Summability in of a vector measure

Abstract

We show a picture of the relations among different types of summability of series in the space L1false(mfalse) of integrable functions with respect to a vector measure m of relatively norm compact range. In order to do that, we study the class of the so‐called m‐1‐summing operators. We give several applications regarding the existence of copies of c0 in L1false(mfalse), as well as on m‐1‐summing operators which are weakly compact, Asplund or weakly precompact.

Isometric factorization of vector measures and applications to spaces of integrable functions

“…In this case, R( m) is relatively norm-compact (because the restriction J| K is a normto-norm homeomorphism and R(m) ⊆ m (Ω)K) and Y is separable. (iii) L 1 ( m) is weakly sequentially complete because Y contains no isomorphic copy of c 0 , see [8, Theorem 3] (cf., [3,25]). In general, L 1 (m) is not weakly sequentially complete, so the equality L 1 ( m) = L 1 (m) can fail.…”

Open problems in Banach spaces and measure theory

“…We stress that, in an arbitrary Banach space, every bounded subset which is compact with respect to the topology of pointwise convergence on a James boundary is weakly compact: this striking result of Pfitzner [101] answered a long-standing question known as "the boundary problem". It is also worth mentioning that James boundaries are useful to study summability in Banach spaces, see [21,52,62].…”