APPROXIMATION BY FINITE RANK OPERATORS WITH RANGES IN co

In this paper the author characterizes all those spaces x, for which Kn(X, co) is proximinal in L(X, co). Some examples were found that satisfy this characterization.

Mach and Saatkamp [1], concerning the proximinality of Kn(X, co) in L(X, co) where c o is the space of all real sequences that converges to zero.The problem is divided into two parts, the first part is to characterize all those spaces X for which Kn(K, co) is proximinal in L(X, co), and the second part is to show whether Kn(X, co) is proximinal in L(X, co) or not, whenX e or loo.Kamal [7] showed that Kn(c, Co) is not proximinal in L(c, Co), given a partial solution for the second part of the mentioned problem.Deutsch, Mach, and Saatkamp [1] showed that if x c o or if X* is uniformly convex, then Kn(X, eo) is proximinal in L(X, eo) Kamal [6] showed that Kn(11,co) is not proximinal in L([l,Co) also Kamal [7] showed that if Q is a compact Hausdorff space that contains an infinite convergent sequence, then Kn(C(Q),eo) is not proximinal in L(C(Q),Co).In this paper a theorem is proved to characterize all those spaces x, for which Kn(X, co) is proximinal in L(X, eo), this characterization includes X c o, x for which x* is uniformly convex, and X such that the metric projection PN from X* onto any of its n-dimensional subspaces N, has a selection which is o*- continuous at zero.A point worth mentioning is that although c o is a one codimensional subspace of e, there are spaces X for which Kn(X, eo) is proximinal in L(X, eo) meanwhile Kn(X,e is not proximinal in L(X,c), for example Deutsch, Mach and Saatkamp [1] showed that Kn(eo, eo) is proximinal in L(eo, eo) meanwhile Kamal [7] showed that Kn(eo, e is not proximinal in L(eo, e).
The rest of introduction will cover some definitions, and known results that will be used later in Section 2.   If x is a normed linear space then co(X*,w* denotes the Banach space of all bounded sequences {zi} in x* that converge to zero in the w*-topology induced on x* by x, c o (x*) is the Banach space of all sequences {zi} in X* that converge to zero in the topology defined on x* by its norm, and if n> is any positive integer, then eo(X*,n denotes the union of all co(N), where N is an n- dimensional subspace of X*.The norm on eo(X*,w* is the suprimum norm.If {xi} is an element in eo(X*,w* then for any positive integer n > 1, define an({Zi} in.f{ {zi} {yi} ;{Yi} E eo(X*,n)} The following theorem can be obtained as a corollary, from the theorem of Dunford and Shwartz [2, p. 490].
As corollary of the Theorem 1.1, one can obtain the following: COROLLARY 1.2.If x is a normed linear space then for any positive integer n > 1, the set Kn(X, eo) is proximinal in L(X, eo) (resp.Kn(X, eo) if and only if eo(X*,n is proximinal in eo(X*,w* (rp.co(X*)).
In this paper if {zi} is an element in co(X*,w*), then dn({Zi},X* (resp.({zi},N) denotes the n- width (resp.the deviation from N) of the subset {=1, =2, x3 of X*.THEOREM 2.1 Let x be a normed linear space, and let n > be any positive integer.If {zi} is a bounded sequence in x* then an({Zi} rnaz{dn({Zi} X*), i-i }- Furthermore there is an n-dimensional subspace N o of X*, such that an({Zi} d({zi}, co(N)).
PROOF.First it will be shown that an({Zi})>max{dn({Xi}, N*), /-7"-m]]zill}.By Garkavi [4], there is an n-dimensional subspace N O of X* such that 5({xi},N)=dn({Xi},X* ).For each i= 1,2 let z be a best approximation for i from N o, and let > 0 be given, there is a positive integer n >_ such that for each i> m, zi < i + e. Define the sequence {Yi} in co(No) as follows.
yi={i ifi<rn if i>m.
PROOF.Let {xi} be an element in co(X*), by Corollary 1.2, it is enough to find an element {ui} in eo(X*,n such that {xi} {ui} an({Xi}).Since lim xi 0 it follows that/-x o, thus by Theorem 2.1, an({Xi} dn({a:i},X*).Let N o be an extremal subspace for dn({i},X*), and for eh 1,2 let ui be a best approximation for i from N o.Since lira x/l{ 0, it follows that tim 11Yi 0; that is, {Yi} -Co(No)" Thus {xi} {ti} su,{ i-ui a((i},ro) dn({Xi} )X*) an({ai}).LEMMA 2.3.Let X be a normed linear space, and let {xi} be a bounded sequence in X*. a) If an({xi},x*) > li-" xi {I, then an({i} is attained.b) If dn({Zi},X* </-llzill, and there is an extremal subspace N o for dn({ri},X* such that li---d(i, No)< li-'ll zil then an({zri} is attained.PROOF a) Assume that N is an extremal subspace for dn({Zi},X*), and let o=dn({i},X*)-I-llzilI, then there is a positive integer m> such that for each i>m, one has :i _< ii-:i +.For each i< m, let z be a best approximation for z from N o, and define the sequence {yi} in Co(No) as follows.
Let {ei} be a sequence of positive real numbers, satisfying that lim e_.=0, for each 1,2 d(zi, No) _</3 + e and for each 1,2 i -< + ei.For each 1,2 let z be a best approximation for i from N o, and define the sequence {ui} in N o as follows, /i + , if i < -Since {zi} is a bounded sequence in No, and lira _. 0, it follows that {i} co(No).
LEMMA 2.4.Let X be a normed space, and let {i} be a bounded sequence in X*.Assume that dn({:ri},X* z II, and for each extremal subspace N for dn({ri},X* one has li-" d(t.i,N li'-'-l[:ril =a.Let N be a extremal subspace for dn({Zi},X*), and for each i= 1,2 define { 0 ifllrill<a { 0 6 ot-d(zi, No), and If lirn a: 0 then an({Zi} is attained.PROOF.Let z be a best approximation for t from No, and let Yi ai' zi, then the sequence {Yi} is an element in Co(No).Furthermore for each 1,2   Ilzi-uill <(1-ai)llzill +'o, illzri-zill < (1 -ai)(cr + i) + ci(cr-/ii) cr + i cri(i + 60" converges w*-to zero.But N is of finite dimension, thus {ui} 6 Co(N).Furthermore {xi} {ui} 6({xi}.N) dn({Xi}, X*) an({Zi}).From Corollary 2.8 one concludes that for each positive integer n > 1, if X c o or tp, < then Kn(X, co) is proximinal in L(X, co).Proposition 2.9 clarify that.The fact that Kn(co, Co) is proximinal in L(co, Co) was proved first by Deutsch, Mach, and Saatkamp [1].PROPOSITION 2.9.Let n > be a positive integer and let X c o or lp, < p < o0.The metric projection P N from X* onto any of its n-dimensional subspace N, has a selection which is w*continuous at zero.PROOF.Let N be any n-dimensional subspace of x*, {zi} be any bounded sequence in X* that converges w*-to zero, and let {ui} by any sequence in N, satisfying that ui 6 PN(Xi) for each i.It will be shown that {ui} ( co(N).The sequence {ui} is a bounded sequence in a finite dimensional subspace of x*, so it has a convergent subsequence {u/k that converges to Uo in N, it will be shown that Uo 0. Assume not, and without loss of generality assume that {ui} converges to no, and that x* lp, < p < oo.Let i (ui no), ri ri ui, and let e > 0 be such that e < Uo , then as in Proposition 3 of Mach [9], there is a positive integer m > such that for each > m one has, ti-Uo '-ti -Uo '1 < e, thus ti-Yo p _> ti p + Yo p-e, that is a: u IIp > i % Uo)II v + Uo IIp e > i % Uo)II '-So for each > m one has xi-%-Uo) > :i-Yi II, which contradict the fact that i ui II a(zi,v), therefore Uo 0.