Existence of best approximation elements inC(Q,X)

Abstract

ABSTRACT. Generalizing the result of A. L. Garkavi (the case X = R) and his own previous result concerning X = C), the author characterizes the existence subspaces of finite codimension in the space C(Q, X) of continuous functions on a bicompact space Q with values in a Banach space X, under some assumptions concerning X. Under the same assumptions, it is proved that in the space of uniform limits of simple functions, each subspace of the form where /Jl E C(Q, X)* are vector measures of regular bounded variation, is an existence subspace (the integral is understood in the sense of Gavurin).We characterize the existence subspaces of finite codimension in the space of continuous functions on a bicompact space Q with values in a Banach space X. For X = IR, this was done by A. L. Gaxkavi [1], and for X = C, by the author [2]. G. M. Ustinov [3] announced, for real C(Q, X), assertions similar to the results of the present paper; however, he used a different terminology, different assumptions concerning the space X, and no vector measures. Our exposition is directly based on the methods of [2], which turn out to be appropriate in the abstract case as well.In the present paper we use the following notation: Q is an infinite bicompact space, X is a Banach space, E is the a-algebra of Borel subsets of Q, C = C(Q,X) is the space of continuous mappings g: Q --* X equipped with the norm Ilgll = sup, QIIg(*)ll, =d B = B(Q, X) is the space of uniform limits of simple functions g(t) = Ez~xe~(t) ( t E Q, zi E X, ei E E, X, is the characteristic function ofa set e, and the sum is finite) equipped with the same norm; thus, C is a subspace of B. We consider C and B as real (complex) spaces if X is real (complex)) The dual space C* is the space of vector measures /~ on E with values in X* equipped with the norm I1 11 = I.IQ = var(~, Q) = sup II~e~ll x" : ei e ~, e~ n ej = e (i # j), n e N ;here the measure I~1 is assumed to be finite and regular and (g,/~) = fQ (g(t), d/~) is the Gavurin integral defined for simple functions by the formula (9, #) = E(xi, I~ei) and extended by continuity to the entire B [4]. Let us note the inequalities Angle brackets axe usually omitted in the sequel. The function /J E L 1 (Q, E, U,I, x*) defined by the relation V e E E ~e=~dl~l= f ~(t)l~l (dt) (the Bochner integral) is called the Radon-Nikodym demvative of the vector measure # with respect to its variation I~,1 and is denoted by d~t/dlg I [5]. We also need art analog of the Radon-Nikodym derivative I The symbol ( ) is used for brevity to express a parMlel assertion.

Norm-attaining functionals onC(Q, X)

“…Obviously, /~ is 8-w*-continuous on S,, and II/~(t)'~l -i for all t 6 S,. But t~ and t~ have common limit points, whereas Pla= F~ and FI~ = F~ do not (despite Proposition 6), since p(rsa, rn)= p(F.', Theorem 7 [5]. Let X E (wLD).…”

“…The modulus of the first summand does not exceed e[/i[E, whereas the second summand (in the sum, we omit the terms with [/,[ei = 0) is equal to [5]. For any measure lz E C*, the derivative #' exists and satist~es II~'(t)ll < ~ Ipla/most everywhere.…”

“…If (g, #) = I1#11, or, equivalently, Re(g, #) = I1#11, then (a) g 9 P, (resp., P, NC) and thereexists ~ 9 B (resp., C); (b) IIg(t)ll = 1 I#l-almost everywhere (resp., for M1 t 9 S~, ); (c) if X 9 (R), then g(t) = b(t) I#l-Mmost everywhere (resp., for a11 t 9 S.) and P. = {g 9 SB: Re(g, #) = II#ll} = {g 9 s~: g(t)= ;(t) I#l-a.e. }, Proposition i [5]. If E E E, # E C', and #' (resp., #) exists, then …”

“…For P, = ~, we assume that /~ is not defined (an example of such a measure # can be found in [5]). The notation is due to the fact that ~ is somewhat similar to the derivatives p' and /i (see Proposition 1).…”

“…Ustinov and Shashkin [3,4] studied the characterization of such functionais on C(Q, X) (as well as on the space of affine abstract functions). The present paper deals with the same topic; we give some applications, supplementing the results of [5], to the characterization of finite-codimensional existence subspaces in C(Q, X).We use the following notation: Q = (Q, 8) is an infinite compact Hausdorff space with topology 0; X is a Banach space; n (resp., w or w*) is the strong (resp., weak or weak*) topology of X or X*; E is the a-algebra of Borel subsets of Q; C = C(Q, X) is the space of continuous mappings g: Q ~ X equipped with the norm [[g[[ = sup{[[g(t)H : ~ e Q}; B = B(Q,E,X) is the space of uniform limits of simple functions r = (t E Q, xi E x, ei E ~, xe is the characteristic function of a set e E ~, and the sum is finite) equipped with the same norm, so that C is a subspace of B. We assume that C (as well as B) is real (resp., complex) if X is real (resp., complex).…”