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

CHARACTERIZA TION OF THE RADON-NI K OD YM PR OPER TY IN TERMS OF INV E RSE LIMITS JEFF CHEEGER AND BR UCE KLEINER Abstract. In this pap er w e clarify the r elation betw een inv erse systems, the Radon-Nikody m prop er ty , the Asymptotic Nor ming Prop erty of James-Ho [JH81], and the GFD A spaces in tro duced in [CK06]. 1. Introduction A Banac h space V is said to ha v e the R adon-Niko dym Pr op erty (RNP) if every Lipschitz ma p f : R → V is differen tia ble almost ev erywhere. By no w, there are a num b er o f c hara cterizations o f Ba- nac h spaces with the RNP , the study o f whic h g o es bac k to G elfand [Gel38]; for a dditional references and discussion, see [BL00, Chapter 5], [G M85]. Of particular interest here is the characterization of the RNP in terms of the Asymptotic Norming Prop ert y; [JH81, GM85]. In this paper w e will sho w that a v arian t of the GFDA prop erty in tro duced in [CK06] is actually equiv alen t to the Asymptotic Norming prop erty of Ja mes-Ho, and hence b y [JH81, GM85], is equiv alen t to the RNP . In addition, w e observ e tha t the GFDA spaces of [CK06] are just spaces whic h are isomorphic to a separable dual space. Definition 1. 1. An in v erse system (1.2) W 1 θ 1 ← − W 2 θ 2 ← − . . . θ i − 1 ← − W i θ i ← − . . . , is s tand ar d if the W i ’s a re finite dimensional Banach spaces and the θ i ’s are linear maps o f norm ≤ 1. W e let π j : lim ← − W i → W j denote the pro jection map. Date : Octob er 24, 20 18. The first author was partially supp orted by NSF Grant DMS 0105 128 and the second by NSF Grant DMS 070 1515. 1 2 JEFF CHEEGER AND BR UCE KLEINER Definition 1.3. Let { ( W i , θ i ) } b e a standard in v erse system and V ⊂ lim ← − W i b e a subspace. The pair (lim ← − W i , V ) has the Determining Pr op- erty if a sequenc e { v k } ⊂ V conv erges strongly pro vided the pro jected sequence s { π j ( v k ) } ⊂ W j con v erge for ev ery j , the seq uence {k v k k} is b ounded, and the con v ergence k π j ( v k ) k → k v k k is uniform in k . A Ba- nac h space U has the Determining Pr op erty if there is a pair (lim ← − W i , V ) with D etermining Prop erty , suc h that V is isomorphic to U . W e ha v e: Theorem 1.4. A sep ar able Banach sp ac e has the RNP if and only it has the Determining Pr op erty. Since a Banach space has the RNP if and only if ev ery separable subspace has the RNP , Theorem 1.4 yields a c ha r acterization of the RNP for nonseparable Banac h spaces as w ell. T o pro v e the theorem, we first observ e in Prop osition 2.8 that the in v erse limit lim ← − W i is the dual space of a separable Banac h space. Then, by a completely elemen tary argumen t, w e show t ha t a Banac h space h as the Determining Property if and only if it has the Asymptotic Norming Prop ert y (ANP) of James-Ho [JH81]. Since a separable Ba- nac h space U has the RNP if and only if it has the ANP [JH81, GM85], the theorem follo ws. W e remark that there is a simple direct pro of that if V has the ANP (o r the Determining Prop erty), t hen ev ery Lipsc hitz map f : R → V is differentiable almost ev erywhere, see [CK]. Characterizations of the RNP using inv erse limits are useful for ap- plications; see [CK06], the discussion b elo w concerning metric measure spaces, and [CK]. Relation with previous w ork. In sligh t ly differen t language, our earlier pap er [CK06] also consid- ered pair s (lim ← − W i , V ), where lim ← − W i is t he in v erse limit of a standard in v erse system, and V ⊂ lim ← − W i is a closed subspace. A Go o d F inite Dimensional Approxim ation (GFDA) of a Banac h space V , a notion in tro duced in [CK0 6 ], is a pair (lim ← − W i , V ) with the Determining Prop- ert y suc h that π i | V : V → W i is a quotien t map for ev ery i . INVERSE LIMITS AND THE RNP 3 It follow s immediately from Lemma 3.8 of [CK06] that if (lim ← − W i , V ) is a G F D A of V , then V = lim ← − W i . Since such in v erse limits are dual spaces by Prop o sition 2.8, V is a separable dual space in this case. Con verse ly , using the Kadec-Klee renorming Lemma [Kad59, Kle61], it w a s shown in [CK06] that ev ery separable dual space is isomorphic to a Banac h space whic h admits a GFD A. Thus , a Banach space admits a GFDA if and only if it is isomorphic to a separable dual space. Applications to metric measure spaces. W e will call a metric measure space ( X , µ ) a PI sp ac e if the mea- sure is doubling, a nd a P oincar ´ e inequality ho lds in the sense of upp er gradien ts [HK98, Che99]. In [CK06], differen tia tion and bi-Lipschitz non-em b edding theorems w ere pro v ed fo r maps f : X → V from PI spaces into G FD A ta rgets V , generalizing results of [Che99] for finite dimensional targets. As e xplained ab o v e, it turns out that these targets are just separable dual spaces, up to isomorphism. As an application of the in v erse limit framew ork and the equiv alence b et w een the ANP a nd RNP , w e will sho w in [CK] that the differen- tiation theorem [CK06, Theorem 4.1] and bi-Lipschitz non-em b edding theorem [CK06, Theorem 5.1] hold whenev er the t a rget has the R NP . Ac kno wledgemen t. W e are v ery g rateful to Bill Johnson for sharing an observ ation whic h help ed giv e rise to this pap er. W e are m uc h indebted to Nigel Kalto n for immediately catching a serious error in an earlier vers ion. 2. Inverse syste ms In this section, w e recall some basic facts concerning direct and in- v erse systems, and the duality b etw een them. Then w e sho w that in- v erse limits of standard in verse systems are precisely duals of separable spaces. The following conv en tio ns will b e in for ce througho ut the r emainder of the pap er. Definition 2.1. An standar d dir e ct system is a sequence of finite di- mensional Banac h spaces { E i } and 1- Lipsc hitz linear maps ι i : E i → E i +1 . 4 JEFF CHEEGER AND BR UCE KLEINER Definition 2.2. An standar d inverse system is a sequence of finite di- mensional Banach spaces { W i } and 1-Lipsc hitz linear maps θ i : W i +1 → W i . Definition 2.3. A standard direct system is isometric al ly inje ctive if the maps ι i : E i → E i +1 are isometric injections. Definition 2.4. A standard inv erse system is quotient if the maps θ i : W i +1 → W i are quotien t maps. By a quotient m a p of normed spaces, w e mean a surjectiv e map π : U → V for whic h the norm on the targ et is the quotient norm, i.e. for ev ery v ∈ V , k v k = inf { k u k | u ∈ π − 1 ( v ) } . W e will refer to the maps ι i and θ i as b onding maps . There is a dualit y b et w een the ob jects in Definitions 2.1 and 2.2, resp ectiv ely , 2 .3 and 2.4: if { ( E i , ι i ) } is a standar d direct system, then { ( E ∗ i , ι ∗ i ) } is a standard in v erse system and con ve rsely; similarly , iso- metrically injectiv e direct systems are dual t o quotient systems. T o see this, one uses the fa cts that the adjoint of a 1-Lipschitz map of Banach spaces is 1-Lipsc hitz and the the a dj o in t of an isometric em b edding is a quotien t map. (This follows from the Hahn-Bana c h theorem.) In particular, since the spaces in our syste ms are assumed to b e finite dimensional (hence reflexiv e) ev ery in v erse system a rises as the dual of its dual direct system and conv ersely . The same holds fo r quotien t in v erse systems. W e no w recall the definitions of direct and in verse limits. Giv en a standard direct system { ( E i , ι i ) } we form the direct limit Banac h space lim − → E i as follows . W e b egin with the disjoin t union ⊔ i E i , and declare t w o elemen ts e ∈ E i , e ′ ∈ E i ′ to b e equiv alen t if their images in E j coincide f or some j ≥ max { i, i ′ } . Since the b onding maps are 1- Lipsc hitz, the set of equiv alence classes inherits an ob vious v ector space structure with a pseudo-norm. The direct limit lim − → E i is defined to b e the completion of the quotien t of t his space b y the closed subs pace of elemen ts whose pseudo-norm is zero. Clearly , there are 1-Lipsc hitz maps τ i : E i → lim − → E i , INVERSE LIMITS AND THE RNP 5 whic h in the case of isometrically injectiv e direct systems , are isometric injections. The union S i τ i ( E i ) is dense in lim − → E i . The in v erse limit lim ← − W i of a standard in v erse system { ( W i , θ i ) } is defined as follows. The underlying set consists of the collection of elemen ts ( w i ) ∈ Q i W i whic h are compatible with the b onding ma ps, i.e. θ i ( w i ) = w i − 1 for all i , and whic h satisfy sup i k w i k < ∞ . This is equipped with the ob vious v ector space structure and t he norm (2.5) k{ w i }k := lim j →∞ k w j k . The map (2.6) π j : lim ← − W i → W j giv en by π j ( { w i } ) = w j is 1- Lipsc hitz, and lim j →∞ k π j ( { w i } ) k = k{ w i }k . An inv erse limit lim ← − W i has a natural inverse lim it top olo gy , namely the we ak est top ology suc h that ev ery pro jection map π j : lim ← − W i → W j is con tin uous. Thus a sequence { v k } ⊂ lim ← − W i con v erges in t he in v erse limit top o logy to v ∈ lim ← − W i if and only if for ev ery i , we ha v e π i ( v k ) → π i ( v ) as k → ∞ . If { v k } ⊂ lim ← − W i and { v k } invl im − → v ∈ lim ← − W i , t hen (2.7) k v k ≤ lim inf k k v k k . Also, ev ery norm b o unded sequence { v k } ⊂ lim ← − W i has a subseq uence whic h con v erges with resp ect to t he in v erse limit top ology; this follows from a diagonal arg umen t, b ecause { π i ( v k ) } is con tained in a compact subset of W i , f o r all i . Prop osition 2.8. Given a standa r d inverse system { ( W i , θ i ) } , ther e is an isometric is omorphism (2.9) C : lim ← − W i ≡ (lim − → W ∗ i ) ∗ . In p articular, lim ← − W i is the dual of the sep ar a ble Banach sp ac e lim − → W ∗ i . 6 JEFF CHEEGER AND BR UCE KLEINER Pr o of. Pic k a compatible sequence ( x i ) ∈ lim ← − W i . W e get a map ⊔ W ∗ j → R b y sending φ ∈ W ∗ j to φ ( x j ); b ecause ( x i ) is compatible with b onding maps and | φ ( x j ) | ≤ k φ k k x j k ≤ k φ k k{ x j }k , this defines a linear functional of no rm ≤ k{ x j }k on lim − → W ∗ i . Therefore w e get a 1- Lipsc hitz map C : lim ← − W i − →  lim − → W ∗ i  ∗ . W e no w v erify that C is an isometry . Pic k ( x i ) ∈ lim ← − W i , and c ho ose n ∈ N suc h that k x n k ≥ k ( x i ) k − ǫ . If φ ∈ W ∗ n has norm 1 and φ ( x n ) = k x n k , then k C (( x i )) k k τ n ( φ ) k ≥ C (( x i ))( τ n ( φ )) = φ ( x n ) = k x n k ≥ k ( x i ) k − ǫ, where τ n : W ∗ n → lim − → W ∗ i is the canonical 1- Lipsc hitz map describ ed ab ov e. This sho ws that C is an isometric embedding. If Φ ∈ (lim − → W ∗ i ) ∗ , t hen w e define Φ i ∈ W ∗∗ i = W i to b e the comp osi- tion W ∗ i − → lim − → W ∗ i Φ − → R . This defines a compat ible sequence (Φ i ) ∈ lim ← − W i , suc h that k (Φ i ) k = k Φ k and C ((Φ i )) = Φ. Hence C is on to.  Corollary 2.10. 1) A sep ar able Banach sp ac e Y is isomorphic to the dir e ct limit of an isometric al ly inje ctive dir e ct system ( E i , ι i ) . 2) The dual sp ac e Y ∗ of the sep ar able Banach sp ac e Y (as in 1)) is isometric to the invers e limit lim ← − E ∗ i of the a quotient inverse system { ( E ∗ i , ι ∗ i ) } . Pr o of. T o see tha t 1) holds, start with a coun table increasing sequenc e E 1 ⊂ E 2 ⊂ · · · ⊂ Y of finite dimensional subspaces whose union is dense in Y , and tak e the b onding maps ι i : E i → E i +1 to b e the inclusions. Clearly the inclusion maps E i → Y induce an isometry lim − → E i → Y . Assertion 2 ) follows from 1) and Prop osition 2.8.  INVERSE LIMITS AND THE RNP 7 Let C b e the isometry in Prop o sition 2 .8. Lemma 2.11. 1) Supp ose { v k } ⊂ lim ← − W i is a se quenc e such that { C ( v k ) } ⊂ (lim − → W ∗ i ) ∗ we ak* c onver ges to some y ∈ (lim − → W ∗ i ) ∗ . Then { v k } is c onver gent w ith r esp e ct to the in verse li m it top olo gy, an d its lim it v ∞ ∈ lim ← − W i satisfies C ( v ∞ ) = y ; in p articular, y ∈ C (lim ← − W i ) . 2) If { v k } ⊂ lim ← − W i c onver ges in the inverse limit top olo gy, and has uniformly b ounde d norm, then { C ( v k ) } is we ak* c onver gent. Pr o of. Assertions 1) and 2) f ollo w readily from the assumption that the W i are finite dimensional to gether with the densit y of compatible sequence s in lim ← − W i .  3. The proof of Theorem 1. 4 The pro of of Theorem 1.4 is based on the Asymptotic Norming P rop- ert y , which w e no w recall. Let Y denote a separable Banac h space and V ⊂ Y ∗ a separable subspace of its dual. (Here Y ∗ need not b e separable.) Definition 3.1. The pair ( Y ∗ , V ) has the Asymptotic Norming Pr op- erty (ANP) if a sequence { v k } ⊂ V conv erges strongly provided it is w eak* con v ergen t and the sequence of norms {k v k k} conv erges to the norm of the we ak* limit. A Banach space U is said to hav e the Asymptotic Norming Pr op erty if there is a pair ( Y ∗ , V ) with the ANP suc h that U is isomorphic to V . Theorem 3.2 ([JH81, GM85]) . F or sep ar a b le Bana ch sp ac es, the RNP is e quivalent to the ANP. Hence to pro v e Theorem 1.4, it suffices to sho w that for separable Banac h spaces, the ANP is equiv alen t to the Determining Prop ert y . By Corollary 2.10, ev ery separable Ba nac h space Y is isometric to the direct limit of a standard direct sys tem, and Y ∗ is isometric to the in v erse limit o f the dual in v erse system. Hence the pro of of Theorem 1.4 reduces to: 8 JEFF CHEEGER AND BR UCE KLEINER Prop osition 3.3. L et { ( W i , θ i ) } b e a standar d inve rs e system, and V b e a clo s e d sep ar able subsp ac e of lim ← − W i . Th e n the p air (lim ← − W i , V ) has the ANP if and only if it h as the Determining Pr op erty. Her e we ar e identifying lim ← − W i with the dual of lim − → W ∗ i , se e Pr op osition 2 . 8 . Pr o of. Let { v k } ⊂ V b e a sequence with b ounded norm. By Lemma 2.11, the sequenc e { v k } is w eak* con v ergen t if and only if it con v erges in the inv erse limit top olog y . Therefore, to prov e the equiv alence of the ANP and the Determining Prop erty for the pair (lim ← − W i , V ), it suffices to show that when (3.4) v k w ∗ − → w ∈ lim ← − W i , the seq uence of norms {k v k k} con verges t o the k w k if and o nly if the con v ergence k π j ( v k ) k → k v k k is uniform in k . Although this is com- pletely elemen tary , we will write out the details. W e ha v e (3.5) k v k k − k w k = ( k v k k − k π i ( v k ) k ) + ( k π i ( v k ) k − k π i ( w ) k ) + ( k π i ( w ) k −k w k ) . Assume first that lim k →∞ k v k k = k w k . Given ǫ > 0, there exists I 1 suc h that k w k − k π i ( w ) k < ǫ/ 3, fo r i ≥ I 1 . By (3.4) there exists K 1 suc h that k π I 1 ( v k ) − π I 1 ( w ) k < ǫ/ 3, fo r k ≥ K 1 . Also, there exists K 2 suc h that   k v k k − k w k   < ǫ/ 3, if k ≥ K 2 . Set K = max( K 1 , K 2 ). F rom (3.5), with i = I 1 , w e get k v k k − k π I 1 ( v k ) k < ǫ , for all k ≥ K . Since, k v k k − k π i ( v k ) k is a nonnegativ e decreasing function of i , this implies, k v k k − k π i ( v k ) k < ǫ , for all i ≥ I 1 , k ≥ K . Finally , there exists I 2 suc h that k v k k − k π i ( v k ) k < ǫ for all i ≥ I 2 , k = 1 , . . . , K − 1, Th us, if i ≥ max( I 1 , I 2 ) then k v k k − k π i ( v k ) k < ǫ , for all k . Con verse ly , supp ose the conv ergence k π i ( v k ) k → k v k k is uniform in k . Give n ǫ > 0 , there exists I suc h that k v k k − k π i ( v k ) k < ǫ/ 3, for i ≥ I and all k . Also, there exis ts I 1 suc h that k w k − k π i ( w ) k < ǫ/ 3, for i ≥ I 1 . Set I ′ = max( I , I 1 ). By (3 .4), there exists K suc h that k π I ′ ( v k ) − π I ′ ( w ) k < ǫ/ 3. F rom (3.5), with i = I ′ , we get   k v k | − k w k   < ǫ , for all k ≥ K .  INVERSE LIMITS AND THE RNP 9 4. A v ariant of the De termining Proper ty In this section w e discuss a v arian t of the Determining Prop erty , whic h w as in t r o duced in [CK06] (with a different name). A compact- ness argumen t implies that it is equiv alen t to D efinition 1.3, see Prop o- sition 4 .6. F or the remainder of t his section, w e fix a standard inv erse system { ( W i , θ i ) } and a closed subspace V ⊂ lim ← − W i . Definition 4.1. A p ositiv e nonincreasing finite seq uence 1 ≥ ρ 1 ≥ . . . ≥ ρ N is ǫ -determini n g if for any pair v , v ′ ∈ V , the conditions (4.2) k v k − k π i ( v ) k < ρ i · k v k , k v ′ k − k π i ( v ′ ) k < ρ i · k v ′ k , 1 ≤ i ≤ N , and (4.3) k π N ( v ) − π N ( v ′ ) k < N − 1 · max( k v k , k v ′ k ) , imply (4.4) k v − v ′ k < ǫ · max( k v k , k v ′ k ) . Observ e that b y dividing b y max( k v k , k v ′ k ), it suffices to consider pairs v , v ′ for whic h max( k v k , k v ′ k ) = 1. This leads to the alternate definition of the D etermining Prop erty: Definition 4.5. The pair (lim ← − W i , V ) has the Determining Pr op erty if for ev ery ǫ > 0 and ev ery infinite nonincreasing sequence 1 ≥ ρ 1 ≥ . . . ≥ ρ i ≥ . . . with ρ i → 0, some finite initial segmen t ρ 1 ≥ . . . ≥ ρ N is ǫ -determining. Prop osition 4.6. The p air (lim ← − W i , V ) satisfies Definition 1.3 if and only if it satisfies Definition 4.5. Pr o of. First w e sho w that the prop ert y in D efinition 4.5 implies the prop erty in D efinition 1.3. So assume that the sequence {k v k k} is b ounded and the con v ergence, k π i ( v k ) k → k v k k is uniform in k . Supp ose that there exists a sequence, a p o sitiv e sequence, ρ i ց 0, suc h that k v k k − k π i ( v k ) k ≤ ρ i . By applying the condition in Defi- nition 4.5 to this sequence and using conv ergence in the in vers e limit top ology tog ether with (4.3) it is clear fro m (4.4) that w e obtain strong con v ergence. 10 JEFF CHEEGER AND BR UCE KLEINER Without loss of essen tia l loss of generality , w e can assume k v k k ≤ 1 for a ll k . Since the con vergenc e, k π i ( v k ) k → k v k k is uniform in k , it follo ws that there exists a strictly increasing sequence, N 1 < N 2 < . . . , suc h that for all k , w e ha v e k v k k − k π N ℓ ( v k ) k < 1 ℓ . Then k v k k − k π i ( v k ) k ≤ ρ i , for the sequence, ρ i giv en by ρ i = 1 ℓ ( N ℓ ≤ i < N ℓ +1 ) . Con verse ly , supp ose that the prop erty in D efinition 1.3 holds, but not the pro p ert y in Definition 4.5. Then for some decreasing sequence { ρ i } ⊂ (0 , ∞ ) with ρ i → 0, and some ǫ > 0, there are sequences { v k } , { v ′ k } ⊂ V , suc h that for all k < ∞ , (4.7) k v k k , k v ′ k k ≤ 1 , (4.8) max ( k v k k − k π i ( v k ) k , k v ′ k k − k π i ( v ′ k ) k ) < ρ i for 1 ≤ i ≤ k , (4.9) k π i ( v k ) − π i ( v ′ k ) k < 1 k , (4.10) k v k − v ′ k k ≥ ǫ . By the Ba na c h- Alaoglu theorem, w e can pass to w eak ∗ con v ergent subseque nces, with resp ectiv e limits v ∞ and v ′ ∞ . F ro m (4.9), it follow s that v ∞ = v ′ ∞ . It follo ws from (4.7), ( 4 .8), that the sequences, k v k k , k v ′ k k , are b ounded and the conv ergence k π i ( v k ) k → k v k k , k π i ( v ′ k ) k → k v ′ k k is uniform in k . Since we assume the pr o p ert y in Definition 1.3, it follow s v k → v ∞ , v ′ k → v ′ ∞ , is actually strong. S ince, v ∞ = v ′ ∞ , this con tradicts (4.10).  W e remark that pro of of the implication Definition 1.3 = ⇒ Defini- tion 4.5 is similar to the pro of of Prop o sition 3 .11 in [CK06]. 5. GFDA ve rsus ANP W e conclude w ith some remarks ab out the relation b et we en the ANP and GFDA’s. INVERSE LIMITS AND THE RNP 11 Supp ose Y is a separable Banac h space and ( Y ∗ , V ) has the ANP . By Lemma 2.10, w e ma y realize Y ∗ – up to isometry – as the in v erse limit o f a quotien t sys tem { ( W i , θ i ) } . Viewing V as a subspace of lim ← − W i , one migh t b e tempted to mo dify the inv erse system to pro duce a GFDA of V . F o r instance, one could restrict the pro jection maps π j : lim ← − W i → W j to V , and replace W j with π j ( V ) ⊂ W j . How ev er, the resulting maps π j | V : V → π j ( V ) will usually not b e quotien t maps. One could also tr y renorming the spaces π j ( V ) ⊂ W j so that the restrictions π j | V : V → π j ( V ) b ecome quotien t maps. This will t ypically destroy the Determining Prop ert y , ho wev er. In an y case, V will not admit a ny GFDA unless it is a separable dual space, whereas man y Ba na c h spaces with the RNP a re not separable dual spaces. Reference s [BL00] Y. Beny amini and J. Lindenstrauss. Ge ometric nonline ar fun ctional a naly- sis. Vol. 1 , volume 48 of Americ an Mathematic al So ciety Col lo quium Pub- lic ations . American Mathematical So ciety , Pr ovidence, RI, 200 0. [Che99] J. Cheeger. Differen tiability of Lipschitz functions on metr ic measure spaces. Ge om. F unct. Anal. , 9(3):428– 5 17, 1999. [CK] J. Cheeger and B. K leiner. In prepar ation. [CK06] J. Cheeger a nd B. K leiner. O n the differentiabilit y of Lipsch tz maps from metric meas ur e spaces into Banach spaces. In Inspir e d by S .S. Chern, A Memorial volume in honor of a gr e at mathematician , volume 11 o f Nankai tr acts in Mathematics , pages 129–15 2. W or ld Scientific, Singap or e, 2006. [Gel38] I. M. Gelfand. 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