A COUNTEREXAMPLE TO A QUESTION OF R. HAYDON, E. ODELL AND H. ROSENTHAL G. ANDROULAKIS Abstract: We give an example of a compact metric space K, anopen dense subset U of K, and a sequence (f n )inc(k) which is pointwise convergent to a non-continuous function on K, such that for every u U there exists n N with f n (u) =f m (u) for all m n, yet (f n ) is equivalent to the unit vector basis of the James quasi-reflexive space of order. Thus c 0 does not embed isomorphically in the closed linear span [f n ]of(f n ). This answers in negative a question asked by H. Haydon, E. Odell and H. Rosenthal.. Introduction A result of J. Elton [E], which was also proved later by R. Haydon, E. Odell and H. Rosenthal [HOR], states that if K is a compact metric space, and (f n ) is a uniformly bounded sequence in C(K) such that f n+ (k) f n (k) <, k K n= and the pointwise limit of (f n )onkis a non-continuous function, then c 0 embeds isomorphically in the closed linear span [f n ]of(f n ). Thus the following question was naturally raised by R. Haydon, E. Odell and H. Rosenthal: Question 4.7 in [HOR]: Let K be a compact metric space, R be a residual subset of K (i.e. K\R is a first category set), and (f n )bea sequence in C(K) which converges pointwise on K to a non-continuous function, and f n+ (r) f n (r) <, for all r R. n= Does c 0 embed in the closed linear span [f n ]of(f n )? Date: October 3, 996. 99 Mathematics Subject Classification. 46B25. This work is part of the author s Ph.D. thesis which was completed at the University of Texas at Austin in August 996 under the supervision of Professor H. Rosenthal.
2 G. ANDROULAKIS We will construct K a compact metric space, U an open dense subset of K and a sequence (g n ) C(K) such that (a) ( n i= g i) n is a uniformly bounded and pointwise convergent sequence on K to a non-continuous function; (b) For every u U there exists n N such that g m (u) = 0 for every m n; (c) [g n ] is isomorphic to the James quasi-reflexive of order space J. Since, of course, c 0 does not embed isomorphically in J, this answers in the negative Question 4.7 of [HOR]. Our construction is very elementary and explicit even though a shorter proof of the existence of a counterexample to Question 4.7 of [HOR] can be given along similar lines using more advanced machinery. 2. The construction We recall the definition of the James space J and some simple facts. Let c 00 denote the finitely supported sequences of real numbers. For (x n ) c 00 we define (x n ) J =sup{[x 2 p +(x p2 x p ) 2 + +(x pk x pk ) 2 ] /2 : k N, p <p 2 < <p k <p k }. Then the James space J is the completion of (c 00, J ). If (e n )isthe unit vector basis of c 00,then(e n ) becomes the unit vector basis of J, which is monotone and shrinking. Also, ( n i= e n) n is a weak-cauchy sequence which is not weakly convergent in J. If (a n ) c 0 such that (a n ) is a monotone sequence of real numbers (i.e. non-increasing, or non-decreasing) then (a n ) J = a (this is because if a, b R with ab 0, then a 2 + b 2 (a + b) 2 ). Notation: For (a n ), (b n ) c 00, we define (a n ) (b n ) c 00,by (a n ) (b n )=(a n b n ). Lemma 2.. For (a n ), (b n ) c 00 we have (a n ) (b n ) J (a n ) J (b n ) + (a n ) (b n ) J. Proof For some k N and some finite sequence of positive integers p <p 2 < p k we have: (a n ) (b n ) J = [(a p b p ) 2 +(a p2 b p2 a p b p ) 2 + +(a pk b pk a pk b pk ) 2 ] /2 = [(a p b p ) 2 +(a p2 (b p2 b p )+(a p2 a p )b p ) 2 + + (a pk (b pk b pk )+(a pk a pk )b pk ) 2 ] /2.
COUNTEREXAMPLE 3 Therefore by the triangle inequality in l 2 we have that (a n ) (b n ) J [a 2 p b 2 p +(a p2 a p ) 2 b 2 p + +(a pk a pk ) 2 b 2 p k ] /2 + [a 2 p 2 (b p2 b p ) 2 + +a 2 p k (b pk b pk ) 2 ] /2 [a 2 p +(a p2 a p ) 2 + +(a pk a pk ) 2 ] /2 (b n ) + (a n ) [(b p2 b p ) 2 + +(b pk b pk ) 2 ] /2 (a n ) J (b n ) + (a n ) (b n ) J which finishes the proof of the lemma. 2 Now we are ready to see the counterexample. Let K := {(a, b) R 2 : 0 a, 0 b }. Since C[0, ] is universal for the class of separable spaces, there exists a sequence (f n ) C[0, ], and M > 0 such that (f n )ism-equivalent to the unit vector basis of J. Forn N set K n := {(a, b) R 2 :0 a,/2 n b },L n := {(a, b) R 2 : 0 a, b=/2 n }L:= {(a, 0) : 0 a } and U = K\L. Now, for n N define g n : K R by g n K n 0, for every 0 a, g n restricted to the segment connecting the points (a, /2 n )and(a, 0), is linear, g n L f n. g n is continuous on K. We will show that (g n ) is equivalent to the unit vector basis (e i )ofthe James space. This will imply that ( n i= g i) n is a weak Cauchy sequence which is not weakly convergent, which will finish the proof. Let n N and (λ i ) n i= R. We want to estimate λ g + + λ n g n. For (a, b), (c, d) K, let[(a, b), (c, d)] denote the linear segment connecting the points (a, b) and(c, d). For every 0 a wehavethat (λ g + +λ n g n ) [(a, ), (a, /2)] 0, (λ g + +λ n g n ) [(a, /2 i ), (a, /2 i+ )] is linear, for every i =,...,n, (λ g + +λ n g n ) [(a, /2 n ), (a, 0)] is linear, λ g + +λ n g n is continuous on K. Therefore we obtain: λ g + +λ n g n = max 2 k n (λ g + +λ n g n ) L k (λ g + +λ n g n ) L = max 2 k n (λ g + λ k g k ) L k λ f + +λ n f n.
4 G. ANDROULAKIS Therefore we obtain immediately the lower estimate: λ g + +λ n g n λ f + +λ n f n M λ e + +λ n e n J. For the upper estimate we need to estimate (λ g + +λ n g n L k ) for 2 k n. Note that for 0 a and2 k nwe have that (λ g + +λ n g n )(a, /2 k ) = λ 2 2 k f (a)+λ 2 2 2 k =λ 2 2 2 k f 2 (a)+ +λ k 2 2 2 k 2 k f k (a) 2 k 2 k 2 2 f 2 k (a)+λ 2 f 2 k 2 2 (a)+ +λ k 2 f k (a). Thereforewehavethat λ g + +λ k g k L k 2 k 2 k 2 2 = λ f 2 k +λ 2 f 2 k 2 2 + +λ k 2 f k 2 k 2 k 2 2 M λ e 2 k +λ 2 e 2 k 2 2 + +λ k 2 e k J =M (λ,λ 2,,λ k,0,...) ( 2k, 2k 2,..., 2 2 k 2 k 2 2,0,...) J M λ e + +λ k e k J +M (λ i ) k i= ( 2k,..., 2 2 k 2,0,...) J(by Lemma 2.) 2 k M λ e + +λ k e k J +M (λ i ) (since the 2 k sequence ( 2k, 2k 2,..., 2,0,...,) is decreasing) 2 k 2 k 2 2 2M λ e + +λ k e k J (since (λ i ) k i= (λ i ) k i= J). Also, since λ f + +λ n f n J M λ e + +λ n e n J,weobtain that λ g + +λ n g n 2M λ e + +λ n e n J. This finishes the proof. 2 References [E] J. Elton, Extremely weakly unconditionally convergent series, Israel J. Math. 40 (98), 255-258. [HOR] R. Haydon, E. Odell, H. Rosenthal, On certain classes of Baire- functions with applications to Banach space theory, Lecture Notes in Mathematics Vol. 470, Springer-Verlag, Berlin 99.
COUNTEREXAMPLE 5 Math. Sci. Bldg., University of Missouri-Columbia, Columbia MO 652 E-mail address: giorgis@math.missouri.edu