A THEOREM ON SCHAUDER DECOMPOSITIONS IN BANACH SPACES

In this paper we prove that in a Banach space all Schauder decompositions are slirinking iff all Schauder decompositions are boundedly complete . 1 . Definitions and preliminary results A sequence (x )m in a Banach space X is called a Schauder n n=1 basis if for every x E X there exists a unique sequence (an)n=1 in R such that x = a nxn , and this series converges with respéct the n=1 norm of X . A sequence (yn)mn--1 is called a basic sequence if it is a basis of his closed linear span . tions Pn : X -. X defined by Miguel A . Ariño A Schauder decomposition of X is a sequence (X .)m of closed i i=1 subspaces of X such that for every x in X there exists a unique se-. quence (xi)i=1 mth x iE Xi for all i and x = ~ xi . Every Schauder i=1 decomposition of X is related with a sequence of continuous projecPn(x) -_ Pn( ~ xi ) _ i=1 In all this paper, the linear span of an element x E X is denoted m by [x] and the closed linear span of the subspaces (Xi ) i=n (l< n <m< m) is denoted by [X . ] m i i=n' The following theorem characterizes the Schauder decompositions and it can be found in [5] . 1 . Theorem : Let X be a Banach space and (Xn ) n=1 a sequence of closed subspaces of X . The following are equivalent : i) (Xn)n=1 is a Schauder decomposition of X . ii) There exists a sequence (Pn)n=1 of continuous projections Pn : n such that Pn pm = Pmin(m n) and lim P (x)=x for . , X -" [Xi] i=1 every x in X . iii) There exists a sequence (Pn )ñ 1 of continuous projections Pn : n ~ such that Pn Pm = Pmin(m,n) and (Pn ) n=1 is uniformly bounded .


. Definitions and preliminary results
A sequence (x )m in a Banach space X is called a Schauder n n=1 basis if for every x E X there exists a unique sequence (an)n=1 in R such that x = a n xn , and this series converges with respéct the n=1 norm of X .A sequence (y n )m n--1 is called a basic sequence if it is a basis of his closed linear span .
tions P n : X -.X defined by Miguel A .Ariño A Schauder decomposition of X is a sequence (X .)m of closed i i=1 subspaces of X such that for every x in X there exists a unique se-.quence (xi)i=1 m th x i E X i for all i and x = ~ x i .Every Schauder i=1 decomposition of X is related with a sequence of continuous projec-

Pn(x) -_ Pn( ~x i ) _ i=1
In all this paper, the linear span of an element x E X is denoted The following theorem characterizes the Schauder decompositions and it can be found in [5] .
1 .Theorem : Let X be a Banach space and (X n ) n=1 a sequence of closed subspaces of X .The following are equivalent : i) (Xn)n=1 is a Schauder decomposition of X .
ii) There exists a sequence (Pn)n=1 of continuous projections P n : n such that Pn pm = Pmin(m n) and lim P (x)=x for ., X --" [X i] i=1 every x in X .
iii) There exists a sequence (P n )ñ 1 of continuous such that the topology of X is originated by a p-norm .In this case X is called p-Banach space (cf .[1] and [4]) .Let X be a p-Banach space ** such that X separates points of X and let J : X -X be the canonical imbedding of X into its bidual .We define in X the norm The Mackey topology of X is criginated by this norm (cf . [2]) and it is called the Mackey norm of X .The Mackey completion of X is denoted by J(X) .
All the above definitions for Banach spaces can be extended to p-Banach spaces .
2 .Lemma .Let (Xn)n=1 be a Schauder decomposition of a Banach space X and let (Pn)n=l be its sequence of projections .We suppose that each Because of the theorem 1 we only need to prove that (A n ) n=1 is uniformly bounded, and, because of the last considerations, it shall be proved if we prove that sñp II A2n 1 I < .] parallel to the other subspaces, the existente of x* E X* satisfying a) and b) is necessary .We suppose that there-exists a such x* .We define (y*) n =1 as in the preceding lemma, and if we consider the sequence Let (An)n=1 be a sequence of projections as in the preceding lemma .
We must prove that sup II A n 1I < m .Por every m, x*1W = 0, and hence m x*(x) _ y * (x) for every x in X and so (vn)n-1 converges weakly to n=1 n 0 and sup II Z Y k II = M 1 < °°.lso sñ p II vn II < 2M, again we must only k=1 ii = i .Let (X n ) n=1 be a non shrinking Schauder decomposition of X .
There exist_ then x* E X with Ilx*I) = 1, s > 0, a strictly increasing sequence of índex (mk)k=1 and a sequence (Yk)k=1 with m We can choose the hyperplane Wk=Yk n Ker x* and using the lemma We must point out that if (Xn)n=1 is boundedly complete then it is also almost boundedly complete .Almost boundedly complete basis of X are defined in a similar way .
7 .Theorem .Let X be a p-Banach space .The follow ing are equivalent : i) All Schauder decompositions of X are shri nki np_ .
ii) All Schauder decompositions of X are almost boundedly complete . [3] [4] [ 5] [ 6] [ 7] Proof : Similar to the proof of Theorem 6, and it can be found in <m< m) is denoted by [X .] m i i=n' > ii .If (Qn)n=1 is the sequence of projections of (Y 1 ,Z 1 , . . . . .,Yn'Zn, . ..)' as An = Q 2n-1'X ' the statement ii is proved .n i .If sup lIA 11 <bounded sequence of projections which n n=1 defines the decomposition (Yn,Zn)n=1 because of theorem 1 .// Remark that if any of the Drevious subspaces is 0, it must be taken away in the decomposition . .Let (X n ) nm be a Schauder decomposition of a Banach --_1 space X and (Xn)n=1 a normalized sequence in X with xn E X n .For every there exists an hyperplane Wn of X n such that ([xll,Wl' . . . .[xnl,Wn' . ..) is a Schauder decomposition -of X .Proof : As 11x 11 = 1, we can define A (x) = u* (x)x , where n n n n u * E X* and u*(xn) uñII = 1 .4 .Lemma .Let X be a Banach space and a Schauder decomposition of the form v1],W1, . . . .lvnl'Wn' . ..) ls a SchauderProof : Let (P n ) n=1 be the sequence of projections of (X n ) n=1 and let K be its norm .Each P 2n-1 P2n-2 (the projection over [y n 1) is originated by a y* e X * according to n n-1 be defined be vñ = Y*-Y*+l' It is easy to check that We define the sequence of projections . Let X be a Banach space and ([Y1]'W1'"''[Yn]    ,Wn' . ..) a Schauder decomposition of X, where (y n )n -l satisfies sup II y n ll = , for every n, there is a continuous projection from X into [v n ~~ 11 P2 n 1111 X II + 11 x * 11 11 XII 11 Y n " + 111 Yk 11 11 x 1111 YnII k=1 11 A 2n 11 < K + M 11 x*11 + M1M Now we can prove the main result : n 6 .Theorem .Let X be a Banach space .The following statements are equivalent : i) All Schauder decompositions of X are shrinking ii) All Schauder decompositions of X are boundedly complete Proof .i ~ii .Let (X n )ñ =1 be a non boundedly complete Schauder decomposition of X .There exists then a sequence (xiSchauder decomposition of X with yk E Y k .Because of the corollary 3, for each k there exists a hyperplane Wk of Y k such that ([Yl],W1' . . ." .., [ Y n) , Wn, . ..) is a Schauder decomposition .Because of the lemma n 4, the sequence (vk)k=1 defined by v n = Y i originates the Schau-i=1 der decomposition([vl],W1, """ ,[vn],Wn, . ..) which is not shrinking because of y * (v k ) = 1 for every k> 1 .
the norm of (Xn)n=1' II vk II = II Y k -Yk-1 1 I ?k II Y k-1 11 With certain modifications, this theorem has an extension to p-Banach spaces (if i .ts dual separates points) .The Mackey topology of this spaces plays an important role in this extension .xk il<°°, the sequence (~ x k ) m converges in (J(X) , II .II * ) .k=1 k=1 n=1 12]I would like to express my acknowledgment to my research supervisor Professor M .A .Canela for his help and encouragement during my work and for the diffusion given to this paper in his lectures .