Weyl numbers in sequence spaces and sections of unit balls

AbstractUsing 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, MϵN with k ⩽ M2, c1k1q − 1p ⩽ xk(Iqp(M)) ⩽ c2k1q − 1p, the upper estimate being even valid for all k ϵ {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