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Let E be a Banach space with a separable dual. Zippin’s theorem asserts that E embeds in a Banach space with a shrinking basis, and W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński have shown that E is a quotient of a Banach space with a shrinking basis. These two results use the interpolation theorem established by W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński. Here, we prove that the Szlenk indices of and can be controlled by the Szlenk index of E, where the Szlenk index...
When the set of closed subspaces of C(Δ), where Δ is the Cantor set, is equipped with the standard Effros-Borel structure, the graph of the basic relations between Banach spaces (isomorphism, being isomorphic to a subspace, quotient, direct sum,...) is analytic non-Borel. Many natural families of Banach spaces (such as reflexive spaces, spaces not containing ℓ₁(ω),...) are coanalytic non-Borel. Some natural ranks (rank of embedding, Szlenk indices) are shown to be coanalytic ranks. Applications...
∗ Supported by D.G.I.C.Y.T. Project No. PB93-1142
Let X be a separable Banach space without the Point of
Continuity Property. When the set of closed subsets of its closed unit ball
is equipped with the standard Effros-Borel structure, the set of those which
have the Point of Continuity Property is non-Borel. We also prove that,
for any separable Banach space X, the oscillation rank of the identity on
X (an ordinal index which quantifies the Point of Continuity Property) is
determined...
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