Viemam J Madi (2014) 42:191-203 DOI 10.1007/s 10013-014-0059-1
The Admissibility and the AR-Property of Some Unbounded Convex Sets in a Class of Non-locally Convex Spaces Containing lp {0 < p <1)
Nguyen Hoang Thanh
Received- 14 May 2012 /Accepted: 23 September 2013 / Published online 28 January 2014
® Viemam Academy of Science and Technology (VAST) and Spnnger Science+Business Media Smgapore 2014
Abstract The aim of this paper is to show the admissibility and the AR-property of some unbounded convex sets in a class of non-locally convex linear metiic spaces.
Keywords Non-locally convex space • Convex set • AR-property • Admissibility Mathematics Subject Classification (2010) Primary 54C55 • Secondary 57N17
1 Introduction
Throughout this paper, a linear metric space X means a metnzable topological linear space.
By Kakutani's theorem (see [2]), there is a monotonic invariant meCric p on X. We denote
\ix-y\\ = p(x.y).
Recall that for p € (0, I) the linear metric space lp is defined by
/p ^ L - (.v„) I ^ \x„I" < CO,.v„ € E, n e N i
wiUi metric j o ( j : . y ) ^ ^ ^ i \x„ - y„\'' for each A-= (-v„).y ^ (v„) e/;,. The spaces/,, are non-locally convex linear metnc spaces.
A topological space X is said lo have the fixed point property' if for even,- conUnuous map f:X-^X, there exists a point xoSX such that /(.vo) = xo (see [4))
Lel X, Y be topological spaces, A connnuous map f • X ^ Y is compact if f(X) is contained in a compacC subset of Y (see [4]).
N.H, Thanh (Ei)
Faculty of Mathemadcs, College of Education. Unnersity of Danjiig, 459 Ton Due -Thang, Da Nang, Vietnam
e-mail, nhthanh@ud,edu vn N H Thanh
e-mail, nguyenhoangthanhS'gmdil ci
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192 N,H. Thanh A topological space X is said to have the fixed point property for compact maps if for every compact map f :X -^ X, there exists a point xo^ X such that /(JTO) — ^o (see [4]).
Let Z be a topological space and A be a subset of X. A continuous map r • X —* A is called a retraction if r(x) — :>: for all J: E A, A is then called a retract of X (see [4]),
A metnzable topological space Y is called an absolute retract if for any medizable space X and any closed set A(zX, every continuous map / : A ^^ T is extendable over X.
The class of absolute retracts is denoted by AR (see [2,4]) Observe that the AR-property is invanant under homeomorphJsms.
In 1935, Schauder proved that every compact convex set in a locally convex linear metric space has the fixed point property (see [4]) Schauder conjectured that his result holds trae for non-locally convex metnc spaces as well. About some partial answers of Schauder's conjecture we can see for instance [7-9], In 1937, Borsuk proved the following theorera (see [4]).
Theorem 1 (Borsuk) Every absolute retract has the fixed point property for compact maps.
By Borsuk's theorem, we can state the Schauder's conjecture in the following problem.
Problem 1 ([1, 11]) Is every compact convex set in a linear metric space an AR?
In 1951, Dugundji proved the following theorem (see [2]).
Theorem 2 Every convex subset of a locally convex linear metric space is an AR.
However, Problem 1 remains open for non-Iocally convex linear metnc space and is one of the most resistant open problems in infinite-dimensional topology.
Problem 2 ([11]) Is every convex set in a non-locally convex linear metric space an AR?
Following Klee [5, 6], a convex subset A of a linear metric space X is said to be admis- sible if and only if for every compact subset K of A, id^ is the uniform hmit of a sequence of continuous maps f„:K^A such that each span f„(K) is finite-dimensional (where span f„(K) denotes the linear subspace generated by fn(K)). We know that an admissible CT-compact convex set is an AR (see [3]). Moreover, we have the following lemma.
Lemma 1 ([10]) Every convex AR of a linear metnc space is admissible.
Proof Let A be a convex AR m an arbitirary linear metric space (X, d) By Arens-Eells theorem ([3]), there exists an isometiic embedding cp: A^ A', where A' is a closed set of a nonned linear space E.
Let /<: be a compact set in A. We denote K' ^ cp(K), then K' is a compact set m A' Let r £ -^ A' be a retract.
Let s, 5 > 0 Because E is anormed linear space, E is admissible (using Schauder projec- tion [4]). Therefore there exists a continuous map f : K' ^ E such that f(K') is contained in a finite-dimensional linear subspace of E and || f(x) - A: || < 3 for each x e A".
Consider the map
<fi~'°i-\nK'):f{K')^A.
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The Admissibility and the AR-Property of Some Unbounded Conven Sets 193 Since fiK') is a finite-dimensional compact set and A is convex, there exists a continuous
map g : f(K') - * A such that \\g(x) - ip-^ o r(x)\i < 5 for each JC e f{K') and g o f(K') IS contained in a finite-dimensional linear subspace of X.
Note that go f o ^ j ^ : A' —* A is a map with image contained in a finite-dunensional linear subspace of X and for every x€K,
\8°fo<P\K(x)-x\ < \\gofo(p\K(x)-(p-^orofo(p(x)\\
-\-\cp-'orofoip{x)-x\\
<S + \\<p-'orofo,p(x)-,p~'ocp(x)\\
= S-\-\\rofc,^(x)-,p(x)\\
= S + \\rofo<p(x)-roip(x)\\.
We will show that there exists 5o>0 such that for each i5 e (0, So),
\\rofocp{x)-rocp(x)\\<^-. (1) Suppose on the contrary that Chis claim is false. Then there is a continuous map f„ : K' ->• E
and a sequence (y„| e K' such that
Since K' is compact, without loss of generality, assume that there exists yo £ K' such thai yn - * yo- Thus, /„(>'„) -^ yo^ K'. Since r is continuous, r • /„(>-„) - • r(yo) and r(y„) -*
r(yo)- This contradiction proves claim (1),
Choose 5 sufficiently small and smaller than | . Then the proof of Lemma I is com-
plete, n From the relationship of the absolute retrace property and the admissibihcy, we have the
following problem.
Problem 3 Is every convex subset of a non-locally convex linear metric space admissible?
In this paper we provide partial answers for the above problems.
Tet { p „ ) c (0, I) be a decreasing sequence such that or = inf„E[i/>„ > 0. Denote X = (x„)\'f2,\x,i\'"' < o o , A : „ e R , n e N !
and
p{x. y) = ^ )-v„ - yA"" foreach.v - (x„). v = (y„) e T,
\t is easy lo verify that (T. p) is a linear metnc space. Denote
||;r||=/>(jr.O) = X^|-r„|''"
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for each AT — (j:„) € r , soforeach^: = (ji:„). y = (y„) e T we have p(x,y) = \\x-y\\.
By i2 we denote the class of all linear metric spaces T. It can be shown that i2 contains some non-locally linear metric spaces. In fact, let pi = p2 = • • • = pa = - - - = P ^ (0,1), then
r - I ; C - ( A : „ ) | ^ | A : „ | ' ' " <CO, A : „ e l , « e N i = / p (0 < p < 1) is a non-locally convex Imear metric space.
Moreover, we have the foUowing claim.
Claim Suppose that p„ — 5 -I- -^for every n e N . Then
T=\x = (x„)\'Y^\x„Y'' < o o , : t „ e M , « e N
is a non-locally convex linear metric space
Proof We assume, on the contrary, that T is a locally convex linear metric space. Then there exists a convex neighborhood U of zero such that U C S(0, 1) = (A: 6 7" j ||A:|| < 1], Since [/ IS a convex neighborhood of zero, there exists r >0 such that B(0, r) = {x eT \
\\x\\ <r]c [/. For every « e N , assume ^„ is of the form j„ = ( 0 , . , . , 0 , ( 5 ) ' / ' ' " , 0 , 0 , . . , ) € r , that is, 2„ is the vector with ('jY'''" in the «th slot and zeros elsewhere. Then
We have
Since \\m„^^ ^ = -j-co, there exists n e N such that ^ > 1. Hence ||A: jj > 1.
On che other hand, since U is convex, Z|,Z2 ;„ e fi(0, r) c U and 1
x=-{zi+Z2+ • • - l - z „ ) e f / c B ( 0 . l ) , I U ||< 1.
This contiradiction proves the assertion. D From now on we assume that 7 is a non-locally convex linear metric space in i2. In diis
paper we will consider die admissibility and the AR-property of some unbounded convex sets in Ihe space T, Section 2 in this paper investigates the AR-property of some subspaces of T, and die admissibility of the standard infinite-dimensional simplex of T will be shown in Sect. 3.
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-Ibe Adraissibiliiy and the AR-Properly of Some Unbounded Convex Sets 195 2 The AR-Property of Some Subspaces of T
We begin this section by proving Ihe following lemma.
Lemma 2 There exists a homeomorphism between T and /i.
Proof het <i):T —>• l\ be a map such Chat
t^(A:) = ((sgnx,)|A:,|P',(sgnA:2)|A:2|'^ (sgnx„)|;c„|P",...) e / , for each x — (x^ € T. It is easy to show that 0 is a bijection and the inverse map of (f is p : /i -> 7" satisfying the condition
<p(x) = {{s%nx{)\x^\^\,{s%nx-^)\x2\n ( s g n x J J A r J ^ . , , , ) e T for each J: = {.x„) e / | .
Now we prove that 0 is continuous Suppose that [J:'*'} C T and J:'*' —>- jr"" € T Let A'*' = ( x ! ' ' , ; c f ' , , , , ) and ; c ™ - ( ; r ; ' " , A f ' , . . . ) . Then we have
Y}x^^-xf\'''^^ asq^cc.
Given an arbitrary £ > 0, because Y,™^^ \x'^' j ' ' " is convergent, there exists no € N such thai
EKT < J.
Since E ^ , txi" -1™!'- -> 0. choose 90 £ f* such that
for every q > q^.
Since i : , ^ i | A : ; ' " - . r i ' > V " ^ 0 ,
and
( s g n x l ' O k r r ' - f s S n ' D I ^ ' n " ' ^ - ^ - ^ for every yt e {1,2 no}, it follow.s that there is t/i e N such that for every q>q\.
i;i(sgn4")i»-;'T--(si"-'"')K°T-i<i-
Denote 92 = max(?o. 9i T To"" every (/ > <?2, we have
E K'T"s E (!'ri + K"-'r!)'"
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n=no+l n=no+l
s E^w°r-+Ekr-^»'r<=|+|=j.
Hence
||*(x>«) - ^ ( , m ) | | = f ; | ( s g n . , ; « ) | ; r » l | ' - - (sEn.t;»')l;r™|''-[
= EINU'"')!'?'!"' - (sgn'DKT'l
+ Y. \{^st'''^')\4'T-l^s''4°')K'\"'\
n=no+l
< | + E i(sEnx;")W'T"i+ E i-(sg"^r)k;°r"i
Therefore 0 is contmuous.
We proceed to show that ip is continuous. Suppose that [A:'*'} C l\ and A:'^' - * A:"" e i.
Let A<^' = (;r;'', x f ' , . . . ) and .v'"' ^ (jr,"", jc^"",...); we then have
Given an arbitrary ^ > 0, because A;"" = (.^t*", A:^"', ) e ;, and ^,^^1 |A:f'| is convergent, there exists «i e N such that
Since YT., I*!" - *S"\ -* 0, choose 93 s N such that for every q>q,,
Ek"-'ri<|-
We thus get
^_Ei'ri+EK"-.ri< 1+1=1
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The Admissibility and the AR-Proper^ of Some Unbounded Convex Sets Since i : r = , U i " - - t r i ^ O ,
and
( s 8 n * « ) W " | » - ^ ( s g n ; r f ' ) | ; t ™ | i a s , ^ o o t ( i r e v e r y * 6 | l , 2 m ) , and it follows that there is 94 E N such that for every 9 > „ .
T,H<'4")W,'"\^ - (sgn:r;»')|:r™|i|'- < £
n=l 8 Let 95 - max(93, q^]. For every q > q^, we have
s E(W"l + W»l)<f.
So
\Mx'")-<p{x'«)\\ = f;i(sgn.,,;")|..i"|" - (sgn.r,™)|.,;»|ii|'-
= El(sgnx,',")|.r;«|ii - (sgn,,™)|,,™jii|'-
+ E l ( s s n x « ' ) K " l " - ( s g n . . » ' ) k r i " r -
3E e 8
" ¥ + 8 = 5"'-
Therefore ^ Is continuous. G Denote
7 " - { ( . r „ ) e 7 - | A : 2 „ + i - 0 , / t e N ) ,
Observe that T' is a subspace of T. Consider the map g . T' -* T defined by g{x) = {x2n) foreachji: = (0, A:2,0,A-4,0 ) e 7", In order lo show lhal T' is an AR, it suffices to prove that g is a homeomorphism. Indeed, we have that g is a bijection and the inverse map of g\sh:T -y 7" defined by/!(A-)^(0,A-|.0.A-2.. ) for each A: ^ (j:„) e T. Suppose that Ix'-"] c r and A:"" -»- x"^' e T'. Lel v^' = (O.-vl"'. 0 , , . . ) and x"^' = (0. x'°\ 0 , . , . ) . we then have
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Hence
\g{x'<') - s{x«") II = f ; | : t « - :t™|'''- ^ 0 as 5 ^ 00.
So g IS continuous. Let [x<"] = {{xf\xf,. .)] c T . i ™ = {xf.xf,...) e T 1
; r ' « ' ^ i " " . T l l u s .
E W " - ' " T " ^ 0 a s « ^ o o .
||ft(:t'«>) - *(i«») II = f ; | x » ' - xf>\'- ^ 0 as <; ^ 00.
Therefore ft is continuous, hence jj is a homeomorphism. By Leirrma 2, T' is homeomorphic with l\ Since ^i is a locally convex space and it is an AR, T' is an AR.
Let {i„} be a strictly increasing sequence of PJ. and denote T\ — {(x„) E T \ Xi... =0.
« e N | By a sinrilar argument as above, we have;
Theorems The subspace r, ofT isanAR.
Denote
r " = { ( i . ) e r | j : , + i , = 0).
Consider the map g : T" ^ T defined by g{x) = (X2,xi,...) for each x = {A:I,;C2, ...) e T" (satisfymg the condition x, ^ -X2). Let {.c'«'j = {ix[^\ x'^\ ...)} c T'\ :r"" = (x™, x f , x f , ) e T" and x<« ^ x»>. We then have
E K " - < f ^ O a s , ^ 0 0 .
| | s ( x ' " ) - S ( ' ™ ) l l = 'E\4"-'''°T ^ 0 as , -» 00.
Hence g is contmuous.
Observe that themverse map of g is ft : r ^ T" defined by A(A:) = (-A:],A:I,A:2, . .)for each X = (x„) e T. Suppose tiiat {x'?') ^ {ix\''\x^^\ ...)]CT, ;c"" - (A:'"' A:™, , . , ) € T and ; r ' ' " ^ A:™*. So
> 0 as ^ ^ 00,
The Admissibility and the AR-Property of Some Unbounded Convex Sets 199 Hence
\\h{x"'>)-h{x^°^)\\=I-A:{''+^rr'+EI^«" -^n"" ^ 0 as^^ =c.
n=l
Therefore h is continuous, hence g is a homeomorphism. So T" is an AR.
Generally, let kf,k2,... ,k„ eN, and denote Tj ^ {{x„) e T j 527=1 ^i, ^ 01- '^^ ^hen have
Theorem 4 The subspace T2 of T is an AR.
3 The Admissibility of the Standard Infinite-Dimensional Simplex of the Space T Assume that e„ is of the form
e„=(0 0 . 1 , 0 . . , 0, . . ) Vn e N, that is, e„ is the vector of the space
A: = (-t„)|E|.v„|' c cc. x„€R. n € 1
with 1 in the fith slot and zeros elsewhere.
Let A = cl convl^i, ^2 e„, } and A call it the .standard infimle-dimensioncil sim plex in the space T. We see that A is an unbounded convex set. In fact, consider u =
;(e, -{-e2-i-• • •-\-e„) s A. Then
I 11''' 111"' lll"^
Because lim„^oo(«'"''= ' i?^'*^ + ° ^ - ^ '^ ^" unbounded set.
To prove the admissibility of A we need the following lemma.
A ^ .v = ( . v , ) e 7 - | X ; v „ - T v „ > 0 , «
Pmof For every .v ^ (.v,. A->, ., A„, ,, ) € A. lei [A-'"') be a sequence in conv(e,, .2. • ^^
e„,, 1 such that {.^'"M converges to the elemeni .v. Now, for every ,/ e N, we put A - (A]"', x i " , , , , ) . It IS easily seen that 0 < x'n" < 1 for every n e N.
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Since hm.j^atx'*'' =x.Um,.^a=xi^^ =x„. SoO < x „ ;£ 1 forall n e N . Since Um,^„„ p ( x « l . x ) = 0. Bm,^,,, J ^ J . ! I^i" - «l™ = "• T'len
ImEk?'-'*!""" ^"«5''-
From xf''. x; € [0.1] we have | x f - x^l < 1 for every it e N. Hence X ] | x ; " - X 4 | < X ^ | x i " - x , | » V „ e N .
1L-=1 li=]
Therefore,
E ' ' = Ei«i=sEW"-"l+EW"l
< ^ I x f - X i | + I < ^ | x < " - x , | » + 1 VneN.
Since hm^^co Y.1=] l^t*' - Xk\^^ — 0, X^^^, Xi < 1 for every K e N. Hence
Since (p,) c (0. 1) and | x * ' - X, I < 1 for every neN.
| |E ' .| - |E'?'|| ^ |E('. -?')| i Ek. --?'l <Ek -'i'T-
Thus
It follows that
i-Eki"-^.r-<|Ex.|=f;x..
Since h m ^ ^ ^ ^ ^ , j x f - x* I'''-- ^ 0, we obtain
From (2) and (3). YlT.i x. = 1. Hence
A c x = ( x , ) 6 r l ^ x . = 1. X. > 0 . n e N l .
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•RK Admissibihty and the AR-Property of Some Unbounded Convex Sets 201 Now let X = (x„) be an arbiti-ary element of T such Uiat 0 < x„ < 1 and YlT=] ^n — 1
(Vn e N). For every « e N , weput M„ = (XI,X2, ...,Xn_|, 1 - x j - x i x „ _ ] , 0 , . . . ) . It is clear that
«„ econv(e],e2, .••,£„} Cconv(ei,e2, ---.en. •• }
I I " ; , - ^ II - (1 - -tl - A:2 A T j " " -1- | x „ + , jP"^' - I - - - -
Since ^ ^ | X n — 1, timn^(i;(l — X| — X2 x„) = 0 . Because [p„] c (0.1) is a decreas- ing sequence and infn^N pn =ex >0,we have
0 < ( 1 - X i - X 2 Xn)''"<{{-Xt^X2 x„)°.
Thenlmin^ooCl - xi - X2 x„)''" = 0 .
Since x G T, Iim„_,cc(k„+il''''-^' -F -••) = 0. Thus, lim„^3;llM„ - xjj = 0, i e..
lira„_oo «n — X. This implies that x e A . Therefore
x = ( x „ ) e 7 ' I ^ x „ - l , X, > 0 , j i e N J c A. (5)
From (4) and (5), the assertion is proved D The mam result of this section is:
Theorem 5 The standard infinite-dimensional simplex in the space T is admissible.
Proof Let K be an arbiti-ary compacc set in A, For every neN and for every x = (xi,X2 x „ , . . , ) e / i : , let P „ ( x ) ^ ( x i , X 2 x . , 0 , , , , ) a n d / „ ( x ) = (x|,X2 .v„_,, 1 - J ; 1 _ J : 2 ;c„„,,0, . ) .
Wehave/„(x) econv{g|,e2 e„\ C span(<?i,e2.. ,e„l.Then f„(K) is contained in a finite-dimensional Hnear subspace of the space T and /^(A') cconv|ei.e2 . . . | C A.
For every x = (X|,X2,...,x )€ K.we have
| | / „ ( x ) - x | | ^ ( l - x , - X 2 A:„)''"-H.W,|'"-'-F--- By Lemma 3,
l/.w-I = (£-•-')"+_£;."•
Since {/j„)c(0, 1) is a decreasing sequence and 0 < .v„ < 1 for every AI G N , we obtain
Thus
!/.(')-'II = (E^->-') + _ E - ' ' "
c f ] X," + J2 -<<" =2|l''.<-')--'ll-
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Therefore the admissibihty of A is reduced to the following claim:
For a given s > 0. there exists nosf^ such that || P„^ (x) — x | < e for each x ^ K. (6) Indeed, assume, on the contrary, that SQ IS a positive real number such that for every n sN wecanfindx — ( X I , X 2 , . . . , X „ , A ; „ + | , . . . ) € K satisfying
Yl M'">£o. (7)
It follows from (7) that there is an element x ' " ^ (xl^',x^'\ ...,xl^\x"_li,...) e K such that
Ek"'l»>%-
Because ^ ^ , jx^'V' is convergent, choose «, G N such that
L
^ I 0)\Pk ^ 0 \xl \ < —. 1 * 1 4From (7), we see that there exists x'^' = (A:p',xf' xj,^^,x^^\^,...) e K and «2 e N,
«2 > All, such that E ^ „ i + i Ixj^'l"' > fio. T.Zn-,+1 l ^ f I"* < T- We have
ijx'^'-x<>'ii=f;ixf'-x<"r> E k f - ^ n "
*="l + ! *=«, + ! So 3so 4 4
We now proceed by choosing x"< = ( x P ' . x f xOl.x™.^,....) e K such that
E E . , + 1 \x't' !"• i Eo and 113 E N. 03 > n,. such that E £ . j + i |Jtf' I " < J We thus get
ik'"-''"ii=EK"-';»r> Ekf-rr*
i^E_kri"- Eki'T
^__Ei'°'r-^Ei';'T*:=o-j=^.
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The Adnussibility and the AR-Property of Some Unbounded Convex Sets 203
l|x"'-x«|| = f;|xf'-xf|»> Ekf-xfi"
t=l i=n2+l
> Ekf'r- Ek?r>%-?=^.
*=n2+l *=n2+) '^
Continuing this process, we obtain a sequence (A:'*'} C K such that for / > i then jjx'*' — j.li)|| > .ifl_ Thus, K is not totally bounded, hence K is not compact. This contradiction
proves the assertion, • Finally, we ask:
Question 1 Is every unbounded convex subset of the space T admissible?
Question 2 Is every unbounded convex subset of Che space T an absolute retract?
References
1. Bessaga, C , Dobrowolski, T,; Some open problems in the border of fiinctional analysis and topology In Proc, Int, Conf Geom Topol,, Warsaw (1978)
2. Bessaga, C , Pelczynski, A.; Selected Topics in Infinite-Dimensional Topology Polish Sci., Warszawa (1975)
J. Dobrowolski, T,; On extending mappings into nonlocally convex linear metric spaces. Proc. Am Ivlath Soc, 93, 555-560 (1985)
4, Granas, A„ Dugundji, J,: Fixed Point Theory, Spnnger, Berlin (2003)
5, Klee, V'Shrinkable neighborhoods in Hausdorff linear spaces Math Ann 14L 281-285 (1960) 6, Klee, V • Leray-Schauder theory without local convexity Math, Ann 141,286-296(1960) 7, Nhu, N.T,, Tri, L,H., No Roberts space is a counter-example to Schauder's conjecture Topology 33,
371-378(1994)
a. Nhu, N,T,. Tri. L,H,, Every needle point space contains a compact convex AR-set with no extreme point.
Proc, Am, Math, Soc, 120. 1261-1265(1994)
9. Thanh, N,H., The fixed point property of the Cartesian product of Roberts spaces. Fixed Point Theory 13.267-272(2012)
10, Tri, L,H,. Thanh, N.H.. Some remarks on the AR-probiem Acta Math, Vietnam 3 4 , 3 8 9 ^ 0 0 ( 2 0 0 9 ) 11 West, J,E,. Problems in mfinite-dimensional topology In' van Mill. J . Reed, G.M (eds ) Open Problems
mTopology. North-Holland, Amsterdam (1990)
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