Banach Envelopes of Non-locally Convex Spaces

Then Xc is the completion of {X, \\ \\c). Alternatively || ||c is the Minkowski functional of the convex hull of the unit ball. Xc has the property that any bounded linear operator L:X —> Z into a Banach space extends with preservation of norm to an operator L\XC —» Z. The Banach envelope of / (0 < p < 1) is, of course, lx. In 1969, Duren, Romberg and Shields [3] identified the dual space of H_ (0 < p < 1) and thus its Banach envelope (cf. [14] ). The Banach envelope of H is a Bergman space which turns out to be isomorphic again to lx (see [18] for a recent direct proof of this). These examples and others prompted Joel Shapiro to ask what special properties a Banach envelope of a non-locally convex space (with separating dual) must have. An example of Pelczynski, the space l2(l n p) (0 < p < 1) has a reflexive Banach envelope (/2(/") ). This suggests that the answer to Shapiro's question lies with the finite-dimensional structure of the space. We say that a Banach space Y contains l{s uniformly (or lx is finitely representable in Y) if for every n e N and € > 0 there is a subspace F of Y with dim F = n and a linear isomorphism T:f\ —» F with