On the adaptive and continuous information problems

In this paper we bound the infimum of the ratio of adaptive to nonadaptive information for linear problems in Banach spaces. This result resolves the conjecture on adaption, showing that adaption can help for linear problems. Letting (Y denote the above infimum, and or the same infimum over all linear problems with Hilbert space range, we show that ) 5 a 5 V%%% and V’?/2 I a2 5 e. Analogous results are presented for classes of problems with Lp and finite-dimensional range spaces. Additionally it is shown that continuous information can yield smaller error (radius of information) than linear information in a Banach space setting. This resolves an open question of B. Kacewicz and G. W. Wasilkowski, who showed that this cannot occur in Hilbert space settings. 8 1989 Academic PWSS. Inc.