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The following relations between
a Banach space E and a Banach space X are, roughly speaking, generalizations of
the relation “E is a closed subspace of X.”
(LIX) The finite dimensional subspaces of E are uniformly isomorphic to
subspaces of X under ṡ omorphisms which extend to all of E without increase of
norm.
(SpX) Finite rank mappings from any Banach space into E can be uniformly
factored through subspaces of X.
(ASX) The continuous linear mappings from E into X distinguish the absolutely
summing mappings Srom any Banach space into E.
(SIX) For each absolutely divergent series ∑
nxn in E there is a continuous linear
mapping T from E into X such that ∑
nTxn diverges absolutely.
Our main result is that these four conditions are equivalent if X contains a
subspace isomorphic to λ[X] where λ is a normal BK-space. A related result of some
interest is that the class of continuous linear mappings which factor through
spaces which contain a complemented copy of λ[X] forms a Banach operator
ideal.
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