On the topology of pointwise convergence on the ... - CARMA

The raison d’être for rich families is revealed next.Proposition 1 Suppose that X is a topological space. If{F n : n ∈ N} are rich families <strong>of</strong> X <strong>the</strong>n so is ⋂ n∈N F n.Throughout this talk we will be primarily working with Banachspaces and so a natural class <strong>of</strong> rich families, given a Banachspace X, is <strong>the</strong> family <strong>of</strong> all closed separable linear subspaces<strong>of</strong> X, which we denote by S X . There are however many o<strong>the</strong>rinteresting examples <strong>of</strong> rich families.Theorem 1 Let X be an L 1 -predual. Then <strong>the</strong> set <strong>of</strong> allclosed separable linear subspaces <strong>of</strong> X that are <strong>the</strong>mselvesL 1 -preduals forms a rich family.