ← E) ’ C ω (U± ← E|U± ) ’ ’ ’ ’ C ω (U± , V ),

’ 2 ’’

’

where the spaces C ω (U± , V ) are equipped with the structure of (27.17).

(2) If M is smoothly regular, choose a smooth closed embedding E ’ M — F

into a trivial vector bundle with ¬ber a convenient vector space F . Then

C ω (M ← E) can be considered as a closed linear subspace of C ω (M, F ).

The space C ω (M ← E) satis¬es the uniform boundedness principle for the point

evaluations evx : C ω (M ← E) ’ Ex .

If the base manifold is compact ¬nite dimensional real analytic, and if the standard

¬ber is a ¬nite dimensional vector space, then C ω (M ← E) is smoothly paracom-

pact.

Proof. We use the following diagram

wC

inj—

C ω (M ← E) ω

(M, F )

u u

(pr2 —¦ψ± )—

wC

C ω (U± , V ) ω

(U± , F ),

30.6

30.9 30. Spaces of sections of vector bundles 299

where the bottom arrow is a push forward with the vector bundle embedding h :

U± ’ L(V, F ) of trivial bundles, given by h§ := pr2 —¦ inj —¦ψ± : U± — V ’ F , which

’1

is bounded by (30.2). The uniform boundedness principle follows from (11.12).

For proving that C ω (M ← E) is smoothly paracompact we use the second descrip-

tion. Then C ω (M ← E) is a direct summand in a space C ω (M, V ), where M is

a compact real analytic manifold and V is a ¬nite dimensional real vector space.

The function space C ω (M, V ) is smoothly paracompact by (11.4).

30.7. C ∞,ω -mappings. Let M and N be real analytic manifolds. A mapping

f : R — M ’ N is said to be of class C ∞,ω if for each (t, x) ∈ R — M and each

real analytic chart (V, v) of N with f (t, x) ∈ V there are a real analytic chart

(U, u) of M with x ∈ U , an open interval t ∈ I ‚ R such that f (I — U ) ‚ V , and

v —¦ f —¦ (I — u’1 ) : I — u(U ) ’ v(V ) is of class C ∞,ω in the sense of (11.20), i.e., the

canonical associate is a smooth mapping (v —¦ f —¦ (I — u’1 ))∨ : I ’ C ω (u(U ), v(V )).

The mapping is said to be C ω,∞ if the canonical associate is a real analytic mapping

(v —¦ f —¦ (I — u’1 ))∨ : I ’ C ∞ (u(U ), v(V )), see (11.20.2).

These notions are well de¬ned by the composition theorem for C ∞,ω -mappings

(11.22), and the obvious generalization of (11.21) is true.

We choose one factor to be R because we need the c∞ -topology of the product to

be the product of the c∞ -topologies, see (4.15) and (4.22).

30.8. Lemma. Curves in spaces of sections.

(1) For a smooth vector bundle p : E ’ M a curve c : R ’ C ∞ (M ← E) is

smooth if and only if c§ : R — M ’ E is smooth.

(2) For a holomorphic vector bundle p : E ’ M a curve c : D ’ H(M ← E)

is holomorphic if and only if c§ : D — M ’ E is holomorphic.

(3) For a real analytic vector bundle p : E ’ M a curve c : R ’ C ω (M ← E)

is real analytic if and only if the associated mapping c§ : R — M ’ E is

real analytic.

(4) For a real analytic vector bundle p : E ’ M a curve c : R ’ C ω (M ← E)

is smooth if and only if c§ : R — M ’ E is C ∞,ω , see (30.7). A curve

c : R ’ C ∞ (M ← E) is real analytic if and only if c§ : R — M ’ E is

C ω,∞ , see (11.20).

Proof. By the descriptions of the structures ((30.1) for the smooth case, (30.5) for

the holomorphic case, and (30.6) for the real analytic case) we may assume that M

is open in a convenient vector space F , and we may consider functions with values

in the standard ¬ber instead of sections. The statements then follow from the

respective exponential laws ((3.12) for the smooth case, (7.22) for the holomorphic

case, (11.18) for the real analytic case, and the de¬nition in (11.20) for the C ∞,ω

and C ω,∞ cases).

30.9. Lemma (Curves in spaces of sections with compact support).

(1) For a smooth vector bundle p : E ’ M with ¬nite dimensional base manifold

M a curve c : R ’ Cc (M ← E) is smooth if and only if c§ : R — M ’ E

∞

30.9

300 Chapter VI. In¬nite dimensional manifolds 30.11

is smooth and satis¬es the following condition:

For each compact interval [a, b] ‚ R there is a compact subset

K ‚ M such that c§ (t, x) is constant in t ∈ [a, b] for all x ∈

M \ K.

(2) For a real analytic ¬nite dimensional vector bundle p : E ’ M a curve

c : R ’ Cc (M ← E) is real analytic if and only if c§ satis¬es the condition

∞

of (1) above and c§ : R — M ’ E is C ω,∞ , see (30.7).

Compare this with (42.5) and (42.12).

∞

Proof. By lemma (30.4.1) a curve c : R ’ Cc (M ← E) is smooth if it factors

∞

locally as a smooth curve into some step CK (M ← E) for some compact K in M ,

and this is by (30.8.1) equivalent to smoothness of c§ and to condition (1). An

analogous proof applies to the real analytic case.

30.10. Corollary. Let p : E ’ M and p : E ’ M be smooth vector bundles

with ¬nite dimensional base manifold. Let W ⊆ E be an open subset, and let

∞

f : W ’ E be a ¬ber respecting smooth (nonlinear) mapping. Then Cc (M ←

∞

W ) := {s ∈ Cc (M ← E) : s(M ) ⊆ W } is open in the convenient vector space

∞ ∞ ∞

Cc (M ← E). The mapping f— : Cc (M ← W ) ’ Cc (M ← E ) is smooth with

∞ ∞ ∞

derivative (dv f )— : Cc (M ← W ) — Cc (M ← E) ’ Cc (M ← E ), where the

d

vertical derivative dv f : W —M E ’ E is given by dv f (u, w) := dt |0 f (u + tw).

∞ ∞

If the vector bundles and f are real analytic then f— : Cc (M ← W ) ’ Cc (M ←

E ) is real analytic with derivative (dv f )— .

If M is compact and the vector bundles and f are real analytic then C ω (M ←

W ) := {s ∈ C ω (M ← E) : s(M ) ⊆ W } is open in the convenient vector space

C ω (M ← E), and the mapping f— : C ω (M ← W ) ’ C ω (M ← E ) is real analytic

with derivative (dv f )— .

∞ ∞

Proof. The set Cc (M ← W ) is open in Cc (M ← E) since its intersection with

∞

each CK (M ← E) is open therein, see corollary (4.16), and the colimit is strict