WEYL NUMBERS IN SEQUENCE-SPACES AND SECTIONS OF UNIT BALLS

Using techniques of a geometrical nature, the following result is proved: denoting by Iqp(M): lpM → lqM the natural embedding and by xk the kth Weyl number, if 0 < p ≤ q ≤ 2 there are positive constants c1 and c2 such that, for all k, Mε{lunate}N with k ≤ M 2, c1k 1 q - 1 p ≤ xk(Iqp(M)) ≤ c2k 1 q - 1 p, the upper estimate being even valid for all k ε{lunate} {1,..., M}. As a consequence of the approach used, some results about sections of unit balls are also derived, namely VolH(H ∩ BpM) ≤ Volk(Bpk) for 0 < p ≤ 2, where BpM, Bpk are the closed unit balls centred at zero of the spaces lpM and lpk, respectively, H is a k-dimensional subspace of lpM, and Volk, VolH denote Lebesgue measures in Rk and H, respectively. © 1992.