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English Pages [62] Year 1967
T h is d issertatio n h as b e e n m ic ro film e d e x a c tly as re c e iv e d
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JENKINS, Thomas Morton, 1935B A N A C H SPACES O F UPSCHITZ FUNCTIONS O F A N A B S T R A C T METRIC SPACE. Vnlo juu a v >■».«».«/>.-.»>>■«.j>j TJU T> 10CQ Mathematics
University Microfilms, Inc., A n n Arbor, Michigan
^Copyright by
THOMAS MORTON JENKINS 1968
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Banach Spaces of Lipschitz Functions on an Abstract Metric Space. By.' Thomas M? Jenkins
1967
A Dissertation Presented to the Faculty of the Graduate School of Yale University in Candidacy for the Degree of Doctor of Philosophy.
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Summary Given a metric space (X,d) and a positive number a < 1, let Lip(X,da ) denote the space of all bounded continuous com plex-valued functions f on X such that | |f | | =. d sup { jf*(x)-f (y) j/da (x,y) : x,y £ X, x ^ y] < oo. Lip(X,da ) is a Banach space under the norm
jj j | O
given by
J[f*j{ = IIiloo^ Ill’ll
C£
^
The closed linear subspace lip (X,d ) consists of those f in Lip(X,da ) which vanish at infinity and which satisfy the con dition
jf(x)-f(y)J/da (x,y) --- > 0
as
da (x,y) --- > 0.
A metric space (X,d) is said to satisfy the Lipschitz four-point property if every complex-valued function g defined on three points of X has a Lipschitz preserving extension to any fourth point of X; i.e., if |g ( x -)-g(x -)j < K d(x-,x.) -J-
J
J*
J
for i,j = 1,2,3 then g(x^) can be defined so that the inequality holds for i,j = 1,2,3,4. It has long been known that for every subset Y of an arbitrary metric space (X,d) and every real-valued function f £ Lip(Y,d), f has an extension f* £ Lip(X,d) such that 1 1 ^ 1 1 ^ * 1 I^ I Id and I I I too = ||f| 1^. We show that such extensions exist for every subset Y C X and every complex-valued f £ Lip(Y,d) if and only if (X,d) satisfies the Lipschitz four-point property. Our second main result states that if (X,d) satisfies the Lipschitz four-point property, if every closed bounded subset of X is compact and if 0 < a < 1, then Lip(X,da ) is isometrically isomorphic to the second dual of lip°(X,da ). A third result identifies the linear isometries of Lip(X,da ) onto itself and of lip°(X,da ) onto itself where we assume that 0 < a < 1 and that (X,d) is a compact, connected metric space with diam (X,da ) < 2. These results generalize some work of K. de Leeuw who proved the second result and the latter half of the third for the particular metric space (X,d) where X is a circle of unit circumference with distance measured along the circum ference.
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Table of Contents Summary § 1a _
TntrnHnptinn
± '
§ 2.
Preliminaries
5«
§3.
Extensions of Lipschitz functions which preserve the Lipschitz condition
10.
§
Lip a
as a second dual
19-
§ 5-
Isometries of Lipschitz spaces
35.
§ 6.
Some related results
53-
Bibliography
57.
Acknowledgments I wish to thank my advisor, Professor Charles E. Rickart, for his encouragement over the past several years and for many helpful suggestions in the preparation of this paper. am indebted to Dr. Edward J. Barbeau for many fruitful dis cussions as the results of this paper took shape. I also wish to express my appreciation for financial support from the National Science Foundation in the form of a Science Faculty Fellowship.
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I
1. 5 1-
Introduction.
0 < a < 1,
If
L i p (X ,da )
(X,d)
is a metric space and if
denotes the space of all bounded
continuous complex-valued functions
f
which satisfy a
Lipschitz condition with respect to the metric |f(x)-f(y)| < K(d(x,y))a
for all
Banach space with the norm max ( I If I loo* I |f I I a )
|| ||
where
da ;
i.e.,
x,y £ L
Lip(X,da )
given by
|jfJ J =
is a
||f|j a = sup { jf (x)-f (y) |/da (x,y) :
x,y 9 X, x i y}. We denote by those functions
f
lip(X,da ) in
the closed linear subspace of
Lip(X,da )
which satisfy the con
dition |f(x)-f (y) |/da (x,y) --- > 0 When
(X,d)
subspace
consisting of all functions in
which vanish at infinity.
In 1961 K. de Leeuw ClO] Lip(R-psa ) modulo 1 n
da (x,y) --- > 0.
is not assumed to be compact, we often need the
lip °(X,da )
lip(X,da )
as
where
0 < a < 1
proved two results for and
with the usual distance
is an integer}).
R-j_
is the real line
(s(x,y) = inf { jx-y+nj:
(As is common, we often refer to
as a circle with distance equal to arc length).
(R^,s)
We prove
similar results for a wide class of abstract metric spaces. Specifically de Leeuw showed that for Lip(R 2 »sa ) of
0 < a < 1,
is isometrically isomorphic with the second dual
lip(R-^,sa ).
every metric space
We prove the same result (Theorem 4.1) for (X,d)
in which closed' bounded sets are
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2. compact and which satisfies a condition we call the Lipschitz four-point property (Def. 3.4). not be assumed if
a < 1/2
The latter condition need
or if
(X,d)
satisfies the
Euclidean four-point property (Def. 3.3).
Moreover, for
(real) spaces of real-valued Lipschitz functions, the result is valid without the Lipschitz four-point property. De Leeuw*s proof involves constructing a sequence of approximations to a given function with
Fejer kernels.
different method.
h
by
convoluting
h
The general case calls for an entirely
Our construction involves extending the
domain of a Lipschitz function without increasing the Lipschitz constant.
This is always possible for real valued functions,
while in the complex case, the extra conditions mentioned above come into play.
§ 3 is devoted to this extension pro
blem. Next consider a distance preserving map space
(Y, ^>)
number
\
(*)
onto a metric space
with
|>.| = 1.
for every
a
lip(X,da )
onto lip(Y,
circle lip(R^,sa )
with
and
f€C(X), T
of
0 < a < 1. ^»a ).
0 < a < 1,
of a metric
and acomplex
Then the equation
Tf(y) * Xf(?y),
defines a linear isometry
(X,d)
?
y € Y,
Lip(X,da ) Furthermore,
onto T
Lip(Y,^=>a ) maps
De Leeuw showed that for the every linear isometry
T
;
of
onto itself has the form (*=).
We show in Theorem 5.2 that if
X
and
Y
are compact
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3. and if one of the spaces satisfies certain connectedness properties, then every xsoiustry of either Lip(Y,
or of
lip(X,da )
form (=!=) where the map smaller than 2. for
?
onto
Lip{X,d )
lip(Y,^>^)
onto
has the
preserves distances which are
The latter result contains de L e euw9s result
lipff^ ,sa ). A problem similar to the isometry problem was considered
by D.R. Sherbert whose Ph.D. dissertation was devoted to the study of III
Lip(X,d)
III given by
as a Banach algebra with the norm
|||f||| = I |fI 1^+I |fI ld -
showed that every algebra homomorphism Lip(Y,
)
of
Lip(£,d)
into
is of the form
Tf(y) = f ( ? y ) , where
T
Sherbert [13]
: Y
>
f€Lip(X,d),
X
< * ( ? ■ y±, ^ y 2 )
y2 € y ,
for some positive constant K. if and only if
y£Y,
?“ is onto.
Furthermore T
is one-to-one
T is onto if and only if
satisfies the additional condition (°*yl ’y 2* - yl> ^ y 2 * ’ for some
positive constant K .
yl ’y2 Though not
€ Y» explicitly stated
by Sherbert, it follows from his proof that if isometric algebra-isomorphism, then for all
y1 ,Y2 ^ y *
T
is an
^ yi>y2^ = ?'y 2^
While Sherbert9s result is properly set
in a Banach algebra framework, our theorem involves a linear
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4isometry of Banach spaces with no assumption that the map preserves multiplication.
We have not succeeded in character
izing the linear-homomorphisms of Lipschitz spaces.
This
appears to be more difficult than the linear-isomorphism result. Since the Banach algebra norm j j J |J J || ||,
is equivalent to
many of Sherbertvs results carry over to our situation.
For our purposes, however,
II II
is more appropriate.
While
the theorems of sections 4 and 5 are an outgrowth of de Lee u w ?s work [10], our point of view and some of our techniques were strongly influenced by the work of Sherbert [13,14]. This introduction is followed by some definitions and preliminary results in § 2.
After considering the extension
problem in § 3, we treat the two main results in Sections 4 and 5. In § 6 references are given to other research on Lipschitz functions.
Several authors, by suppressing the con
stant functions in various ways, have considered (often denoted by
as norm. In this d context, for example, it was known by 1956 (Arens and Eells [1]) that linear space.
H )
with
Lip(X,da )
Lip(X,d)
|| ||
is always the dual of a normed
The first recognition of
Lip(X,da )
as a
second dual appears to be de Leeuw*s result for the circle.
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5. S 2. f
Preliminaries.
For any real or complex-valued function
on a metric space
"Lip- norm"
| |f |
(X,d),
tne sup norm
||x||
and
are defined by
N f I loo = S U P
(x) | :x G X]
{
Ilf Ild = sup { |f(x)-f(y)|/d(x ,y ):x,y € X, x / y}. We define tions
Lip(X,d)
to be the class of complex-valued func
f : X --- > C
(Actually,
j| |(^
functions).
for which both norms are finite. is a semi-norm since it vanishes on constant
Then every
f
in
Lip(X,d)
is a bounded con
tinuous function satisfying a Lipschitz condition with respect to the metric
d.
Denote by
lip(X,d)
the subspace
lip(X,d) = {f € Lip(X,d) :
— > 0
as
d(x,y) — > 0}.
The condition is to be interpreted "uniformly"; i.e., for every
e > 0,
implies
there exists
6 > 0
jf(x)-f(y)|/d(x,y) < e.
such that
Unless
d(x,y) < 8
(X,d)
is compact,
this interpretation is not equivalent to assuming that the condition holds in a neighborhood of each point.
When
(X,d)
is not assumed to be compact, we have occasion to consider the subspace
lip °(X,d)
functions
f
f(x)
which we define to be the set of
in lip(X,d)
is small for
x
outside sufficiently large compact sets).
It is well known that the norm
|| ||
which vanish at infinity (i.e.,
given by
Lip(X,d)
is a Banach space with
S if || = I \f I 1^ V | |f | ld
notes the maximum of the real numbers
a
clear that
are closed linear
lip(X,d)
and
lip °(X,d)
and
b).
(avb It is
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de
6. subspaces of If
Lip(X,d).
G < a < I,
then
(da (x,y) = Cd(x,y)]a ). the metric
da
for
aa
is also a metric on
X
A Lipschitz condition with respect to
0 < a < 1
is often referred to as a
«•
Holder condition with respect to the metric (X,d)
d.
Whenever
is a fixed metric space, we will abbreviate
(lip(X,da ), lip ° (X ,da )) subscript
R
by
Lip a
the norm
(lip a, lip°a).
The
will be used when we wish to consider spaces of
real-valued Lipschitz functions. lipR (X,da )
Lip(X,da )
and || ||
lipR°(X,da )
Then
Lip^(X,da ),
are (real) Banach spaces with
defined above.
Of course the real spaces are
contained in the corresponding complex spaces. We now state several results for reference later in this paper.
Throughout the rest of this section,
an arbitrary metric space. real-valued function
f,
(X,d)
For each real number the functions
the results of truncating
f
fa a
a
and
above and below at
denotes and f va
a.
are
The
results in the following lemma are obvious. Lemma 2.1.
If
f £ Lip^(X,d)
and if
a
is a real number,
then (1)
I If+a||d = ||f||d
(2)
||f/\a||d
s to approximate any function
by a sequence of functions
uniformly on compact sets and be the element of
(lip a) ""
through the canonical embedding of
gn
in
lim sup
lip°a
J|gn lI
F is onto and isometric. We now proceed to the first of our lemmas.
Lemma 4.2. I|F|| < Proof. r
If
F G lip°(X,da )**,
then
F G Lip(X,da )
and
jIF||. Let
F G (llp°a)**.
Since
9*A G ( l i p ^ ) *
for each
A
x G X,
F(x)
is well defined and
|F(x)| = |F(x )|