Topological classification of closed convex sets in Frechet spaces

TOPOLOGIC AL CLASSIFIC A TION OF CLOSED CO NVEX SETS I N FR ´ ECHET SP ACES T ARAS BANAKH AND ROB ER T CA UTY Abstra ct. W e prov e that eac h non-separable completely metrizable con vex subset of a F r ´ ec het space is homeomorphic to a Hilb ert space. This resolves an old (more than 30 ye ars) problem of infinite- dimensional top ology . Com bined with the t opological classification of separable con ve x sets due to Klee, Dobrow oslki and T oru ´ nczyk, th is result implies that each closed conv ex sub set of a F r ´ ec het space is homeomorphic to [0 , 1] n × [0 , 1) m × ℓ 2 ( κ ) for some cardinals 0 ≤ n ≤ ω , 0 ≤ m ≤ 1 and κ ≥ 0. The problem of top ological classification o f con v ex sets in linear metric sp aces traces its history bac k to found ers of functional analysis S .Banac h and M.F r´ ec het. F or separable closed conv ex sets in F r´ ec het spaces this problem was resolv ed b y com bined efforts of V.Klee [8] (see [3, II I.7.1]), Dobro wolski and T oru ´ nczyk [4], [5]: Theorem 1 (Klee-Dobro w olski-T oru ´ nczyk) . Each sep ar able close d c onvex subse t C of a F r´ echet sp ac e is hom e omorphic to [0 , 1] n × [0 , 1) m × (0 , 1) k for some c ar dinals 0 ≤ n, k ≤ ω and 0 ≤ m ≤ 1 . In p articular, C is home omorphic to the sep ar able H ilb ert sp ac e l 2 if and only if C is not lo c al ly c omp act. By a F r ´ echet sp ac e w e mean a lo cally co n vex complete linear metric space. A line ar metric sp ac e is a linear top ological space end o wed with an inv arian t metric that generates its top olog y . A top ological space is called c ompletely metrizable if its topology is generated by a complete m etric. In this pap er w e stud y the top ologica l structure of non-separable (completely metrizable) conv ex sets in F r ´ ec het spaces a nd pro v e the follo wing theorem that answ ers a n old prob lem LS10 p osed in Geoghega n ’s list [7] and then rep eated in [11] and [2]. Theorem 2. Each non-sep ar able c ompletely metrizable c onvex su bset of a F r´ echet sp ac e i s home o- morphic to a Hilb ert sp ac e. Theorems 1 and 2 imply the follo wing top ologica l c lassifi cation of closed con vex subs et in F r´ ec het spaces. Theorem 3. Each close d c onvex subset C of a F r´ e chet sp ac e is home omorphic to [0 , 1] n × [0 , 1) m × ℓ 2 ( κ ) for some c ar dinals 0 ≤ n ≤ ω , 0 ≤ m ≤ 1 and κ ≥ 0 . In p articular, C is ho me omorphic to an infinite- dimensional Hilb e rt sp ac e if and only if C is not lo c al ly c omp act. Here ℓ 2 ( κ ) stands for th e Hilb ert space that has an orthonormal basis of cardin alit y κ . The top ology of in finite-dimensional Hilb ert spaces was c h aracterized b y T oru ´ nczyk [9], [10]. Th is c h aracterizat ion w as u s ed in the pro of of the follo win g criterion from [2] whic h is our main to ol for th e pro of of Theorem 2. Theorem 4 (Banakh-Zaric hnyy) . A c onvex subset C i n a line ar metric sp ac e is home omorph ic to an i nfinite-dimensional Hilb ert sp ac e if and only if C is a c ompletely metrizable absolute r etr act with LF AP. A top ological space X is defined to hav e the lo c al ly finite appr oximation pr op erty (briefly , LF AP), if for eac h op en co ve r U of X there is a sequen ce of maps f n : X → X , n ∈ ω , suc h that eac h f n 1991 Mathematics Subje ct Cl assific ation. 57N17; 46A04. Key wor ds and phr ases. Conv ex set, F r´ ec het space, non-separable Hilb ert space. 1 2 T ARAS BANAK H AND ROBER T CAUTY is U -near to the ident ity id X : X → X an d th e family  f n ( X )  n ∈ ω is locally finite in X . The latter means that eac h p oin t x ∈ X has a n eighb orh o o d O ( x ) ⊂ X that meets only finitely m an y sets f n ( X ), n ∈ ω . Theorem 2 follo ws immediately from T heorem 4, the Borsuk-Dugundji Theorem [3, I I .3.1] (sa yin g that conv ex subsets of F r ´ ec het sp aces are absolute retracts) and the f ollo wing theorem that will b e pro ved in S ection 3. Theorem 5. Each non-sep ar able c onvex subset of a F r´ echet sp ac e h as LF AP. 1. Sep ara ted Appro xima tion Proper ty Theorem 5 establishing LF AP in non-separable con vex sets will b e pro ved with h elp of the metric coun terpart of LF AP , called SAP . A metric sp ace ( X, d ) is d efined to hav e the sep ar ate d appr oximation pr op erty (briefly , SAP) if for eac h ε > 0 there is a sequence of maps f n : X → X , n ∈ ω , suc h that eac h f n is ε -homotopic to id X and the family ( f n ( X )) n ∈ ω is sep ar ate d i n the sense that inf n 6 = m d ( f n ( X ) , f m ( X )) > 0. Here for t wo non-empt y su bsets A, B ⊂ X w e p u t d ( A, B ) = in f { d ( a, b ) : a ∈ A, b ∈ B } . Two maps f , g : A → X are called ε -homotopic if th ey can b e linke d b y a homotop y ( h t ) t ∈ I : A → X suc h that h 0 = f , h 1 = g and diam { h t ( a ) : t ∈ I } ≤ ε for all a ∈ A . By I w e denote the unit inte r v al [0 , 1]. The follo wing lemma is pro ved b y analogy with Lemm a 1 of [5] and Lemma 5.2 of [2]. Lemma 1. Each metric sp ac e with SAP satisfies LF AP . Pr o of. Assu me that a metric sp ace ( X, d ) has SAP . T o sh o w that X has LF AP , fix an op en co v er U of X and find a non-expanding function ε : X → (0 , 1) suc h that the co v er { B d ( x, ε ( x )) : x ∈ X } refines the co ver U . F or ev ery k ∈ ω consider the closed s u bset X k = { x ∈ X : ε ( x ) ≥ 2 − k } of X . Pu t ε k = 1 / 4 k +2 for k ≤ 1 and let f 0 : X × ω → X , f 0 : ( x, n ) 7→ x , b e the pro jection. By in duction w e shall construct a sequence ( ε k ) k ∈ ω of p ositiv e real num b ers and a sequence of maps f k : X × ω → X , k ∈ ω , such that the follo win g cond itions are satisfied: (1) ε k ≤ 1 4 ε k − 1 ≤ 1 4 k +2 ; (2) f k is ε k -homotopic to f k − 1 ; (3) f k | X k − 3 × ω = f k − 1 | X k − 3 × ω ; (4) f k | ( X \ X k +1 ) × ω = f 0 | ( X \ X k +1 ) × ω ; (5) inf n 6 = m d ( f k ( X k × { n } ) , f k ( X k × { m } )) ≥ 4 ε k +1 . Assume that maps f i : X × ω → X and n u m b ers ε i +1 satisfying the conditions (1)–(4) hav e b een constructed for all i < k . By S AP , there is an ε k -homotop y ( h t ) t ∈ I : X × ω → X suc h that h 0 = f 0 and δ = in f n 6 = m d ( h 1 ( X × { n } ) , h 1 ( X × { m } ) > 0. Cho ose a con tin uous function λ : X → [0 , 1] such that X k − 3 ∪ ( X \ X k +1 ) ⊂ λ − 1 (0) and X k \ X k − 2 ⊂ λ − 1 (1). T ak e any p ositiv e n u m b er ε k +1 ≤ 1 4 min { δ , ε k } and defin e a function f k : X × ω → X by f k ( x, n ) = h λ ( x ) ( f k − 1 ( x, n ) , n ) . It is clear that the conditions (1)–(4) are satisfied. The condition (5) w ill follo w as so on as w e c hec k that d ( f k ( x, n ) , f k ( y , m )) ≥ 4 ε k +1 for any p oin ts x, y ∈ X k and distinct n umb ers n 6 = m . Find un ique n umb ers i, j ≤ k suc h that x ∈ X i \ X i − 1 and y ∈ X j \ Y j − 1 . If i, j < k , then d ( f k ( x, n ) , f k ( y , m )) ≥ d ( f k − 1 ( x, n ) , f k − 1 ( y , m )) − 2 ε k ≥ 4 ε k − 2 ε k = 2 ε k ≥ 4 ε k +1 . It remains to consider the case max { i, j } = k . W e lose no generalit y assumin g th at i = k . If j ≥ k − 1, th en d ( f k ( x, n ) , f k ( y , m )) = d ( h 1 ( f k − 1 ( x, n ) , n ) , h 1 ( f k − 1 ( y , m ) , m )) ≥ δ ≥ 4 ε k +1 . Next, assume that j ≤ k − 2. In this case k ≥ j + 2 ≥ 3. Then ε ( x ) < 2 − i +1 = 2 − k +1 < 2 − k +2 ≤ 2 − j ≤ ε ( y ) TOPOLOGICAL CLASSIFICA TION OF CLOSED CONVEX SETS IN FR ´ ECHET SP ACES 3 and th e n on-expanding prop ert y of ε imply that d ( x, y ) ≥ | ε ( x ) − ε ( y ) | ≥ 2 − j − 2 − k +1 ≥ 2 − j − 1 . It follo ws from (4) and (2) that d ( x, f k ( x, n )) = d ( f i − 2 ( x, n ) , f k ( x, n )) = d ( f k − 2 ( x, n ) , f k ( x, n )) ≤ ε k − 1 + ε k ≤ 2 ε k − 1 ≤ 2 4 k +1 and d ( y , f k ( y , m )) = d ( f j − 2 ( y , m ) , f k ( y , m )) ≤ ε j − 1 + · · · + ε k ≤ 2 ε j − 1 ≤ 2 4 j +1 . Then d ( f k ( x, n ) , f k ( y , m )) ≥ d ( x, y ) − d ( x, f k ( x, n )) − d ( y , f k ( y , m )) ≥ 1 2 j +1 − 2 4 k +1 − 2 4 j +1 ≥ 1 2 j +1 − 2 4 j +3 − 2 4 j +1 ≥ 4 4 j +5 ≥ 4 4 k +3 ≥ 4 ε k +1 . This completes th e indu ctiv e s tep. After c omp leting the inductiv e construction, let f ∞ = lim k →∞ f k : X × ω → X . The conditions (1)-(3) guaran tee that the limit function f ∞ is w ell-defined and con tinuous. Let us show that f ∞ is ε -near to f 0 . Giv en a ny p oin t ( x, n ) ∈ X × ω , find a un ique num b er i ∈ N such that x ∈ X i \ X i − 1 . By (3) and (4), f ∞ ( x, n ) = f i +2 ( x, n ) and f 0 ( x, n ) = f i − 2 ( x, n ). Th en d ( f ∞ ( x, n ) , x ) = d ( f ∞ ( x, n ) , f 0 ( x, n )) = d ( f i +2 ( x, n ) , f i − 2 ( x, n )) ≤ e i +2 + · · · + ε i − 1 ≤ 2 ε i − 1 ≤ 2 4 i < 1 2 i ≤ ε ( x ) . The c h oice of the function ε guaran tees that f ∞ is U -near to the pro jection f 0 : X × ω → X . It remains to p ro ve that the family ( f ∞ ( X × { n } ) n ∈ ω is discrete in X . Giv en an y p oint x ∈ X , let i ∈ N b e th e u nique n u m b er su c h that x ∈ X i \ X i − 1 . Con s ider the ball B ( x ; 1 / 2 i +2 ) = { x ′ ∈ X : d ( x, x ′ ) < 1 / 2 i +2 } cen tered at x . Claim 1. B ( x ; 1 / 2 i +2 ) ∩ f ∞ ( X × ω ) ⊂ f ∞ ( X i +1 × ω ) . Pr o of. Assu me con v ersely that f ∞ ( y , m ) ∈ O ( x ) for some y ∈ X \ X i +1 and m ∈ ω . Let j ∈ ω b e a unique num b er with y ∈ X j \ X j − 1 . It follo ws fr om y / ∈ X i +1 that j ≥ i + 2. Since d ( f ∞ ( y , m ) , y ) = d ( f j +3 ( y , m ) , f j − 2 ( y , m )) ≤ 2 ε j − 1 ≤ 2 4 j +1 ≤ 1 2 i +2 , and ε ( y ) < 1 2 j − 1 < 1 2 i ≤ ε ( x ), b y the non-expanding p rop erty of ε , w e get a cont r adiction: 1 2 i +1 ≤ 1 2 i − 1 2 j − 1 ≤ | ε ( x ) − ε ( y ) | ≤ d ( x, y ) ≤ d ( x, f ∞ ( y , m )) + d ( f ∞ ( y , m ) , y ) < 1 2 i +2 + 1 2 i +2 = 1 2 i +1 .  No w the condition (5) and the inequalit y ε i +2 ≤ 1 4 i +4 ≤ 1 2 i +2 implies that the b all B ( x ; ε i +2 ) meets at most one set f ∞ ( X i +1 × { n } ) and hence at most one set f ∞ ( X × { n } ), which means that the f amily  f ∞ ( X × { n } )  n ∈ ω is discrete in X and h ence X has LF AP .  2. SAP in non-se p arabl e conve x cones In this sectio n we shall prov e that non-separable conv ex cones in F r´ ec h et spaces hav e SAP . A subset C of a linear metric s pace ( L, d ) is c alled a c onvex c one if it is conv ex and R + · C = C where R + = [0 , ∞ ). The principal r esult of th is section is Lemma 2. Each non-sep ar able c onvex c one C in a F r´ echet sp ac e L ha s SAP. F or the pro of of this lemma w e shall u se an op erator version of Josefson-Nissenzw eig Theorem pro ved in [1]: 4 T ARAS BANAK H AND ROBER T CAUTY Lemma 3. F or any dense c ontinuous non-c omp act line ar op er ator S : X → Y b etwe en norme d sp ac es ther e is a line ar c ontinuous op er ator T : Y → c 0 such that the op er ator T S : X → c 0 is not c omp act. Let us recall that an op erator T : X → Y b etw een linear top ologica l s paces is • dense if T X is dense in Y ; • c omp act if the image T ( U ) of some op en neigh b orho o d U ⊂ X of zero is tota lly b ound ed in Y . A s u bset B of a linear top ological space Y is total ly b ounde d if for eac h op en neigh b orho o d V ⊂ Y of zero there is a finite su bset F ⊂ Y suc h that B ⊂ V + F . Pr o of of L emma 2. Assume that C is a non-separable con vex cone in a F r ´ ec h et space L . By [3, I.6.4], the top ology of the F r´ ec het space L is generated by an in v arian t metric d L suc h that for ev ery ε > 0 the ε -ball B L ( ε ) = { x ∈ L : d L ( x, 0) < ε } cen tered at the origin is con v ex. W e lose no generalit y assuming that the linear sub space C − C is dense in the F r´ ec het space L . Giv en an y ε > 0, w e need to construct maps f k : C → C , k ∈ ω , such that eac h f k is ε -homotopic to id C and inf k 6 = n d ( f k ( C ) , f n ( C )) > 0. Since the metric d has con v ex balls, any t wo ε -near m ap s in to C are ε -homotopic. Claim 2. Ther e is a line ar c ontinuous op er ator R : L → Y onto a norme d sp ac e Y such that the image R ( C ) is not sep ar able and R − 1 ( ¯ B Y ) ⊂ B L ( ε/ 2) wh er e ¯ B Y = { y ∈ Y : k y k ≤ 1 } is the close d unit b al l in the norme d sp ac e Y . Pr o of. By [3 , I.6.4], the F r ´ ec het sp ace L can b e identified with a closed linear sub space of the count able pro du ct Q i ∈ ω X i of Banac h s paces. F or ev ery n ∈ ω let Y n = Q i 0. Finally , consider the linear space Y = pr m ( L ) ⊂ Y m endo wed with the norm k y k = 1 r k y k m where k · k m is the norm of the Banac h space Y m . T h en the op erator R = pr m : L → Y h as the d esir ed prop erties.  In the con vex cone C consid er the co nv ex subset B C = C ∩ R − 1 ( ¯ B Y ) and observe that C = R + · C 1 and hence T ( C ) = R + · T ( B C ). Since the sp ace T ( C ) is not separable, T ( B C ) is not separable to o. Consid er the con vex b ounded symmetric su bset D = T ( B C ) − T ( B C ) ⊂ Y and obs erv e that R · D = R ( C ) − R ( C ) = R ( C − C ). T hen the Minko w ski fu nctional k x k Z = inf { λ > 0 : x ∈ λD } is a w ell-defin ed norm on the linear space Z = R · D = R ( C − C ) and the identit y inclusion I : Z → Y is a b ounded linear oper ator from the normed space ( Z, k · k Z ) to the Ba n ac h sp ace Y . S ince I ( Z ) = Z i s non-separab le, the op erator I is not compact. By Lemm a 3, there is a b ou n ded op erator T : Y → c 0 suc h that the comp osition T I : Z → c 0 is not compact. The latter means that the image T ( D ) = T R ( B C ) − T R ( B C ) is not totally b ounded in c 0 and hence the b ounded set T R ( B C ) is n ot totally b ou n ded in c 0 . Consequent ly , ther e is δ ∈ (0 , 1] su c h th at for ev ery n ∈ ω (1) T R ( B C ) 6⊂ { ( x i ) i ∈ ω ∈ c 0 : max i ≥ n | x i | < δ } . F or ev er y n ∈ ω let e ∗ n ∈ c ∗ 0 , e ∗ n : ( x i ) i ∈ ω 7→ x n , b e th e n th coord inate functional of c 0 and let z ∗ n = ( T R ) ∗ ( e ∗ n ) ∈ L ∗ . Claim 3. Ther e ar e an incr e asing numb er se quenc e ( m k ) k ∈ ω and a se quenc e ( z k ) k ∈ ω ⊂ B C such that for every k ∈ ω ; (1) | z ∗ m k ( z k ) | ≥ δ ; TOPOLOGICAL CLASSIFICA TION OF CLOSED CONVEX SETS IN FR ´ ECHET SP ACES 5 (2) | z ∗ m i ( z k ) | < δ 3 / 100 for al l i > k . Pr o of. Th e sequences ( m k ) and ( z k ) will b e constru cted by ind u ction. By (1) there are a p oin t z 0 ∈ B C and a n umb er m 0 ∈ ω suc h that | e ∗ m 0 ( z 0 ) | ≥ δ . No w assu me that for some k ∈ ω p oin ts z 0 , . . . , z k and num b ers m 0 < m 1 < · · · < m k ha ve b een constructed. Sin ce the p oin ts T R ( z i ), i ≤ k , b elong to the Ba n ac h space c 0 , there is a num b er m > m k so l arge that | e ∗ n ( T R ( z i )) | < δ 3 / 100 for all n ≥ m and i ≤ k . By (1), there are a p oint z k +1 ∈ B C and a n umb er m k +1 ≥ m suc h that | z ∗ m k ( z k +1 ) | = | e ∗ m k ( T R ( z k +1 )) | ≥ δ . Th is complete the inductiv e step.  Divide ω int o the coun table union ω = S k ∈ ω N k of pairwise disjoin t infin ite subsets and b y induction define a function ξ : ω × ω → ω such that ξ ( i, k ) ∈ N k and ξ ( i + 1 , k ) > ξ ( i ) > i for all i, k ∈ ω . F or an y n u m b ers i, k ∈ ω let z i,k := z ξ ( i,k ) and z ∗ i,k := z ∗ m ξ ( i,k ) = ( T R ) ∗ ( e ∗ m ξ ( i,k ) ) , where ( z i ) i ∈ ω and ( m k ) k ∈ ω are giv en by Claim 3. It follo ws that the double sequences ( z i,k ) i,k ∈ ω and ( z ∗ i,k ) i,k ∈ ω ha ve the follo wing prop erties (that will b e used in th e pro of of Claim 8 b elo w): Claim 4. If ( i, k ) , ( j, n ) ∈ ω × ω , then (1) | z ∗ i,k ( z i,k ) | ≥ δ ; (2) | z ∗ j,k ( z i,n ) | < δ 3 / 100 pr ovide d ξ ( j, k ) > ξ ( i, n ) ; (3) | z ∗ i,k ( z ) | ≤ 1 for any z ∈ B C . Claim 5. Ther e is a map f : C → C such that d ( f , id) < ε/ 2 and e ach p oint x ∈ C has a neighb orho o d O ( x ) whose image f ( O ( x )) lies in the c onvex hul l con v ( F x ) of some finite subset F x ⊂ C . Pr o of. Using the paracompactness of the metrizable space C , find a lo cally finite op en co ve r U of X that r efines the co v er of C by op en ε 4 -balls. In eac h set U ∈ U pic k up a p oin t c U ∈ U . Let { λ U : C → [0 , 1] } U ∈U b e a partition of the u nit y , sub ordinated to the co ver U in the sens e that λ − 1 U ((0 , 1]) ⊂ U for all U ∈ U . Finally , define a map f : C → C by the formula f ( x ) = X U ∈U λ U ( x ) c U . It is s tandard to c hec k that f h as the d esired prop ert y .  F or ev ery k ∈ Z b y C k denote the set of p oin ts x ∈ C that h a v e neighborh o o d O ( x ) ⊂ C s u c h that for ea ch point x ′ ∈ O ( x ) and a non-negativ e num b er m ≥ k w e get | z ∗ m f ( x ′ ) | < δ 3 / 100. It is cl ear that eac h set C k is op en in C and lies in C k +1 . Claim 6. C = S k ∈ ω C k . Pr o of. By Claim 5, eac h p oin t x ∈ C has a neigh b orho o d O ( x ) ⊂ C su c h that f ( O ( x )) ⊂ con v ( F ) for some finite subset F ⊂ C . T aking in to account that T R ( F ) is a fi nite s u bset of the Ba nac h space c 0 , w e can find a n umber m ∈ ω suc h that | e ∗ n T R ( z ) | < δ 3 / 100 for a ll n ≥ m a n d all z ∈ F . T hen also | e ∗ n T R ( z ) | < δ 3 / 100 for all z ∈ con v ( F ), in particular, | e ∗ n T Rf ( x ′ ) | < δ 3 / 100 for any x ′ ∈ O ( x ). This means that x ∈ C m b y the definition of the set C m .  Claim 7. Ther e is a n op en c over ( U k ) k ∈ ω of t he sp ac e C such that U k ⊂ ¯ U k ⊂ C k − 1 ∩ U k +1 for al l k ∈ ω . Pr o of. By Theorem 5.2.3 of [6], there is an op en co ver ( V k ) k ∈ ω of X suc h that ¯ V k ⊂ C k ∩ V k +1 for all k ∈ ω . F or eac h x ∈ X fin d the smallest num b er k ∈ ω with x ∈ C k and the largest num b er n ≤ k with x / ∈ ¯ V n and p ut O ( x ) = C k \ ¯ V n . Consider the op en co ver W 0 = { O ( x ) : x ∈ C } and observe that S t ( ¯ V k , W 0 ) ⊂ C k for ev ery k ∈ ω . Here S t ( A, W 0 ) = ∪{ W ∈ W 0 : W ∩ A 6 = ∅} for a subset A ⊂ C . 6 T ARAS BANAK H AND ROBER T CAUTY Using the paraco m pactness of the sp ace C , for ev ery n ∈ ω by indu ction find an op en co v er W n of C wh ose star S t ( W n +1 ) = {S t ( W , W n +1 ) : W ∈ W n +1 } is inscrib ed in to the co v er W n . Then the op en sets U k = S t ( ¯ V k − 1 ∪ U k − 1 , W k +1 ) , k ∈ ω , ha ve the requir ed prop erty: U k ⊂ ¯ U k ⊂ C k − 1 ∩ U k +1 for all k ∈ ω .  By Theorem 5.1.9 of [6] th er e us a partition of unit y { λ k : C → [0 , 1]) k ∈ ω , sub ord inated to the co v er { U k +1 \ ¯ U k − 1 } k ∈ ω of C in the sense that λ − 1 k (0 , 1] ⊂ U k +1 \ ¯ U k − 1 for all k ∈ ω (here w e assume that U k = ∅ for k < 0). No w, for ev ery k ∈ ω d efine a map f k : C → C by the form u la f k ( x ) = f ( x ) + X i ∈ ω λ i ( x ) z i,k = f ( x ) + λ i ( x ) z i,k + (1 − λ i ( x )) z i +1 ,k , where i is the uniqu e num b er such that x ∈ U i +1 \ U i . Since f k ( x ) − f ( x ) ∈ B C ⊂ B d ( ε/ 2), w e conclude that d ( f ( x ) , f k ( x )) < ε/ 2 and hence d ( x, f k ( x )) ≤ d ( x, f ( x )) + d ( f ( x ) , f k ( x )) < ε 2 + ε 2 = ε for all x ∈ C . S o, eac h function f k : C → C is ε -near and ε -homotopic to the identit y id C : C → C . Claim 8. The fa mily ( f k ( C )) k ∈ ω is sep ar ate d. Pr o of. By the con tinuit y of the op erator T R : L → c 0 , there is η > 0 su c h that T R ( B L ( η )) ⊂ B c 0 ( δ 3 / 20). W e claim that inf n 6 = k d ( f n ( C ) , f k ( C )) ≥ η . Fix any d istinct num b ers n , k ∈ ω and p oin ts x , y ∈ C . By the c hoice of η , the in equalit y d ( f k ( x ) , f n ( y )) ≥ η will follo w as so on as w e c heck that k T R ( f k ( x ) − f n ( y )) k > δ 3 / 20. The latte r inequalit y w ill follo w as so on as w e fi nd m ∈ ω suc h that | e ∗ m T R ( f k ( x ) − f n ( y )) | > δ 3 / 20. Since e ∗ m T R ( z ) = z ∗ m ( z ) for all z ∈ L , it suffices to show that | z ∗ m ( f k ( x ) − f n ( y )) | > δ 3 / 20 for some m ∈ ω . Since C = S i ∈ ω U i +1 \ U i , there are unique n u m b ers i, j ∈ ω such that x ∈ U i +1 \ U i and y ∈ U j +1 \ U j . Then f k ( x ) = f ( x ) + λ i ( x ) z i,k + λ i +1 ( x ) z i +1 ,k , f n ( y ) = f ( y ) + λ j ( y ) z j,n + λ j +1 ( y ) z j +1 ,n . Without loss of generalit y , ξ ( i + 1 , k ) < ξ ( j + 1 , n ). Since x, y ∈ U max { i,j } +1 ⊂ C max { i,j } , w e conclude that (2) max {| z ∗ m ( f ( x )) | , | z ∗ m ( f ( y )) |} < δ 3 100 for all m ≥ max { i, j } according to the definition of the set C max { i,j } . W e s h all consider fi v e cases. 1) λ j +1 ( y ) > δ 2 / 10. In this case, put m = m ξ ( j +1 ,n ) and observe that | z ∗ m ( z j +1 ,n ) | = | z ∗ j +1 ,n ( z j +1 ,n ) | ≥ δ . Since max { ξ ( j, n ) , ξ ( i + 1 , k ) , ξ ( i, k ) } < ξ ( j + 1 , k ), we conclude that max {| z ∗ m ( z j,n ) | , | z ∗ m ( z i +1 ,k ) | , | z ∗ m ( z i,k ) |} < δ 3 / 100 b y Claim 5. I t follo ws from (2) and max { i, j } ≤ max { ξ ( i, n ) , ξ ( j, k ) } t h at max {| z ∗ m ( f ( x )) | , | z ∗ m ( f ( y )) |} < δ 3 100 . No w w e see that | z ∗ m ( f n ( y ) − f k ( x )) | = | z ∗ m ( λ j +1 ( y ) z j +1 ,n + λ j ( y ) z j,n + f ( y ) − f ( x ) − λ i +1 ( x ) z i +1 ,k − λ i ( x ) z i,k ) | ≥ λ j +1 ( y ) | z ∗ m ( z j +1 ,n ) | − | z ∗ m ( λ j ( y ) z j,n + f ( y ) − f ( x ) − λ i ( x ) z i,k + λ i +1 ( x )) z i +1 ,k ) | > δ 2 10 δ − 5 δ 3 100 ≥ δ 3 20 . TOPOLOGICAL CLASSIFICA TION OF CLOSED CONVEX SETS IN FR ´ ECHET SP ACES 7 2) λ j +1 ( y ) ≤ δ 2 / 10 and ξ ( j, n ) > ξ ( i + 1 , k ). In this case put m = ξ ( j, n ). Arguing as in the preced- ing case, w e c an sho w that max {| z ∗ m ( f ( x )) | , | z ∗ m ( f ( y )) |} < δ 3 / 100 and max { z ∗ m ( z i +1 ,k ) | , | z ∗ m ( z i,k |} < δ 3 / 100. T hen | e ∗ m ( f n ( y ) − f k ( x )) | = | e ∗ m ( λ j ( y ) z j,n + λ j +1 ( y ) z j +1 ,n + f ( y ) − f ( x ) − λ i +1 ( x ) z i +1 ,k − λ i ( x ) z i,k ) | ≥ λ j ( y ) | z ∗ m ( z j,n ) | − λ j +1 ( y ) | z ∗ m ( z j +1 ,n ) | − | z ∗ m ( f ( y ) − f ( x ) − λ i ( x ) z i,k + λ i +1 ( x ) z i +1 ,k ) | ≥ (1 − λ j +1 ( y )) δ − δ 2 10 − 4 δ 3 100 ≥ (1 − δ 2 10 ) δ − δ 2 10 − δ 3 25 > δ 3 20 . 3) λ j +1 ( y ) ≤ δ 2 / 10, ξ ( j, n ) < ξ ( i + 1 , k ), and λ i +1 ( x ) > δ / 4. In this case put m = ξ ( i + 1 , n ) and observ e that | z ∗ m ( f k ( x ) − f n ( y )) | ≥ λ i +1 ( x ) | z ∗ m ( z i +1 ,k ) | − λ j +1 ( y ) | z ∗ m ( z j +1 ,n ) | − | z ∗ m ( f ( x ) + λ i ( x ) z i,k − f ( y ) − λ j ( y ) z j,n ) | > δ 4 δ − δ 2 10 − 4 δ 3 100 > δ 3 20 . 4) λ j +1 ( y ) ≤ δ 2 / 10, ξ ( j, n ) < ξ ( i + 1 , k ), λ i +1 ( x ) ≤ δ / 4, and ξ ( i, k ) < ξ ( j, n ). In this case put m = m ξ ( j,n ) and observ e that λ j ( y ) = (1 − λ j +1 ( y )) > 1 − δ 2 / 10 ≥ 9 / 10 and th u s | z ∗ m ( f k ( x ) − f n ( y )) | ≥ λ j ( y ) | z ∗ m ( z j,n ) | − λ j +1 ( y ) | z ∗ m ( z j +1 ,n ) | − λ i +1 ( x ) | z ∗ m ( z i +1 ,k ) | − | z ∗ m ( f ( x ) − f ( y ) − λ i ( x ) z i,k ) | ≥ 9 10 δ − δ 2 10 − δ 4 − 3 δ 3 100 > δ 3 20 . 5) λ j +1 ( y ) ≤ δ 2 / 10, ξ ( j, n ) < ξ ( i + 1 , k ), λ i +1 ( x ) ≤ δ / 4, and ξ ( i, k ) > ξ ( j, n ). In this case put m = m ξ ( i,k ) and observe that λ i ( x ) = 1 − λ i +1 ( x ) ≥ 1 − δ / 4 ≥ 3 / 4. Then | z ∗ m ( f k ( x ) − f n ( y )) | ≥ λ i ( x ) | z ∗ m ( z i,n ) | − λ i +1 ( x ) | z ∗ m ( z i +1 ,k ) | − λ j +1 ( y ) | z ∗ m ( z j +1 ,n ) | − | z ∗ m ( f ( x ) − f ( y ) − λ j ( y ) z j,k ) | ≥ 3 4 δ − δ 4 − δ 2 10 − 3 δ 3 100 > δ 3 20 .  3. Proof of Theorem 5 Giv en a non-separable conv ex set X is a F r ´ ec het sp ace L , consider the con vex cone C = { ( tx, t ) : x ∈ X , t ∈ [0 , + ∞ ) } ⊂ L × R in L × R with base X × { 1 } whic h w ill b e iden tified w ith X . By pr : C → R + , pr : ( x, t ) 7→ t , we denote the p ro jection on to the second c o ordinate. Observe that the map r : C \ { 0 } → X , r : ( x, t ) 7→ x/t , determines a retraction of C \ { 0 } on to X . Th is retraction r estricted to the set C [ 1 3 , 3] = p r − 1 ([ 1 3 , 3] ) is a p erf ect map. T o pr o ve that X h as LF AP , fix an op en co ver U of X . F or eac h op en set U ∈ U consider the set ˜ U = { ( tx, t ) : x ∈ U, 1 3 < t < 3 } . Then ˜ U = { pr − 1 ( R \ [ 1 2 , 2] ) , ˜ U : U ∈ U } is an op en co ve r of C . By Lemma 2, the con vex cone C h as SAP and by Lemma 1, C has LF AP . Consequently , there is a map f : C × ω → C suc h th at f is ˜ U -n ear to the pro jection f 0 : C × ω → C , f 0 : ( x, n ) 7→ x , and the famil y  f ( C × { n } )  n ∈ ω is lo cally finite in C . Let ˜ f = f | X × ω and ˜ f 0 = f 0 | X × ω . It foll ows from the c hoice of the co v er ˜ U that ˜ f ( X × ω ) ⊂ C [ 1 3 , 3] and the map g = r ◦ ˜ f : X × ω → X is U -near to the pro jection ˜ f 0 : X × ω → X . Since the family ( ˜ f ( X × { n } ) n ∈ ω is locally finite in C 1 3 , 3] and th e m ap r : C [ 1 3 , 3] → X is p erfect, the family ( r ◦ ˜ f ( X × { n } ) n ∈ ω is lo cally fi nite in X , witnessing that X has LF AP . 8 T ARAS BANAK H AND ROBER T CAUTY 4. Ope n Problems The p ro of of Theorem 5 h eavily exploits the mac hin ery of Banac h s p ace theory and does not w ork in the non-locally con v ex case. This lea ves the follo w ing problem op en: Problem 1. Is e ach non-sep ar able c ompletely metriza ble c onvex AR-subset of a line ar metric sp ac e home omorphic to a H i lb ert sp ac e? Ev en a w eake r p roblem seems to b e op en: Problem 2. Is e ach c omplete line ar metr ic AR-sp ac e home omorph ic to a Hilb ert sp ac e? This is tr ue in the s ep arable case, see [4], [5]. Referen ces [1] I. Banakh, T. Banakh, Constructing non-c omp act op er ators into c 0 , preprint. [2] T. Banakh, I. Zaric hnyy , T op olo gic al gr oups and c onvex sets home om orphic to non-sep ar able Hil b ert sp ac es , Cent. Eur. J. Math. 6 :1 (2008), 77–8 6. [3] C. Bessaga, A . Pelczynski, S elected topics in infin ite-dimensional top ology , PWN, W arsaw, 1975. [4] T. D obro wolski, H. T orun czyk, O n metric line ar sp ac es home omorphic to l 2 and c om pa ct c onvex sets home omorphic to Q , Bull. Acad. Po lon. Sci. Ser. Sci. Math. 27 :11-12 (1979), 883–887. [5] T. D obro wolski, H. T oru ´ nczyk, Sep ar able c om plete ANR’s admitting a gr oup st ructur e ar e Hi lb ert m ani folds , T op ol- ogy Appl. 12 (1981), 229–235. [6] R.Engelking, General top ology , Heldermann V erlag, Berlin, 1989. [7] R.Geoghegan, Op en pr oblems in infini te-dimensional top olo gy , T opology Proc. 4 :1 (1979), 287–338 . [8] V. Klee, Some top olo gic al pr op erties of c onvex sets , T rans. Amer. Math. So c. 78 (1955), 30–45. [9] H. T orun czy k, Char acterizing Hilb ert sp ac e top olo gy , F und . Math. 111 :3 (1981), 247–262. [10] H . T orunczyk , A c orr e ction of two p ap ers c onc erning Hilb ert manifolds , F und. Math. 125 :1 (1985), 89–93. [11] J.W est, Op en pr oblems in infini te-dimensional top olo gy , in: Op en problems in top ology , North- Holland, Amsterdam, 1990, P . 523–597. Instytut Ma tema tyki, Uniwersytet Human istyczno-Przyrodniczy Jana Kochano wskiego, Kielce, Poland, and Dep ar tment of Ma thema tics, Iv an Franko Na tional University of L viv, Universyte tska 1, 79000, L viv, Ukraine E-mail addr ess : t.o.banakh@ gmail.com Universit ´ e P aris VI ( France) E-mail addr ess : cauty@math. jussieu.f r