The counter-example for embeddings given for the projective tensor product works

also, since all spaces involved are Banach.

Since the bornological tensor-product preserves coproducts it cannot preserve prod-

ucts. In fact (R —β R(N) )N ∼ (R(N) )N whereas RN —β R(N) ∼ (RN —β R)(N) ∼

= = =

(RN )(N) .

5.7

56 Chapter I. Calculus of smooth mappings 5.9

5.8. Proposition. Projective versus bornological tensor product. If every

bounded bilinear mapping on E — F is continuous then E —π F = E —β F . In

particular, we have E —π F = E —β F for any two metrizable spaces, and for a

normable space F we have Eborn —π F = E —β F .

Proof. Recall that E —π F carries the ¬nest locally convex topology such that

— : E — F ’ E — F is continuous, whereas E —β F carries the ¬nest locally

convex topology such that — : E — F ’ E — F is bounded. So we have that

— : E — F ’ E —β F is bounded and hence by assumption continuous, and thus

the topology of E —π F is ¬ner than that of E —β F . Since the converse is true in

general, we have equality.

In (52.23) it is shown that in metrizable locally convex spaces the convergent se-

quences coincide with the Mackey-convergent ones. Now let T : E — F ’ G be

bounded and bilinear. We have to show that T is continuous. So let (xn , yn ) be

a convergent sequence in E — F . Without loss of generality we may assume that

its limit is (0, 0). So there are µn ’ ∞ such that {µn (xn , yn ) : n ∈ N} is bounded

and hence also T {µn (xn , yn ) : n ∈ N} = µ2 T (xn , yn ) : n ∈ N , i.e. T (xn , yn )

n

converges even Mackey to 0.

If F is normable and T : Eborn — F ’ G is bounded bilinear then T ∨ : Eborn ’

L(F, G) is bounded, and since Eborn is bornological it is even continuous. Clearly,

for normed spaces F the evaluation map ev : L(F, G) — F ’ G is continuous, and

hence T = ev —¦(T ∨ — F ) : Eborn — F ’ G is continuous. Thus, Eborn —π F =

E —β F .

Note that the bornological tensor product is invariant under bornologi¬cation, i.e.

Eborn —β Fborn ∼ E —β F . So it is no loss of generality to assume that both spaces

=

are bornological. Keep however in mind that the corresponding identity for the

projective tensor product does not hold. Another possibility to obtain the identity

E —π F = E —β F is to assume that E and F are bornological and every separately

continuous bilinear mapping on E —F is continuous. In fact, every bounded bilinear

mapping is obviously separately bounded, and since E and F are assumed to be

bornological, it has to be separately continuous. We want to ¬nd another class

beside the Fr´chet spaces (see (52.9)) which satis¬es these assumptions.

e

5.9. Corollary. The following mappings are bounded multilinear.

(1) lim : Nat(F, G) ’ L(lim F, lim G), where F and G are two functors on

the same index category, and where Nat(F, G) denotes the space of all

natural transformations with the structure induced by the embedding into

i L(F(i), G(i)).

(2) colim : Nat(F, G) ’ L(colim F, colim G).

(3)

L : L(E1 , F1 ) — . . . —L(En , Fn ) — L(F, E) ’

’ L(L(F1 , . . . , Fn ; F ), L(E1 , . . . , En ; E))

(T1 , . . . , Tn , T ) ’ (S ’ T —¦ S —¦ (T1 — . . . — Tn ));

5.9

5.9 5. Uniform boundedness principles and multilinearity 57

n

: L(E1 , F1 ) — . . . — L(En , Fn ) ’ L(E1 —β · · · —β En , F1 —β · · · —β Fn ).

(4) β

n n n n

: L(E, F ) ’ L(

(5) E, F ), where E is the linear subspace of all

n

alternating tensors in β E. It is the universal solution of

n

∼ Ln (E; F ),

L E, F = alt

where Ln (E; F ) is the space of all bounded n-linear alternating mappings

alt

E — ... — E ’ F.

n n n n

: L(E, F ) ’ L(

(6) E, F ), where E is the linear subspace of all

n

symmetric tensors in β E. It is the universal solution of

n

∼ Ln (E; F ),

L E, F = sym

where Ln (E; F ) is the space of all bounded n-linear symmetric mappings

sym

E — ... — E ’ F.

n

∞

β : L(E, F ) ’ L( n=0 —β E is the tensor

(7) β E, β F ), where β E :=

algebra of E. Note that is has the universal property of prolonging bounded

linear mappings with values in locally convex spaces, which are algebras with

bounded operations, to continuous algebra homomorphisms:

L(E, F ) ∼ Alg( E, F ).

=

β

∞ n

: L(E, F ) ’ L( E, F ), where E :=

(8) E is the exterior

n=0

algebra. It has the universal property of prolonging bounded linear mappings

to continuous algebra homomorphisms into graded-commutative algebras,

i.e. algebras in the sense above, which are as vector spaces a coproduct

n∈N En and the multiplication maps Ek — El ’ Ek+l and for x ∈ Ek and

y ∈ El one has x · y = (’1)kl y · x.

∞ n

: L(E, F ) ’ L( E, F ) , where E := n=0

(9) E is the symmetric

algebra. It has the universal property of prolonging bounded linear mappings

to continuous algebra homomorphisms into commutative algebras.

Recall that the symmetric product is given as the image of the symmetrizer sym :

E —β · · · —β E ’ E —β · · · —β E given by

1

x1 — · · · — xn ’ xσ(1) — · · · — xσ(n) .

n!

σ∈Sn

Similarly the wedge product is given as the image of the alternator

alt : E —β · · · —β E ’ E —β · · · —β E

1

given by x1 — · · · — xn ’ sign(σ) xσ(1) — · · · — xσ(n) .

n!

σ∈Sn

5.9

58 Chapter I. Calculus of smooth mappings 5.11

Symmetrizer and alternator are bounded projections, so both subspaces are com-

plemented in the tensor product.

Proof. All results follow easily by ¬‚ipping coordinates until only a composition of

products of evaluation maps remains.

That the spaces in (5), and similar in (6), are universal solutions can be seen from

the following diagram:

w E — · · · — E ew

k

— alt

E — ... — E E

˜ ee

β β

f eef˜|

uh

e e

f k

E

F

5.10. Lemma. Let E be a convenient vector space. Then E ’ Pf (E) :=

E alg ⊆ C ∞ (E, R) is the free commutative algebra over the vector space E , i.e. to

every linear mapping f : E ’ A in a commmutative algebra, there exists a unique

˜

algebra homomorphism f : Pf (E) ’ A.

Elements of the space Pf (E) are called polynomials of ¬nite type on E.

Proof. The solution of this universal problem is given by the symmetric alge-

∞ k

bra E := E described in (5.9.9). In particular we have an algebra

k=0

homomorphism ˜ : E ’ Pf (E), which is onto, since by de¬nition Pf (E) is gen-

ι

N

k=1 ±k ∈

erated by E . It remains to show that it is injective. So let E,

k N

i.e. ±k ∈ E , with ˜( k=1 ±k ) = 0. Thus all derivatives ι(±k ) at 0 of this

ι

mapping in Pf (E) ⊆ P (E) ⊆ C ∞ (E, R) vanish. So it remains to show that

k

E ’ L(E, . . . , E; R) is injective, since then by the polarization identity also

k

E ’ Pf (E) ⊆ C ∞ (E, R) is injective. Let ± ∈ E — F be zero as element on

L(E, F ; R). We have ¬nitely many xk ∈ E and y k ∈ F with ± = k xk — y k and

we may assume that the {xk } are linearly independent. So we may choose vectors

xj ∈ E with xk (xj ) = δj . Then 0 = ±(xj , y) = k xk (xj ) · y k (y) = y j (y), so y j = 0

k

for all j and hence ± = 0.

Now for the mapping E1 — · · · — En ’ L(E1 , . . . , En ; R). We proceed by induction.

Let ± = k ±k — xk , where ±k ∈ E1 — En’1 and xk ∈ En . We may assume that

(xk )k is linearly independent. So, as before, we choose xj ∈ En with xk (xj ) = δj k

and get 0 = ±(y 1 , . . . , y n’1 , xj ) = ±j (y 1 , . . . , y n’1 ), hence ±j = 0 for all j and so

± = 0.

5.11. Corollary. Symmetry of higher derivatives. Let f : E ⊇ U ’ F

be smooth. The n-th derivative f (n) (x) = dn f (x), considered as an element of

Ln (E; F ), is symmetric, so has values in the space Lsym (E, . . . , E; F ) ∼ L( E; F )

k

=

5.11

5.13 5. Uniform boundedness principles and multilinearity 59

Proof. Recall that we can form iterated derivatives as follows:

f :E⊇U ’F

df : E ⊇ U ’ L(E, F )

d(df ) : E ⊇ U ’ L(E, L(E, F )) ∼ L(E, E; F )

=

.

.

.

d(. . . (d(df )) . . . ) : E ⊇ U ’ L(E, . . . , L(E, F ) . . . ) ∼ L(E, . . . , E; F )

=

Thus, the iterated derivative dn f (x)(v1 , . . . , vn ) is given by

˜

‚ ‚

‚t1 |t1 =0 · · · ‚tn |tn =0 f (x + t1 v1 + · · · + tn vn ) = ‚1 . . . ‚n f (0, . . . , 0),

˜

where f (t1 , . . . , tn ) := f (x + t1 v1 + · · · + tn vn ). The result now follows from the

¬nite dimensional property.

5.12. Theorem. Taylor formula. Let f : U ’ F be smooth, where U is c∞ -

open in E. Then for each segment [x, x + y] = {x + ty : 0 ¤ t ¤ 1} ⊆ U we

have

n 1

(1 ’ t)n n+1

1k k

f (x + ty)y n+1 dt,

f (x + y) = d f (x)y + d

k! n!

0

k=0

where y k = (y, . . . , y) ∈ E k .

Proof. This is an assertion on the smooth curve t ’ f (x + ty). Using functionals

we can reduce it to the scalar valued case, or we proceed directly by induction on

n: The ¬rst step is (6) in (2.6), and the induction step is partial integration of the

remainder integral.

5.13. Corollary. The following subspaces are direct summands:

L(E1 , . . . , En ; F ) ⊆ C ∞ (E1 — . . . — En , F ),

Ln (E; F ) ⊆ Ln (E; F ) := L(E, . . . , E; F ),

sym

Ln (E; F ) ⊆ Ln (E; F ),

alt

Ln (E; F ) ’ C ∞ (E, F ).

sym

Note that direct summand is meant in the bornological category, i.e. the embedding

admits a left-inverse in the category of bounded linear mappings, or, equivalently,

with respect to the bornological topology it is a topological direct summand.

Proof. The projection for L(E, F ) ⊆ C ∞ (E, F ) is f ’ df (0). The statement on

Ln follows by induction using cartesian closedness and (5.2). The projections for

the next two subspaces are the symmetrizer and alternator, respectively.

The last embedding is given by — , which is bounded and linear C ∞ (E — . . . —

E, F ) ’ C ∞ (E, F ). Here ∆ : E ’ E — . . . — E denotes the diagonal mapping

5.13

60 Chapter I. Calculus of smooth mappings 5.15

x ’ (x, . . . , x). A bounded linear left inverse C ∞ (E, F ) ’ Lk (E; F ) is given by

sym

1k

f ’ k! d f (0). See the following diagram:

y w L(E, . . .y , E; F )

Lk (E; F )

sym

u u

u

∞ ∞

C (E — . . . — E, F )

C (E, F )

∆—

5.14. Remark. We are now going to discuss polynomials between locally convex

spaces. Recall that for ¬nite dimensional spaces E = Rn a polynomial in a locally

convex vector space F is just a ¬nite sum

ak xk ,

k∈Nn

n

where ak ∈ F and xk := i=1 xki . Thus, it is just an element in the algebra gener-

i

ated by the coordinate projections pri tensorized with F . Since every (continuous)

linear functional on E = Rn is a ¬nite linear combination of coordinate projections,

this algebra is also the algebra generated by E — . For a general locally convex space

E we de¬ne the algebra of ¬nite type polynomials to be the one generated by E — .

However, there is also another way to de¬ne polynomials, namely as those smooth

functions for which some derivative is equal to 0. Take for example the square

2

: E ’ R on some in¬nite dimensional Hilbert space E. Its

of the norm

derivative is given by x ’ (v ’ 2 x, v ), and hence is linear. The second derivative

is x ’ ((v, w) ’ 2 v, w ) and hence constant. Thus, the third derivative vanishes.

This function is not a ¬nite type polynomial. Otherwise, it would be continuous

for the weak topology σ(E, E — ). Hence, the unit ball would be a 0-neighborhood

for the weak topology, which is not true, since it is compact for it.

Note that for (xk ) ∈ 2 the series k x2 converges pointwise and even uniformly

k

on compact sets. In fact, every compact set is contained in the absolutely convex

hull of a 0-sequence xn . In particular µk := sup{|xn | : n ∈ N} ’ 0 for k ’ ∞

k

nj

(otherwise, we can ¬nd an µ > 0 and kj ’ ∞ and nj ∈ N with xnj 2 ≥ |xkj | ≥ µ.

Since xn ∈ 2 ⊆ c0 , we conclude that nj ’ ∞, which yields a contradiction to

xn 2 ’ 0.) Thus

K ⊆ xn : n ∈ N ⊆ µn en absolutely convex ,

absolutely convex

and hence k≥n |xk | ¤ max{µk : k ≥ n} for all x ∈ K.

The series does not converge uniformly on bounded sets. To see this choose x = ek .

5.15. De¬nition. A smooth mapping f : E ’ F is called a polynomial if some

derivative dn f vanishes on E. The largest p such that dp f = 0 is called the degree

of the polynomial. The mapping f is called a monomial of degree p if it is of the

˜ ˜

form f (x) = f (x, . . . , x) for some f ∈ Lp (E; F ).

sym

5.15

5.18 5. Uniform boundedness principles and multilinearity 61

5.16. Lemma. Polynomials versus monomials.

(1) The smooth p-homogeneous maps are exactly the monomials of degree p.

(2) The symmetric multilinear mapping representing a monomial is unique.

(3) A smooth mapping is a polynomial of degree ¤ p if and only if its restriction

to each one dimensional subspace is a polynomial of degree ¤ p.

(4) The polynomials are exactly the ¬nite sums of monomials.

Proof. (1) Every monomial of degree p is clearly smooth and p-homogeneous. If

f is smooth and p-homogeneous, then

(dp f )(0)(x, . . . , x) = ( ‚t )p f (tx) = ( ‚t )p tp f (x) = p!f (x).

‚ ‚

t=0 t=0

(2) The symmetric multilinear mapping g ∈ Lp (E; F ) representing f is uniquely

sym

p

determined, since we have (d f )(0)(x1 , . . . , xp ) = p!g(x1 , . . . , xp ).

(3) & (4) Let the restriction of f to each one dimensional subspace be a polynomial

k

p

of degree ¤ p, i.e., we have (f (tx)) = k=0 t ( ‚t )k t=0 (f (tx)) for x ∈ E and ∈

‚

k!

p 1

F . So f (x) = k=0 k! dk f (0.x)(x, . . . , x) and hence is a ¬nite sum of monomials.

For the derivatives of a monomial q of degree k we have q (j) (tx)(v1 , . . . , vj ) =

k(k ’ 1) . . . (k ’ j + 1)tk’j q (x, . . . , x, v1 , . . . , vj ). Hence, any such ¬nite sum is a

˜

polynomial in the sense of (5.15).

Finally, any such polynomial has a polynomial as trace on each one dimensional

subspace.

5.17. Lemma. Spaces of polynomials. The space Polyp (E, F ) of polynomi-

k

als of degree ¤ p is isomorphic to k¤p L( E; F ) and is a direct summand in

C ∞ (E, F ) with a complement given by the smooth functions which are p-¬‚at at 0.

k

Proof. We have already shown that L( E; F ) embeds into C ∞ (E, F ) as a di-

rect summand, where a retraction is given by the derivative of order k at 0. Fur-

thermore, we have shown that the polynomials of degree ¤ p are exactly the di-

k

rect sums of homogeneous terms in L( E; F ). A retraction to the inclusion

k

E; F ) ’ C ∞ (E, F ) is hence given by k¤p k! dk |0 .

1

k¤p L(

Remark. The corresponding statement is false for in¬nitely ¬‚at functions. I.e.

the sequence E ’ C ∞ (R, R) ’ RN does not split, where E denotes the space of

smooth functions which are in¬nitely ¬‚at at 0. Otherwise, RN would be a subspace

of C ∞ ([0, 1], R) (compose the section with the restriction map from C ∞ (R, R) ’

C ∞ ([0, 1], R)) and hence would have a continuous norm. This is however easily

seen to be not the case.

5.18. Theorem. Uniform boundedness principle. If all Ei are convenient

vector spaces, and if F is a locally convex space, then the bornology on the space

L(E1 , . . . , En ; F ) consists of all pointwise bounded sets.

So a mapping into L(E1 , . . . , En ; F ) is smooth if and only if all composites with

evaluations at points in E1 — . . . — En are smooth.