∈ E — . Then c0 is Lipn and (c0 )(k) = ck .

Proof. For n = 0 this was shown in (1.5). For n ≥ 1, by (1.7) applied to —¦ c we

have that

1 c0 (t) ’ c0 (0)

’ c1 (0)

t t

is locally bounded, and hence by (52.19) the set

c0 (t) ’ c0 (0)

1

’ c1 (0) :t∈I

t t

0 0

is bounded. Thus c (t)’c (0) converges even Mackey to c1 (0). Now the general

t

statement follows by induction.

2. Completeness

Do we really need the knowledge of a candidate for the derivative, as in (1.9)? In

¬nite dimensional analysis one often uses the Cauchy condition to prove conver-

gence. Here we will replace the Cauchy condition again by a stronger condition,

which provides information about the quality of being Cauchy:

A net (xγ )γ∈“ in E is called Mackey-Cauchy provided that there exist a bounded

(absolutely convex) set B and a net (µγ,γ )(γ,γ )∈“—“ in R converging to 0, such

that xγ ’ xγ ∈ µγ,γ B. As in (1.6) one shows that for a net xγ in EB this is

equivalent to the condition that xγ is Cauchy in the normed space EB .

2.1. Lemma. The di¬erence quotient is Mackey-Cauchy. Let c : R ’ E be

scalarly a Lip1 -curve. Then t ’ c(t)’c(0) is a Mackey-Cauchy net for t ’ 0.

t

Proof. For Lip1 -curves this is a immediate consequence of (1.7) but we only as-

sume it to be scalarly Lip1 . It is enough to show that t’s c(t)’c(0) ’ c(s)’c(0) is

1

t s

bounded on bounded subsets in R \ {0}. We may test this with continuous linear

functionals, and hence may assume that E = R. Then by the fundamental theorem

of calculus we have

1

c(t) ’ c(0) c(s) ’ c(0) c (tr) ’ c (sr)

1

’ = dr

t’s t’s

t s 0

1

c (tr) ’ c (sr)

= r dr.

tr ’ sr

0

Since c (tr)’c (sr) is locally bounded by assumption, the same is true for the integral,

tr’sr

and we are done.

2.1

2.4 2. Completeness 15

2.2. Lemma. Mackey Completeness. For a space E the following conditions

are equivalent:

(1) Every Mackey-Cauchy net converges in E;

(2) Every Mackey-Cauchy sequence converges in E;

(3) For every absolutely convex closed bounded set B the space EB is complete;

For every bounded set B there exists an absolutely convex bounded set B ⊇

(4)

B such that EB is complete.

A space satisfying the equivalent conditions is called Mackey complete. Note that a

sequentially complete space is Mackey complete.

Proof. (1) ’ (2), and (3) ’ (4) are trivial.

(2) ’ (3) Since EB is normed, it is enough to show sequential completeness. So let

(xn ) be a Cauchy sequence in EB . Then (xn ) is Mackey-Cauchy in E and hence

converges in E to some point x. Since pB (xn ’ xm ) ’ 0 there exists for every

µ > 0 an N ∈ N such that for all n, m ≥ N we have pB (xn ’ xm ) < µ, and hence

xn ’ xm ∈ µB. Taking the limit for m ’ ∞, and using closedness of B we conclude

that xn ’ x ∈ µB for all n > N . In particular x ∈ EB and xn ’ x in EB .

(4) ’ (1) Let (xγ )γ∈“ be a Mackey-Cauchy net in E. So there is some net µγ,γ ’ 0,

such that xγ ’ xγ ∈ µγ,γ B for some bounded set B. Let γ0 be arbitrary. By (4)

we may assume that B is absolutely convex and contains xγ0 , and that EB is

complete. For γ ∈ “ we have that xγ = xγ0 + xγ ’ xγ0 ∈ xγ0 + µγ,γ0 B ∈ EB , and

pB (xγ ’ xγ ) ¤ µγ,γ ’ 0. So (xγ ) is a Cauchy net in EB , hence converges in EB ,

and thus also in E.

2.3. Corollary. Scalar testing of di¬erentiable curves. Let E be Mackey

complete and c : R ’ E be a curve for which —¦ c is Lipn for all ∈ E — . Then c

is Lipn .

Proof. For n = 0 this was shown in (1.5) without using any completeness, so let

n ≥ 1. Since we have shown in (2.1) that the di¬erence quotient is a Mackey-Cauchy

net we conclude that the derivative c exists, and hence ( —¦ c) = —¦ c . So we may

apply the induction hypothesis to conclude that c is Lipn’1 , and consequently c is

Lipn .

Next we turn to integration. For continuous curves c : [0, 1] ’ E one can show

completely analogously to 1-dimensional analysis that the Riemann sums R(c, Z, ξ),

de¬ned by k (tk ’ tk’1 )c(ξk ), where 0 = t0 < t1 < · · · < tn = 1 is a partition

Z of [0, 1] and ξk ∈ [tk’1 , tk ], form a Cauchy net with respect to the partial strict

ordering given by the size of the mesh max{|tk ’ tk’1 | : 0 < k < n}. So under

the assumption of sequential completeness we have a Riemann integral of curves.

A second way to see this is the following reduction to the 1-dimensional case.

2.4. Lemma. Let L(Eequi , R) be the space of all linear functionals on E — which are

—

bounded on equicontinuous sets, equipped with the complete locally convex topology

2.4

16 Chapter I. Calculus of smooth mappings 2.5

of uniform convergence on these sets. There is a natural topological embedding

—

δ : E ’ L(Eequi , R) given by δ(x)( ) := (x).

Proof. Let U be a basis of absolutely convex closed 0-neighborhoods in E. Then

the family of polars U o := { ∈ E — : | (x)| ¤ 1 for all x ∈ U }, with U ∈ U form a

basis for the equicontinuous sets, and hence the bipolars U oo := { — ∈ L(Eequi , R) :

—

| — ( )| ¤ 1 for all ∈ U o } form a basis of 0-neighborhoods in L(Eequi , R). By the

—

bipolar theorem (52.18) we have U = δ ’1 (U oo ) for all U ∈ U. This shows that δ is

a homeomorphism onto its image.

2.5. Lemma. Integral of continuous curves. Let c : R ’ E be a continuous

curve in a locally convex vector space. Then there is a unique di¬erentiable curve

c : R ’ E in the completion E of E such that ( c)(0) = 0 and ( c) = c.

Proof. We show uniqueness ¬rst. Let c1 : R ’ E be a curve with derivative c and

c1 (0) = 0. For every ∈ E — the composite —¦ c1 is an anti-derivative of —¦ c with

initial value 0, so it is uniquely determined, and since E — separates points c1 is also

uniquely determined.

Now we show the existence. By the previous lemma we have that E is (isomorphic

—

to) the closure of E in the obviously complete space L(Eequi , R). We de¬ne ( c)(t) :

t

E — ’ R by ’ 0 ( —¦ c)(s)ds. It is a bounded linear functional on Eequi since for

—

an equicontinuous subset E ⊆ E — the set {( —¦ c)(s) : ∈ E, s ∈ [0, t]} is bounded.

—

So c : R ’ L(Eequi , R).

c is di¬erentiable with derivative δ —¦ c.

Now we show that

( c)(t + r) ’ ( c)(r)

’ (δ —¦ c)(r) ( ) =

t

t+r r

1

( —¦ c)(s)ds ’ ( —¦ c)(s)ds ’ t( —¦ c)(r)

= =

t 0 0

r+t 1

1

( —¦ c)(s) ’ ( —¦ c)(r) ds = c(r + ts) ’ c(r) ds.

=

t r 0

Let E ⊆ E — be equicontinuous, and let µ > 0. Then there exists a neighborhood U

of 0 such that | (U )| < µ for all ∈ E. For su¬ciently small t, all s ∈ [0, 1] and ¬xed

1

r we have c(r + ts) ’ c(r) ∈ U . So | 0 (c(r + ts) ’ c(r))ds| < µ. This shows that

the di¬erence quotient of c at r converges to δ(c(r)) uniformly on equicontinuous

subsets.

It remains to show that ( c)(t) ∈ E. By the mean value theorem (1.4) the di¬erence

—

quotient 1 (( c)(t) ’ ( c)(0)) is contained in the closed convex hull in L(Eequi , R)

t

of the subset {c(s) : 0 < s < t} of E. So it lies in E.

De¬nition of the integral. For continuous curves c : R ’ E the de¬nite integral

b b

c ∈ E is given by a c = ( c)(b) ’ ( c)(a).

a

2.5

2.8 2. Completeness 17

2.6. Corollary. Basics on the integral. For a continuous curve c : R ’ E we

have:

b b

∈ E—.

(1) ( a c) = a ( —¦ c) for all

b d d

(2) a c + b c = a c.

b •(b)

c for • ∈ C 1 (R, R).

(c —¦ •)•

(3) =

a •(a)

b

(4) c lies in

the closed convex hull in E of the set

a

{(b ’ a)c(t) : a < t < b} in E.

b

(5) a : C(R, E) ’ E is linear.

(6) (Fundamental theorem of calculus.) For each C 1 -curve c : R ’ E we have

s

c(s) ’ c(t) = t c .

We are mainly interested in smooth curves and we can test for this by applying linear

functionals if the space is Mackey complete, see (2.3). So let us try to show that

the integral for such curves lies in E if E is Mackey-complete. So let c : [0, 1] ’ E

be a smooth or just a Lip-curve, and take a partition Z with mesh µ(Z) at most

δ. If we have a second partition, then we can take the common re¬nement. Let

[a, b] be one interval of the original partition with intermediate point t, and let

a = t0 < t1 < · · · < tn = b be the re¬nement. Note that |b ’ a| ¤ δ and hence

|t ’ tk | ¤ δ. Then we can estimate as follows.

(b ’ a) c(t) ’ (tk ’ tk’1 )c(tk ) = (tk ’ tk’1 ) (c(t) ’ c(tk )) = µk bk ,

k k k

c(t)’c(tk )

where bk := is contained in the absolutely convex Lipschitz bound

δ

c(t) ’ c(s)

: t, s ∈ [0, 1]

B :=

t’s abs.conv

of c and µk := (tk ’tk’1 )δ ≥ 0 and satis¬es k µk = (b’a)δ. Hence we have for the

Riemann sums with respect to the original partition Z1 and the re¬nement Z that

R(c, Z1 ) ’ R(c, Z ) lies in δ · B. So R(c, Z1 ) ’ R(c, Z2 ) ∈ 2δB for any two partitions

Z1 and Z2 of mesh at most δ, i.e. the Riemann sums form a Mackey-Cauchy net

with coe¬cients µZ1 ,Z2 := 2 max{µ(Z1 ), µ(Z2 )} and we have proved:

2.7. Proposition. Integral of Lipschitz curves. Let c : [0, 1] ’ E be a

Lipschitz curve into a Mackey complete space. Then the Riemann integral exists in

E as (Mackey)-limit of the Riemann sums.

2.8. Now we have to discuss the relationship between di¬erentiable curves and

Mackey convergent sequences. Recall that a sequence (xn ) converges if and only if

there exists a continuous curve c (e.g. a reparameterization of the in¬nite polygon)

and tn 0 with c(tn ) = xn . The corresponding result for smooth curves uses the

following notion.

De¬nition. We say that a sequence xn in a locally convex space E converges fast

to x in E, or falls fast towards x, if for each k ∈ N the sequence nk (xn ’ x) is

bounded.

2.8

18 Chapter I. Calculus of smooth mappings 2.10

Special curve lemma. Let xn be a sequence which converges fast to x in E.

Then the in¬nite polygon through the xn can be parameterized as a smooth curve

1

c : R ’ E such that c( n ) = xn and c(0) = x.

Proof. Let • : R ’ [0, 1] be a smooth map, which is 0 on {t : t ¤ 0} and 1 on

{t : t ≥ 1}. The parameterization c is de¬ned as follows:

for t ¤ 0,

x

±

1

t’ 1 1

c(t) := xn+1 + • 1 ’n+1 (xn ’ xn+1 ) for n+1 ¤ t ¤ n , .

1

n n+1

for t ≥ 1

x1

1 1

Obviously, c is smooth on R \ {0}, and the p-th derivative of c for ¤t¤ is

n+1 n

given by

1

t ’ n+1

c(p) (t) = •(p) 1 (n(n + 1))p (xn ’ xn+1 ).

1

n ’ n+1

Since xn converges fast to x, we have that c(p) (t) ’ 0 for t ’ 0, because the ¬rst

factor is bounded and the second goes to zero. Hence c is smooth on R, by the

following lemma.

2.9. Lemma. Di¬erentiable extension to an isolated point. Let c : R ’ E

be continuous and di¬erentiable on R \ {0}, and assume that the derivative c :

R \ {0} ’ E has a continuous extension to R. Then c is di¬erentiable at 0 and

c (0) = limt’0 c (t).

Proof. Let a := limt’0 c (t). By the mean value theorem (1.4) we have c(t)’c(0) ∈ t

c (s) : 0 = |s| ¤ |t| closed, convex . Since c is assumed to be continuously extendable

to 0 we have that for any closed convex 0-neighborhood U there exists a δ > 0 such

that c (t) ∈ a + U for all 0 < |t| ¤ δ. Hence c(t)’c(0) ’ a ∈ U , i.e. c (0) = a.

t

The next result shows that we can pass through certain sequences xn ’ x even

with given velocities vn ’ 0.

2.10. Corollary. If xn ’ x fast and vn ’ 0 fast in E, then there are smoothly

parameterized polygon c : R ’ E and tn ’ 0 in R such that c(tn + t) = xn + tvn

for t in a neighborhood of 0 depending on n.

Proof. Consider the sequence yn de¬ned by

1 1

y2n+1 := xn ’

y2n := xn + 4n(2n+1) vn and 4n(2n+1) vn .

It is easy to show that yn converges fast to x, and the parameterization c of the

polygon through the yn (using a function • which satis¬es •(t) = t for t near 1/2)

has the claimed properties, where

1 1 1

4n+1

tn := = + .

4n(2n+1) 2 2n 2n + 1

As ¬rst application (2.10) we can give the following sharpening of (1.3).

2.10

2.13 2. Completeness 19

2.11. Corollary. Bounded linear maps. A linear mapping : E ’ F between

locally convex vector spaces is bounded (or bornological), i.e. it maps bounded sets

to bounded ones, if and only if it maps smooth curves in E to smooth curves in F .

Proof. As in the proof of (1.3) one shows using (1.7) that a bounded linear map

preserves Lipk -curves. Conversely, assume that a linear map : E ’ F carries

smooth curves to locally bounded curves. Take a bounded set B, and assume that

f (B) is unbounded. Then there is a sequence (bn ) in B and some » ∈ F such

that |(» —¦ )(bn )| ≥ nn+1 . The sequence (n’n bn ) converges fast to 0, hence lies on

some compact part of a smooth curve by (2.8). Consequently, (» —¦ )(n’n bn ) =

n’n (» —¦ )(bn ) is bounded, a contradiction.

2.12. De¬nition. The c∞ -topology on a locally convex space E is the ¬nal topol-

ogy with respect to all smooth curves R ’ E. Its open sets will be called c∞ -open.

We will treat this topology in more detail in section (4): In general it describes

neither a topological vector space (4.20) and (4.26), nor a uniform structure (4.27).

However, by (4.4) and (4.6) the ¬nest locally convex topology coarser than the

c∞ -topology is the bornologi¬cation of the locally convex topology.

Let (µn ) be a sequence of real numbers converging to ∞. Then a sequence (xn ) in

E is called µ-converging to x if the sequence (µn (xn ’ x)) is bounded in E.

2.13. Theorem. c∞ -open subsets. Let µn ’ ∞ be a real valued sequence.

Then a subset U ⊆ E is open for the c∞ -topology if it satis¬es any of the following

equivalent conditions:

(1) All inverse images under Lipk -curves are open in R (for ¬xed k ∈ N∞ ).

(2) All inverse images under µ-converging sequences are open in N∞ .

(3) The traces to EB are open in EB for all absolutely convex bounded subsets

B ⊆ E.

Note that for closed subsets an equivalent statement reads as follows: A set A is c∞ -

closed if and only if for every sequence xn ∈ A, which is µ-converging (respectively

M -converging, resp. fast falling) towards x, the point x belongs to A.

The topology described in (2) is also called Mackey-closure topology. It is not the

Mackey topology discussed in duality theory.

Proof. (1) ’ (2) Suppose (xn ) is µ-converging to x ∈ U , but xn ∈ U for in¬nitely

/

many n. Then we may choose a subsequence again denoted by (xn ), which is fast

falling to x, hence lies on some compact part of a smooth curve c as described in

1

(2.8). Then c( n ) = xn ∈ U but c(0) = x ∈ U . This is a contradiction.

/

(2) ’ (3) A sequence (xn ), which converges in EB to x with respect to pB , is Mackey

convergent, hence has a µ-converging subsequence. Note that EB is normed, and

hence it is enough to consider sequences.

(3) ’ (2) Suppose (xn ) is µ-converging to x. Then the absolutely convex hull B of

{µn (xn ’ x) : n ∈ N} ∪ {x} is bounded, and xn ’ x in (EB , pB ), since µn (xn ’ x)

is bounded.

2.13

20 Chapter I. Calculus of smooth mappings 2.14

(2) ’ (1) Use that for a converging sequence of parameters tn the images xn := c(tn )

under a Lip-curve c are Mackey converging.

Let us show next that the c∞ -topology and c∞ -completeness are intimately related.

2.14. Theorem. Convenient vector spaces. Let E be a locally convex vector

space. E is said to be c∞ -complete or convenient if one of the following equivalent

(completeness) conditions is satis¬ed:

(1) Any Lipschitz curve in E is locally Riemann integrable.

(2) For any c1 ∈ C ∞ (R, E) there is c2 ∈ C ∞ (R, E) with c2 = c1 (existence of

an anti-derivative).

(3) E is c∞ -closed in any locally convex space.

(4) If c : R ’ E is a curve such that —¦ c : R ’ R is smooth for all ∈ E — ,

then c is smooth.

(5) Any Mackey-Cauchy sequence converges; i.e. E is Mackey complete, see

(2.2).

(6) If B is bounded closed absolutely convex, then EB is a Banach space. This

property is called locally complete in [Jarchow, 1981, p196].

(7) Any continuous linear mapping from a normed space into E has a continu-

ous extension to the completion of the normed space.

Condition (4) says that in a convenient vector space one can recognize smooth

curves by investigating compositions with continuous linear functionals. Condition

(5) says via (2.2.4) that c∞ -completeness is a bornological concept. In [Fr¨licher,

o

Kriegl, 1988] a convenient vector space is always considered with its bornological

topology ” an equivalent but not isomorphic category.

Proof. In (2.3) we showed (5) ’ (4), in (2.7) we got (5) ’ (1), and in (2.2) we

had (5) ’ (6).

(1) ’ (2) A smooth curve is Lipschitz, thus locally Riemann integrable. The

inde¬nite Riemann integral equals the “weakly de¬ned” integral of lemma (2.5),

hence is an anti-derivative.