∞ k

k=0 Lsym (E, R). Since E is Banach, the space of formal power series is a Fr´chet

e

∞

space and since E is Cb (E, R)-regular the last mapping is injective by (27.19).

By E. Borel™s theorem (15.4) every bounded subset of the space of formal power

series is the image of a bounded subset of C ∞ (E, R). Hence this mapping is a

bornological isomorphism and the inductive limit C ∞ (E ⊇ {a}, R) is regular.

27.20

276 Chapter VI. In¬nite dimensional manifolds 28.1

27.21. Lemma. If M is smoothly regular then each germ at a point of a smooth

function has a representative which is de¬ned on the whole of M .

If M is smoothly paracompact then this is true for germs along closed subsets.

For germs of real analytic or holomorphic functions this is not true.

If M is as in the lemma, C ∞ (M ⊇ {x}, R) is the quotient of the algebra C ∞ (M, R)

by the ideal of all smooth functions f : M ’ R which vanish on some neighborhood

(depending on f ) of x.

The assumption in the lemma is not necessary as is shown by the following example:

By (14.9) the Banach space E := C([0, 1], R) is not C ∞ -regular, in fact not even

C 1 -regular. For h ∈ C ∞ (R, R) the push forward h— : C ∞ (R, R) ’ C ∞ (R, R) is

smooth, thus continuous, so (h— )— : C([0, 1], C ∞ (R, R)) ’ C([0, 1], C ∞ (R, R)) is

continuous. The arguments in the proof of theorem (3.2) show that

C([0, 1], C ∞ (R, R)) ∼ C ∞ (R, C([0, 1], R)),

=

thus h— : E ’ E is smooth. Let h(t) := t for |t| ¤ 1 and |h(t)| ¤ 1 for all t ∈ R.

2

1

In particular h— is the identity on {f ∈ E : f ¤ 2 }. Let U be a neighborhood

of 0 in E. Choose µ > 0 such that the closed ball with radius µ > 0 is contained

in U . Then hµ := µ h— 1 : E ’ E has values in U and is the identity near 0.

µ

Thus (hµ ) : C (U, R) ’ C ∞ (E, R) is a bounded algebra homomorphism, which

— ∞

respects the corresponding germs at 0.

28. Tangent Vectors

28.1. The tangent spaces of a convenient vector space E. Let a ∈ E. A

kinematic tangent vector with foot point a is simply a pair (a, X) with X ∈ E.

Let Ta E = E be the space of all kinematic tangent vectors with foot point a. It

consists of all derivatives c (0) at 0 of smooth curves c : R ’ E with c(0) = a,

which explains the choice of the name kinematic.

For each open neighborhood U of a in E (a, X) induces a linear mapping Xa :

C ∞ (U, R) ’ R by Xa (f ) := df (a)(X), which is continuous for the convenient vector

space topology on C ∞ (U, R) and satis¬es Xa (f · g) = Xa (f ) · g(a) + f (a) · Xa (g), so

it is a continuous derivation over eva . The value Xa (f ) depends only on the germ

of f at a.

An operational tangent vector of E with foot point a is a bounded derivation

‚ : C ∞ (E ⊇ {a}, R) ’ R over eva . Let Da E be the vector space of all these

derivations. Any ‚ ∈ Da E induces a bounded derivation C ∞ (U, R) ’ R over eva

for each open neighborhood U of a in E. Moreover any family of bounded deriva-

tions ‚U : C ∞ (U, R) ’ R over eva , which is coherent with respect to the restriction

maps, de¬nes an ‚ ∈ Da E. So the vector space Da E is a closed linear subspace of

the convenient vector space U L(C ∞ (U, R), R). We equip Da E with the induced

28.1

28.2 28. Tangent vectors 277

convenient vector space structure. Note that the spaces Da E are isomorphic for all

a ∈ E.

Taylor expansion induces the dashed arrows in the following diagram.

u

C ∞ (E, R) Lk (E, R)

g

e

eg

sym

e

ee

dk |0 prk

ee

u

“ ww

d

(U, R) '

{0} ‘

''

v‘

∞

Lk (E, R)

C

“ pr u

sym

'' “

’’

‘

k=1

’ '(

‘ ’'

“

‘

u u ’

{∞-¬‚at} v w C (E ⊇ {a}, R)‘ w L (E, R)

∞

v

∞ k

‘

(

& u

sym

& ‘“

k=1

& “

‘

u&

( ∞

{d-¬‚at} C (E ⊇ {a}, R)/{0}

Note that all spaces in the right two columns except the top right corner are

algebras, the ¬nite product with truncated multiplication. The mappings are

algebra-homomorphisms. And the spaces in the left column are the respective

∞

kernels. If E is a Cb (E, R)-regular Banach space, then by (27.20) the vertical

dashed arrow is bibounded. Since R is Hausdor¬ every ‚ ∈ Da E factors over

Taya (E, R) := C ∞ (E ⊇ {a}, R)/{0}, so in this case we can view ‚ as derivation on

the algebra of formal power series. Any continuous linear functional on a countable

product is a sum of continuous linear functionals on ¬nitely many factors.

28.2. Degrees of operational tangent vectors. A derivation ‚ is said to have

d

order at most d, if it factors over the space k=0 Lk (E, R) of polynomials of

sym

degree at most d, i.e. it vanishes on all d-¬‚at germs. If no such d exists, then it will

be called of in¬nite order; this may happen only if ‚ does not factor over the space

of formal power series, since if it factors to a bounded linear functional on the latter

space it depends only on ¬nitely many factors. If ‚ factors it must vanish ¬rst on

the ideal of all ¬‚at germs, and secondly the resulting linear functional on Taya (E, R)

must extend to a bounded linear functional on the space of formal power series.

For a results and examples in this direction see (28.3), (28.4), and (28.5). An open

question is to ¬nd operational tangent vectors of in¬nite order.

An operational tangent vector is said to be homogeneous of order d if it factors over

Ld (E, R), i.e. it corresponds to a continuous linear functional ∈ Ld (E, R) via

sym sym

(d)

‚(f ) = ( f d!(0) ). In order that such a functional de¬nes a derivation, we need

exactly that

d’1

Lj (E, R) — Ld’j (E, R) = {0},

Sym sym sym

j=1

28.2

278 Chapter VI. In¬nite dimensional manifolds 28.2

i.e. that vanishes on on the subspace

j’1

Li (E; R) ∨ Lj’i (E; R)

sym sym

i=1

of decomposable elements of Lj (E; R). Here Li (E; R) ∨ Lj’i (E; R) denotes

sym sym sym

the linear subspace generated by all symmetric products ¦∨Ψ of the corresponding

elements. Any such de¬nes an operational tangent vector ‚ j |a ∈ Da E of order j

by

‚ j |a (f ) := ( j! dj f (a)).

1

Since vanishes on decomposable elements we see from the Leibniz rule that ‚ j

is a derivation, and it is obviously of order j. The inverse bijection is given by

‚ ’ (¦ ’ ‚((¦ —¦ diag)( ’a))), since the complete polarization of a homogeneous

1

polynomial p of degree j is given by j! dj p(0)(v1 , . . . , vj ), and since the remainder

of the Taylor expansion is ¬‚at of order j ’ 1 at a.

Obviously every derivation of order at most d is a unique sum of homogeneous

[d]

derivations of order j for 1 ¤ j ¤ d. For d > 0 we denote by Da E the lin-

ear subspace of Da E of operational tangent vectors of homogeneous order d and

(d) d [j]

by Da E := j=1 D the subspace of (non homogeneous) operational tangent

vectors of order at most d.

In more detail any operational tangent vector ‚ ∈ Da E has a decomposition

k’1

‚ [i] + ‚ [k,∞] ,

‚=

i=1

which we obtain by applying ‚ to the Taylor formula with remainder of order k,

see (5.12),

k’1 1

(1 ’ t)k’1 k

1i

d f (a)y i + d f (a + ty)y k dt.

f (a + y) =

(k ’ 1)!

i! 0

i=0

Thus, we have

1i

‚ [i] (f ) := ‚ x ’ d f (a)(x ’ a)i ,

i!

1

(1 ’ t)(k’1) k

[k,∞]

d f (a + t(x ’ a))(x ’ a)k dt .

(f ) := ‚ x ’

‚

k ’ 1!

0

A simple computation shows that all ‚ [i] are derivations. In fact

k

1 k

‚ [k] (f · g) = ‚ (f (j) (0) —¦ ∆) · (g (k’j) (0) —¦ ∆)

k! j

j=0

k

f (j) (0) —¦ ∆ g (k’j) (0)(0(k’j) )

·

= ‚

(k ’ j)!

j!

j=0

k

f (j) (0)(0j ) g (k’j) (0) —¦ ∆

·‚

+

(k ’ j)!

j!

j=0

= g(0) · ‚ [k] (f ) + 0 + · · · + 0 + f (0) · ‚ [k] (g).

28.2

28.4 28. Tangent vectors 279

Hence also ‚ [k,∞] is a derivation. Obviously, ‚ [i] is of order i, and hence we get a

decomposition

d

[j] [d+1,∞]

Da • Da

Da E = ,

j=1

[d+1,∞]

where Da denotes the linear subspace of derivations which vanish on polyno-

mials of degree at most d.

28.3. Examples. Queer operational tangent vectors. Let Y ∈ E be an

element in the bidual of E. Then for each a ∈ E we have an operational tangent

vector Ya ∈ Da E, given by Ya (f ) := Y (df (a)). So we have a canonical injection

E ’ Da E.

Let : L2 (E; R) ’ R be a bounded linear functional which vanishes on the subset

[2]

E — E . Then for each a ∈ E we have an operational tangent vector ‚ |a ∈ Da E

[2]

given by ‚ |a (f ) := (d2 f (a)), since

(d2 (f g)(a)) = (d2 f (a)g(a) + df (a) — dg(a) + dg(a) — df (a) + f (a)d2 g(a))

= (d2 f (a))g(a) + 0 + f (a) (d2 g(a)).

Let E = ( 2 )N be a countable product of copies of an in¬nite dimensional Hilbert

space. A smooth function on E depends locally only on ¬nitely many Hilbert space

[kn ]

variables. Thus, f ’ n ‚Xn (f —¦ injn ) is a well de¬ned operational tangent vector

in D0 E for arbitrary operational tangent vectors Xn of order kn . If (kn ) is an

unbounded sequence and if Xn = 0 for all n it is not of ¬nite order. But only for

k = 1, 2, 3 we know that nonzero tangent vectors of order k exist, see (28.4) below.

28.4. Lemma. If E is an in¬nite dimensional Hilbert space, there exist nonzero

operational tangent vectors of order 2, 3.

Proof. We may assume that E = 2 . For k = 2 one knows that the closure of

L( 2 , R) ∨ L( 2 , R) in L2 ( 2 , R) consists of all symmetric compact operators, and

sym

the identity is not compact.

For k = 3 we show that for any A in the closure of L( 2 , R) ∨ L2 ( 2 , R) the

sym

following condition holds:

A(ei , ej , ek ) ’ 0 i, j, k ’ ∞.

(1) for

Since this condition is invariant under symmetrization it su¬ces to consider A ∈

2

—L( 2 , 2 ), which we may view as a ¬nite dimensional and thus compact operator

2

’ L( 2 , 2 ). Then A(ei ) ’ 0 for i ’ ∞, since this holds for each continuous

linear functional on 2 . The trilinear form A(x, y, z) := i xi yi zi is in L3 ( 2 , R)

sym

and obviously does not satisfy (1).

28.4

280 Chapter VI. In¬nite dimensional manifolds 28.7

28.5. Proposition. Let E be a convenient vector space with the following two

properties:

(1) The closure of 0 in C ∞ (E ⊇ {0}, R) consists of all ¬‚at germs.

(2) The quotient Tay0 (E, R) = C ∞ (E ⊇ {0}, R)/{0} with the bornological

topology embeds as topological linear subspace into the space k Lk (E; R)

sym

of formal power series.

Then each operational tangent vector on E is of ¬nite order.

∞

Any Cb -regular Banach space, in particular any Hilbert space has these properties.

Proof. Let ‚ ∈ D0 E be an operational tangent vector. By property (1) it factors

to a bounded linear mapping on Tay0 (E, R), it is continuous in the bornological

topology, and by property (2) and the theorem of Hahn-Banach ‚ extends to a

continuous linear functional on the space of all formal power series and thus depends

only on ¬nitely many factors.

∞

A Cb -regular Banach space E has property (1) by (27.19), and it has property (2)

∞

by E. Borel™s theorem (15.4). Hilbert spaces are Cb -regular by (15.5).

28.6. De¬nition. A convenient vector space is said to have the (bornological)

approximation property if E — E is dense in L(E, E) in the bornological locally

convex topology.

For a list of spaces which have the bornological approximation property see (6.6)“

(6.14).

28.7. Theorem. Let E be a convenient vector space which has the approxima-

tion property. Then we have Da E ∼ E . So if E is in addition re¬‚exive, each

=

operational tangent vector is a kinematic one.

Proof. We may suppose that a = 0. Let ‚ : C ∞ (E ⊇ {0}, R) ’ R be a derivation

at 0, so it is bounded linear and satis¬es ‚(f · g) = ‚(f ) · g(0) + f (0) · ‚(g). Then

we have ‚(1) = ‚(1 · 1) = 2‚(1), so ‚ is zero on constant functions.

Since E = L(E, R) is continuously embedded into C ∞ (E, R), ‚|E is an element

of the bidual E . Obviously, ‚ ’ (‚|E )0 is a derivation which vanishes on a¬ne

functions. We have to show that it is zero. We call this di¬erence again ‚. For

f ∈ C ∞ (U, R) where U is some radial open neighborhood of 0 we have

1

f (x) = f (0) + df (tx)(x)dt,

0

1

thus ‚(f ) = ‚(g), where g(x) := 0 df (tx)(x)dt. By assumption, there is a net

± ∈ E — E ‚ L(E, E) of bounded linear operators with ¬nite dimensional image,

which converges to IdE in the bornological topology of L(E, E). We consider g± ∈

1

C ∞ (U, R), given by g± (x) := 0 df (tx)( ± x)dt.

Claim. g± ’ g in C ∞ (U, R).

1

We have g(x) = h(x, x) where h ∈ C ∞ (U —E, R) is just h(x, y) = 0 df (tx)(y)dt. By

cartesian closedness, the associated mapping h∨ : U ’ E ‚ C ∞ (E, R)) is smooth.

28.7

28.9 28. Tangent vectors 281

Since : L(E, E) ’ L(E , E ) is bounded linear, the net ± converges to IdE in

L(E , E ). The mapping (h∨ )— : L(E , E ) ‚ C ∞ (E , E ) ’ C ∞ (U, E ) is bounded

linear, thus (h∨ )— ( ± ) converges to h∨ in C ∞ (U, E ). By cartesian closedness, the

net ((h∨ )— ( ± ))§ converges to h in C ∞ (U — E, R). Since the diagonal mapping

δ : U ’ U — E is smooth, the mapping δ — : C ∞ (U — E, R) ’ C ∞ (U, R) is

continuous and linear, so ¬nally g± = δ — (((h∨ )— ( ± ))§ ) converges to δ — (h) = g.

Claim. ‚(g± ) = 0 for all ±. This ¬nishes the proof.

n

•i — xi ∈ E — E ‚ L(E, E). We have

Let =

± i=1

1

g± (x) = df (tx) •i (x)xi dt

0 i

1

= •i (x) df (tx)(xi )dt =: •i (x)hi (x),

0

i i

•i · hi =

‚(g± ) = ‚ ‚(•i )hi (0) + •i (0)‚(hi ) = 0.

i i

28.8. Remark. There are no nonzero operational tangent vectors of order 2 on

E if and only if E ∨ E ‚ L2 (E; R) is dense in the bornological topology. This

sym

seems to be rather near the bornological approximation property, and one may

suspect that theorem (28.7) remains true under this weaker assumption.

28.9. Let U ⊆ E be an open subset of a convenient vector space E. The operational

tangent bundle DU of U is simply the disjoint union a∈U Da E. Then DU is in

bijection to the open subset U —D0 E of E—D0 (E) via ‚a ’ (a, ‚—¦( ’a)— ). We use

this bijection to put a smooth structure on DU . Let now g : E ⊃ U ’ V ‚ F be

a smooth mapping, then g — : C ∞ (W, R) ’ C ∞ (g ’1 (W ), R) is bounded and linear

for all open W ‚ V . The adjoints of these mappings uniquely de¬ne a mapping

Dg : DU ’ DV by (Dg.‚)(f ) := ‚(f —¦ g).

Lemma. Dg : DU ’ DV is smooth.

Proof. Via the canonical bijections DU ∼ U — D0 E and DV ∼ V — D0 F the

= =