Characterization of the Radon-Nikodym Property in terms of inverse limits

0.16 MB·English

by  Jeff Cheeger

CHARACTERIZATION OF THE RADON-NIKODYM
PROPERTY IN TERMS OF INVERSE LIMITS
8
JEFF CHEEGER AND BRUCE KLEINER
0
0
2
Abstract. In this paper we clarify the relation between inverse
n
a systems, the Radon-Nikodym property, the Asymptotic Norming
J PropertyofJames-Ho[JH81],andtheGFDAspacesintroducedin

1 [CK06].
1

]
A
F
.
h 1. Introduction
t
a
m
A Banach space V is said to have the Radon-Nikodym Property
[
(RNP) if every Lipschitz map f : R → V is differentiable almost

3 everywhere. By now, there are a number of characterizations of Ba-
v
nach spaces with the RNP, the study of which goes back to Gelfand
9
8 [Gel38]; for additional references and discussion, see [BL00, Chapter
3
5], [GM85]. Of particular interest here is the characterization of the
3
. RNP in terms of the Asymptotic Norming Property; [JH81, GM85].
6
0
In this paper we will show that a variant of the GFDA property
7
introduced in [CK06] is actually equivalent to the Asymptotic Norming
0
: property of James-Ho, and hence by [JH81, GM85], is equivalent to the
v
i RNP. In addition, we observe that the GFDA spaces of [CK06] are just
X
spaces which are isomorphic to a separable dual space.
r
a
Definition 1.1. An inverse system
(1.2) W ←θ1− W ←θ2− ... ←θi−−1 W ←θ−i ... ,
1 2 i
is standard if the W ’s are finite dimensional Banach spaces and the
i
θ ’s are linear maps of norm ≤ 1. We let π : lim W → W denote the
i j i j
←−
projection map.
Date: February 1, 2008.
The first author was partially supported by NSF Grant DMS 0105128 and the
second by NSF Grant DMS 0701515.
1
2 JEFF CHEEGER AND BRUCEKLEINER
Definition 1.3. Let {(W ,θ )} be a standard inverse system and V ⊂
i i
lim W be a subspace. The pair (limW ,V) has the Determining Prop-
i i
←− ←−
erty if a sequence {v } ⊂ V converges strongly provided the projected
k
sequences {π (v )} ⊂ W converge for every j, the sequence {kv k} is
j k j k
bounded, and the convergence kπ (v )k → kv k is uniform in k. A Ba-
j k k
nachspaceU hastheDeterminingPropertyifthereisapair(lim W ,V)
i
←−
with Determining Property, such that V is isomorphic to U.
We have:
Theorem 1.4. A separable Banach space has the RNP if and only it
has the Determining Property.
Since a Banach space has the RNP if and only if every separable
subspace has the RNP, Theorem 1.4 yields a characterization of the
RNP for nonseparable Banach spaces as well.
To prove the theorem, we first observe in Proposition 2.8 that the
inverse limit lim W is the dual space of a separable Banach space.
i
←−
Then, by a completely elementary argument, we show that a Banach
spacehastheDeterminingPropertyifandonlyifithastheAsymptotic
Norming Property (ANP) of James-Ho [JH81]. Since a separable Ba-
nach space U has the RNP if and only if it has the ANP [JH81, GM85],
the theorem follows. We remark that there is a simple direct proof that
if V has the ANP (or the Determining Property), then every Lipschitz
map f : R → V is differentiable almost everywhere, see [CK].
Characterizations of the RNP using inverse limits are useful for ap-
plications; see [CK06], the discussion below concerning metric measure
spaces, and [CK].
Relation with previous work.
In slightly different language, our earlier paper [CK06] also consid-
ered pairs (limW ,V), where limW is the inverse limit of a standard
i i
←− ←−
inverse system, and V ⊂ limW is a closed subspace. A Good Finite
i
←−
Dimensional Approximation (GFDA) of a Banach space V, a notion
introduced in [CK06], is a pair (limW ,V) with the Determining Prop-
i
←−
|
erty such that π : V → W is a quotient map for every i.
i V i
INVERSE LIMITS AND THE RNP 3
It follows immediately from Lemma 3.8 of [CK06] that if (limW ,V)
i
←−
is a GFDA of V, then V = lim W . Since such inverse limits are dual
i
←−
spaces by Proposition 2.8, V is a separable dual space in this case.
Conversely, using the Kadec-Klee renorming Lemma [Kad59, Kle61], it
was shown in [CK06] that every separable dual space is isomorphic to
a Banach space which admits a GFDA. Thus, a Banach space admits
a GFDA if and only if it is isomorphic to a separable dual space.
Applications to metric measure spaces.
We will call a metric measure space (X,µ) a PI space if the mea-
sure is doubling, and a Poincar´e inequality holds in the sense of upper
gradients [HK98, Che99]. In [CK06], differentiation and bi-Lipschitz
non-embedding theorems were proved for maps f : X → V from PI
spaces into GFDA targets V, generalizing results of [Che99] for finite
dimensionaltargets. Asexplained above, itturnsoutthatthesetargets
are just separable dual spaces, up to isomorphism.
As an application of the inverse limit framework and the equivalence
between the ANP and RNP, we will show in [CK] that the differen-
tiation theorem [CK06, Theorem 4.1] and bi-Lipschitz non-embedding
theorem [CK06, Theorem 5.1] hold whenever the target has the RNP.
Acknowledgement. We are very grateful to Bill Johnson for sharing
an observation which helped give rise to this paper. We are much
indebted to Nigel Kalton for immediately catching a serious error in an
earlier version.
2. Inverse systems
In this section, we recall some basic facts concerning direct and in-
verse systems, and the duality between them. Then we show that in-
verse limits of standard inverse systems are precisely duals of separable
spaces.
The following conventions will be in force throughout the remainder
of the paper.
Definition 2.1. An standard direct system is a sequence of finite di-
mensional Banach spaces {E } and 1-Lipschitz linear maps ι : E →
i i i
E .
i+1
4 JEFF CHEEGER AND BRUCEKLEINER
Definition 2.2. An standard inverse system is a sequence of finite di-
mensionalBanachspaces{W }and1-Lipschitzlinearmapsθ : W →
i i i+1
W .
i
Definition 2.3. A standard direct system is isometrically injective if
the maps ι : E → E are isometric injections.
i i i+1
Definition 2.4. A standard inverse system is quotient if the maps
θ : W → W are quotient maps.
i i+1 i
By a quotient map of normed spaces, we mean a surjective map
π : U → V for which the norm on the target is the quotient norm, i.e.
for every v ∈ V,
−1
kvk = inf{ kuk | u ∈ π (v)}.
We will refer to the maps ι and θ as bonding maps.
i i
There is a duality between the objects in Definitions 2.1 and 2.2,
respectively, 2.3 and 2.4: if {(E ,ι )} is a standard direct system, then
i i
∗ ∗
{(E ,ι )} is a standard inverse system and conversely; similarly, iso-
i i
metrically injective direct systems are dual to quotient systems. To see
this, one uses the facts that the adjoint of a 1-Lipschitz map of Banach
spaces is 1-Lipschitz and the the adjoint of an isometric embedding is
a quotient map. (This follows from the Hahn-Banach theorem.) In
particular, since the spaces in our systems are assumed to be finite
dimensional (hence reflexive) every inverse system arises as the dual
of its dual direct system and conversely. The same holds for quotient
inverse systems.
We now recall the definitions of direct and inverse limits.
Given a standard direct system {(E ,ι )} we form the direct limit
i i
Banach space limE as follows. We begin with the disjoint union ⊔ E ,
i i i
−→
′
anddeclaretwoelementse ∈ Ei,e ∈ Ei′ tobeequivalentiftheirimages
′
in E coincide for some j ≥ max{i,i}. Since the bonding maps are 1-
j
Lipschitz, the set of equivalence classes inherits anobvious vector space
structure with a pseudo-norm. The direct limit lim E is defined to be
i
−→
the completion of the quotient of this space by the closed subspace
of elements whose pseudo-norm is zero. Clearly, there are 1-Lipschitz
maps
τ : E → limE ,
i i i
−→
INVERSE LIMITS AND THE RNP 5
which in the case of isometrically injective direct systems, are isometric
injections. The union τ (E ) is dense in limE .
Si i i −→ i
The inverse limit limW of a standard inverse system {(W ,θ )} is
i i i
←−
defined as follows. The underlying set consists of the collection of
elements (w ) ∈ W which are compatible with the bonding maps,
i Qi i
i.e. θi(wi) = wi−1 for all i, and which satisfy supikwik < ∞. This is
equipped with the obvious vector space structure and the norm
(2.5) k{w }k := lim kw k.
i j
j→∞
The map
(2.6) π : limW → W
j i j
←−
given by
π ({w }) = w
j i j
is 1-Lipschitz, and
lim kπ ({w })k = k{w }k.
j i i
j→∞
An inverse limit limW has a natural inverse limit topology, namely
i
←−
the weakest topology such that every projection map π : limW → W
j i j
←−
is continuous. Thus a sequence {v } ⊂ limW converges in the inverse
k i
←−
limit topology to v ∈ limW if and only if for every i, we have π (v ) →
i i k
←−
π (v) as k → ∞.
i
invlim
If {v } ⊂ limW and {v } −→ v ∈ limW , then
k i k i
←− ←−
(2.7) kvk ≤ liminf kv k.
k
k
Also, every norm bounded sequence {v } ⊂ limW has a subsequence
k i
←−
which converges with respect to the inverse limit topology; this follows
from a diagonal argument, because {π (v )} is contained in a compact
i k
subset of W , for all i.
i
Proposition 2.8. Given a standard inverse system {(W ,θ )}, there is
i i
an isometric isomorphism
∗ ∗
(2.9) C : lim W ≡ (limW ) .
←− i −→ i
∗
In particular, limW is the dual of the separable Banach space limW .
←− i −→ i
6 JEFF CHEEGER AND BRUCEKLEINER
Proof. Pick a compatible sequence (x ) ∈ limW . We get a map
i i
←−
∗
⊔ W → R
j
∗
by sending φ ∈ W to φ(x ); because (x ) is compatible with bonding
j j i
maps and
|φ(x )| ≤ kφkkx k ≤ kφkk{x }k,
j j j
∗
this defines a linear functional of norm ≤ k{x }k on lim W . Therefore
j −→ i
we get a 1-Lipschitz map
∗
∗
C : limW −→ limW .
←− i (cid:16)−→ i (cid:17)
We now verify that C is an isometry.
Pick (x ) ∈ limW , and choose n ∈ N such that kx k ≥ k(x )k−ǫ.
i i n i
←−
∗
If φ ∈ W has norm 1 and φ(x ) = kx k, then
n n n
kC((x ))kkτ (φ)k ≥ C((x ))(τ (φ)) = φ(x ) = kx k ≥ k(x )k−ǫ,
i n i n n n i
∗ ∗
where τ : W → limW is the canonical 1-Lipschitz map described
n n −→ i
above. This shows that C is an isometric embedding.
∗ ∗ ∗∗
If Φ ∈ (limW ) , then we define Φ ∈ W = W to be the composi-
−→ i i i i
tion
∗ ∗ Φ
W −→ limW −→ R.
i −→ i
This defines a compatible sequence (Φ ) ∈ limW , such that k(Φ )k =
i i i
←−
kΦk and C((Φ )) = Φ. Hence C is onto. (cid:3)
i
Corollary 2.10.
1) A separable Banach space Y is isomorphic to the direct limit of an
isometrically injective direct system (E ,ι ).
i i
∗
2) The dual space Y of the separable Banach space Y (as in 1)) is
∗
isometric to the inverse limit lim E of the a quotient inverse system
←− i
∗ ∗
{(E ,ι )}.
i i
Proof. To see that 1) holds, start with a countable increasing sequence
E ⊂ E ⊂ ··· ⊂ Y of finite dimensional subspaces whose union is
1 2
dense in Y, and take the bonding maps ι : E → E to be the
i i i+1
inclusions. Clearly the inclusion maps E → Y induce an isometry
i
limE → Y.
i
−→
(cid:3)
Assertion 2) follows from 1) and Proposition 2.8.
INVERSE LIMITS AND THE RNP 7
Let C be the isometry in Proposition 2.8.
Lemma 2.11.
∗ ∗
1) Suppose {v } ⊂ lim W is a sequence such that {C(v )} ⊂ (limW )
k ←− i k −→ i
∗ ∗
weak* converges to some y ∈ (limW ) . Then {v } is convergent with
−→ i k
respect to the inverse limit topology, and its limit v∞ ∈ lim Wi satisfies
←−
C(v∞) = y; in particular, y ∈ C(lim Wi).
←−
2) If {v } ⊂ lim W converges in the inverse limit topology, and has
k i
←−
uniformly bounded norm, then {C(v )} is weak* convergent.
k
Proof. Assertions 1) and 2) follow readily from the assumption that
the W are finite dimensional together with the density of compatible
i
sequences in lim W . (cid:3)
i
←−
3. The proof of Theorem 1.4
The proofofTheorem 1.4isbased ontheAsymptotic Norming Prop-
erty, which we now recall.
∗
Let Y denote a separable Banach space and V ⊂ Y a separable
∗
subspace of its dual. (Here Y need not be separable.)
Definition 3.1. The pair (Y∗,V) has the Asymptotic Norming Prop-
erty (ANP) if a sequence {v } ⊂ V converges strongly provided it is
k
weak* convergent and the sequence of norms {kv k} converges to the
k
norm of the weak* limit.
A Banach space U is said to have the Asymptotic Norming Property
∗
if there is a pair (Y ,V) with the ANP such that U is isomorphic to
V.
Theorem 3.2 ([JH81, GM85]). For separable Banach spaces, the RNP
is equivalent to the ANP.
Hence to prove Theorem 1.4, it suffices to show that for separable
Banach spaces, the ANP is equivalent to the Determining Property.
By Corollary 2.10, every separable Banach space Y is isometric to the
∗
direct limit of a standard direct system, and Y is isometric to the
inverse limit of the dual inverse system. Hence the proof of Theorem
1.4 reduces to:
8 JEFF CHEEGER AND BRUCEKLEINER
Proposition 3.3. Let {(W ,θ )} be a standard inverse system, and V
i i
be a closed separable subspace of limW . Then the pair (limW ,V) has
i i
←− ←−
the ANP if and only if it has the Determining Property. Here we are
∗
identifying limW with the dual of limW , see Proposition 2.8.
←− i −→ i
Proof. Let {v } ⊂ V be a sequence with bounded norm. By Lemma
k
2.11, the sequence {v } is weak* convergent if and only if it converges
k
in the inverse limit topology. Therefore, to prove the equivalence of the
ANP and the Determining Property for the pair (limW ,V), it suffices
i
←−
to show that when
w∗
(3.4) v −→ w ∈ lim W ,
k i
←−
the sequence of norms {kv k} converges to the kwk if and only if the
k
convergence kπ (v )k → kv k is uniform in k. Although this is com-
j k k
pletely elementary, we will write out the details.
We have
(3.5)
kv k−kwk = (kv k−kπ (v )k)+(kπ (v )k−kπ (w)k)+(kπ (w)k−kwk).
k k i k i k i i
Assume first that limk→∞kvkk = kwk. Given ǫ > 0, there exists I1
such that kwk − kπ (w)k < ǫ/3, for i ≥ I . By (3.4) there exists K
i 1 1
such that kπ (v )−π (w)k < ǫ/3, for k ≥ K . Also, there exists K
I1 k I1 1 2
such that kv k−kwk < ǫ/3, if k ≥ K . Set K = max(K ,K ).
k 2 1 2
(cid:12) (cid:12)
(cid:12) (cid:12)
From (3.5), with i = I , we get kv k−kπ (v )k < ǫ, for all k ≥ K.
1 k I1 k
Since, kv k − kπ (v )k is a nonnegative decreasing function of i, this
k i k
implies, kv k−kπ (v )k < ǫ, for all i ≥ I , k ≥ K.
k i k 1
Finally, there exists I such that kv k−kπ (v )k < ǫ for all i ≥ I ,
2 k i k 2
k = 1,...,K−1, Thus, if i ≥ max(I ,I ) then kv k−kπ (v )k < ǫ, for
1 2 k i k
all k.
Conversely, suppose the convergence kπ (v )k → kv k is uniform in
i k k
k. Given ǫ > 0, there exists I such that kv k − kπ (v )k < ǫ/3, for
k i k
i ≥ I and all k. Also, there exists I such that kwk−kπ (w)k < ǫ/3,
1 i
′
for i ≥ I . Set I = max(I,I ). By (3.4), there exists K such that
1 1
kπI′(vk)−πI′(w)k < ǫ/3.
From (3.5), with i = I′, we get kv |−kwk < ǫ, for all k ≥ K. (cid:3)
k
(cid:12) (cid:12)
(cid:12) (cid:12)
INVERSE LIMITS AND THE RNP 9
4. A variant of the Determining Property
In this section we discuss a variant of the Determining Property,
which was introduced in [CK06] (with a different name). A compact-
ness argument implies that it is equivalent to Definition 1.3, see Propo-
sition 4.6.
For the remainder of this section, we fix a standard inverse system
{(W ,θ )} and a closed subspace V ⊂ lim W .
i i i
←−
Definition 4.1. A positive nonincreasing finite sequence 1 ≥ ρ ≥
1
′
... ≥ ρ is ǫ-determining if for any pair v,v ∈ V, the conditions
N
(4.2)
′ ′ ′
kvk−kπ (v)k < ρ ·kvk, kv k−kπ (v )k < ρ ·kv k, 1 ≤ i ≤ N ,
i i i i
and
′ −1 ′
(4.3) kπ (v)−π (v )k < N ·max(kvk,kv k),
N N
imply
′ ′
(4.4) kv −v k < ǫ·max(kvk,kv k).
′
Observe that by dividing by max(kvk,kv k), it suffices to consider pairs
′ ′
v,v for which max(kvk,kv k) = 1.
This leads to the alternate definition of the Determining Property:
Definition 4.5. The pair (limW ,V) has the Determining Property if
i
←−
for every ǫ > 0 and every infinite nonincreasing sequence
1 ≥ ρ ≥ ... ≥ ρ ≥ ...
1 i
with ρ → 0, some finite initial segment ρ ≥ ... ≥ ρ is ǫ-determining.
i 1 N
Proposition 4.6. The pair (limW ,V) satisfies Definition 1.3 if
i
←−
and only if it satisfies Definition 4.5.
Proof. First we show that the property in Definition 4.5 implies the
property in Definition 1.3. So assume that the sequence {kv k} is
k
bounded and the convergence, kπ (v )k → kv k is uniform in k.
i k k
Suppose that there exists a sequence, a positive sequence, ρ ց 0,
i
such that kv k − kπ (v )k ≤ ρ . By applying the condition in Defi-
k i k i
nition 4.5 to this sequence and using convergence in the inverse limit
topology together with (4.3) it is clear from (4.4) that we obtain strong
convergence.
10 JEFF CHEEGER AND BRUCEKLEINER
Without loss of essential loss of generality, we can assume kv k ≤ 1
k
for all k. Since the convergence, kπ (v )k → kv k is uniform in k, it
i k k
follows that there exists a strictly increasing sequence, N < N < ...,
1 2
such that for all k, we have
1
kv k−kπ (v )k < .
k Nℓ k ℓ
Then kv k−kπ (v )k ≤ ρ , for the sequence, ρ given by
k i k i i
1
ρ = (N ≤ i < N ).
i ℓ ℓ+1
ℓ
Conversely, suppose that the property in Definition 1.3 holds, but
not the property in Definition 4.5. Then for some decreasing sequence
{ρ } ⊂ (0,∞) with ρ → 0, and some ǫ > 0, there are sequences
i i
′
{v },{v } ⊂ V, such that for all k < ∞,
k k
′
(4.7) kv k, kv k ≤ 1,
k k
′ ′
(4.8) max(kv k−kπ (v )k, kv k−kπ (v )k) < ρ for 1 ≤ i ≤ k,
k i k k i k i
1
′
(4.9) kπ (v )−π (v )k < ,
i k i k k
′
(4.10) kv −v k ≥ ǫ.
k k
∗
By the Banach-Alaoglu theorem, we can pass to weak convergent
′
subsequences, with respective limits v∞ and v∞. From (4.9), it follows
′
that v∞ = v∞.
′
Itfollowsfrom(4.7),(4.8),thatthesequences, kv k,kv k,arebounded
k k
′ ′
and the convergence kπ (v )k → kv k, kπ (v )k → kv k is uniform in
i k k i k k
k. Since we assume the property in Definition 1.3, it follows v →
k
′ ′ ′
v∞, vk → v∞, is actually strong. Since, v∞ = v∞, this contradicts
(cid:3)
(4.10).
We remark that proof of the implication Definition 1.3 =⇒ Defini-
tion 4.5 is similar to the proof of Proposition 3.11 in [CK06].
5. GFDA versus ANP
Weconclude withsome remarksabout therelationbetween theANP
and GFDA’s.