Strategically reproducible bases and the factorization property

The factorisation property ofl∞(Xk)

“…Also recall the following definition of strategic reproducibility [13] of a Banach space X with a basis (e j ). Definition 2•3.…”

“…We now turn to defining the dyadic Hardy spaces, BMO and VMO. For a more in depth discussion of the biparameter Hardy spaces, we refer to [10]; see also [13]. Let D denote the collection of dyadic intervals given by…”

“…Then the series ∞ n=1 a n e n converges in the norm topology of Y . We now consider the following "simultaneous version" of the game described in [13]. Pregame.…”

“…In particular, we lose the norm convergence of the series expansion of vectors in Z which are not in the c 0 -sum of the X k . Consequently, the arguments given in [13] are not applicable to the space Z .…”

“…In our previous paper [13], we developed an approach to factorisation problems based on two player games; the type of games we are referring to were first considered by Maurey, Milman, Tomczak-Jaegermann in [16] and further developed by Odell-Schlumprecht [21] and Rosendal [25] who coined the term infinite asymptotic games (see also [6,22,23]). Thereby, we were able to unify the proofs of several known factorisation results as well as provide new ones.…”