361
Internat. J. Math. & Math. Sci.
Vo!. 9 No. 2 (1986) 361-364
A NOTE ON THE INVERSE FUNCTION THEOREM
OF NASH AND MOSER
MIKAEL LINDSTROM
Department of Mathematics
bo Akademi
SF-20500 bo, Finland
(Received September 9, 1985)
ABSTRACT.
The Nash-Moser inverse function theorem is proved for different kind of
differentiabilities.
K[’.’Y
Continuous convergence,
WORDS AND PHRASES.
1980 MATHE[ATICS
coreflective
subcategory, graded space,
a*-di
fferentiability.
SUBJECT CLASSIFICATION CODE. 58C15.
smoothing operators, tame map,
INTRODUCT ION
The purpose of this note is to formulate the inverse function theorem of Nash and
Moser for different differentiabilities using a categorical approach. The proof is
based on the inverse function theorem of Nash and Moser in the version of Hamilton [I]
formulated in the category of graded Frchet spaces which admit smoothing operators and
[2] tame maps. Our proof is using the same technique as Schmid [3]
C-differentiable
c
uses when he proves this theorem for a stronger notion of differentiability, called
From our formulation it is
the F-differentiability than the Coo-differentiability.
C
possible to derive the inverse function theorem of Nash and Moser for natural
differentiabilities stronger than the C C differentiability
2.
THE INVERSE FUNCTION THEOREM OF NASH AND MOSER.
Let
LC
Further let
tinuous linear mappings.
K
on morphisms and such that the
identity mapping
endowed with continuous convergence
For any pair
E,F E obj(LC)
E
k-linear mappings from
(ekc(E
k
into
[2],
we let
F
[2] and con-
denote a coreflective subcategory of
which is closed under finite products and the coreflector
F))
.
denote the category of locally convex limit vector spaces
and
Lk(E
C
X
(Cc(X
F))
LC
K s is the identity
C (X,F)
is a limit space and
F)
LC
C C (X F)
is
F E obj(LC)
be the space of all continuous
endowed with continuous convergence.
We write
e k(E,F)
DEFINITION.
Let
E
and
F
be locally convex spaces and let
U
be open in
E
MIKAEL
362
A mapping
F
U
f
LINDSTRO
differentiable of
is said to be
Cp
class
if there exist
f unc t ion s
Dkf
Df
such that
Lk(E,F)
U
x E U
and for each
f
0,I
k
,p
h E E
each
and each
k
we
O,|,...,p-|
have
lid t-|(Dkf(x+th)
t+0
IN
k < p
and such that for each k
(I)
Dkf(u)
is called
f
every
Lk(E,F)
c
Dk+|f(x)h
the following two conditions are satisfied:
Dkf
(2)
dfferentiable of
Dkf(x))
Lk(E,F)
U
Cp
if it is differentiable of class
class C
for
IN
p
Coo(
By Keller [2] the chain rule is valid for
vergence follows that for any continuous map
k
Ux E /F
(x,hl
defined by
,hk)
As the limit structure
since
is a finer limit
a
From the universal property of continuous con-
structure than continuous convergence.
tinuous.
is continuous.
U-
g
g(x)(h
Lk(E,F)
k)
h
the associated map
U
x
is always finer than
h
E
i
is con-
we have that differenti-
c
C a implies differentiability of class C
The latter is exactly
c
the cDncept of differentiability used by Hamilton [I] to prove the inverse function
ability of class
theorem of hash and Moser.
We first recall some definitions that will be needed.
Let
(li’II
E
r)riN
E
on
II’112r )rCIN
and
A grading on
be a Frchet space.
E
are equivalent if for some
II..x.. r2 _< c llx llr+s2
is an increasing sequence of norms
E
which defines the topology on
E
X
with a constant
llXllr
IN
s
<
cllxli2r+s
which may depend on
c
(II’II
Two gradings
rEIN
r
and
A graded space
r
is a Frchet space together with an equivalence class of gradings. We say that a graded
E
space
admits smoothing operators if we can find linear maps
such that for some
all
i,k f IN
Let
E
<
F
and
t
is tame if for every
x
<
E
Oo, x
ctr+kllxll i
<
k
and
and some constant
be graded spaces and
U
c
E
open in
c
may depend on
t
E
xll i <
We say that a map
we have the growth estimate
U, where the constant
S
ctr-kllxlli+i k
which may depend
llSt(x)
E U we can find a neighbourhood
x
n C IN
such that for every
all
liSt(x) lli+
r
U
for
on
and
f
U- F
and a number
f(x)II n < =(llxli n+r
k.
r
+ I)
for
n
In the proof of the inverse function theorem of hash and Moser we shall also need
the following result (Lemma 2, [3]): The composition of two continuous tame maps is
continuous and tame.
U
THEOREM.
Let
be open in
E
(I)
f
(2)
DKf
(3)
For each
U- F
U
(4)
The map
(5)
Vf
U xF
E
and
F
be graded spaces which admit smoothing operators.
and assume that
is differentiable of class
Ek-
F
is tame for every
x f U
U
Vf
E
the derivative
L (F,E)
is tame.
Vf(x)
k
Df(x)
C
and tame.
IN.
E
(Df(x)) -I
F
is an isomorphism.
is continuous.
Let
363
INVERSE FUNCTION THEOREM OF NASH AND MOSER
Then for any
we can find open neighbourhoods of
U
is a bijective map from
f
that
E U
x
C
differentiable of class
Dkf
f(x 0)
of
-1
such
U
V
is
are tame for all k t IN
E
for all
V
y
is differenti-
f
Further the assumptions (4) and (5) imply that
Now we have that
Df(x)
F
U
Vf
Fk
V
f
are continuous and tame, since
are tame,
and the family of inverses
V
and
x
and the inverse map
Vf(f-l(y))
is continuous and tame.
E
and all
Ca
of class
F
and assumption (2).
C
Ux F
Vf
D-
D(f-1)(y)
Ek
U
The maps
able of class
also
onto
and the maps
a
Furthermore we have the formula
PROOF.
V
is differentiable
f
is an isomorphism for every x t U
E -F
are continuous and tame maps.
E
Consequently
the conditions of the inverse function theorem of Nash-Moser are fulfilled (theorem
1.1.1 p. 171 in [I]).
V
and
f(x 0)
of
and tame.
on
Cc
Vf
E U
is bijective and
lim
t/O
t-1(f-1(y
and assumption (4) follow that
to be true for
f-1
Dk+If -I
map
k
V
Df
E
F
ak+1
Vf(f-1(y))w
From the formula
L (F,E)
V
Df
is tame since
-Vf(x)[D2f(x){u,Vf(x)w}]
Thus the derivative
D(Vf)
h(,)
is defined by
0.3.5 in [2]
h
and
Vf
holds,
Df
is continuous
f
Further
Assume now it
are tame.
f-1
?
Vf
U
x
E E
and
F.
w
e oc (E F,E)
L((F,E)
o(idE, ) for
D2f(x)
Vf(x)
and
Since the category
c
is a coreflector it follows that
h
u
can be factorized according to
e c (E,F)
Since
p
and consequently the map
is differentiable of class
Vf
holds for all
F,E)
is continuous for
under finite products and
Thus it is true for
-
Lc(E
U
Uo (D2f--)
theorem
x
By theorem 5.3.1, p. 102 in [I] we have that Vf is weakD) U E F E is continuous and the formula
p
ly differentiable and that
h
of
is continuous
k(
By induction on
where
IN
is clearly tame so we only have to show that
[D(Vf)](x){u,w}
U
From the definition of the c-differentiability follows that the
-I is differentiable of class C
Vf
Since Df -I
if Df
k
C
is
V
f
f-1(y))
+ tw)
Dkf-1
f
U
there exist neighbourhoods
by the proof of theorem 1.1.1 p. 186 in [I]. By induction
V
is differentiable of class
U
w f F
and
x
V
we will prove the remaining part that f
k
V
and
F
E is tame for every k
k
we have that
C
U
f
Furthermore the formula
y E V
for all
Then for every
such that
h
K
By
is closed
is continuous.
C
is bilinear it is differentiable of class
is differentiable of class
C
a
by induction
a
Thus the
theorem is proved
We shall now consider examples of coreflective subcategories of
closed under finite products and the coreflectors
identity mapping
EXAMPLE
Cc(U
Let
C (U,F)
C
F)
Kc
E
Kc
be the category
Let Ka be the category
K
LC
The coreflector ?e
LC
K
e
it is characterized as foilows: a filter
for some filter
which are
fulfill the assumption that the
is continuous.
EXANPLE 2.
spaces [2].
LC
90/. is the identity functor
1LC
?C
of equabIe locaiIy convex Iimit vector
is the identity on morphisms and on objects
F
G which converges
on
E
converges to zero in
to zero in
E
E
e
iff
MIKAEL
364
Let
EXAMPLE 3.
M
KM
LC
G E} E
EXAMPLE 4.
vector spaces.
objects
Eb
E
be the category
KM
F
on
E
converges to zero in
for some filter
Let
E
The coreflector
E
M
LC
b
Kb
for some bounded subset
converges to zero in
Example
G < F
G
and
E
of bornological locally convex limit
is the identity on morphisms and on
it is characterized as follows: a filter
iff B <
iff
it is characterized as
which converges to zero in
be the category
b
[2]. The corflector
of Marinescu spaces
is the identity on morphisms and on objects
follows: a filter
N{G
K
LINDSTRO
B c E
F
on
E
converges to zero in
i.e. some set
B
such that
VB
E
gives us the inverse function theorem of Nash and Moser by Hamilton
[|].
From example 3 we derive the inverse function theorem of Nash and Moser for the
(C
C
M
A in Keller [2]). In [4] Kriegl has discussed
smooth mappings between locally convex spaces, where a mapping is called smooth iff
differentiability of class
its composition with smooth curves are smooth.
He has compared this concept of smooth-
ness with different C -differentiabilities (see
[2]).
From [2] and [4] follow that a
mapping between Frchet spaces is smooth iff it is C c -differentiable. Thus the inverse
function theorem of Nash and Moser is valid for this concept of smoothness.
REFERENCES
I.
HAMILTON, R.S. The Inverse Function Theorem of Nash and Moser.
New ser. vol. 7, 65-222 (1982).
2.
KELLER, H.H. Differential calculus in locally convex spaces. Lecture Notes in
Math. 417, Springer Berlin, 1974.
SCHMID, R. The inverse function theorem of Nash and Moser for the F-differentiability. Proc. of the 2nd school on convergence spaces, Schwerin, 1983,
3.
Conversence
4.
Bulletin of AMS,
Structures and Applications, I__I, Berlin 1984.
KRIEGL, A. Eine kartesisch abgeschlossene Kategorie glatter Abbildungen zwischen
beliebigen lokalkonvexen Vektorrumen. Mh. Math. 9__5, 287-309 (1983).