We examine the symmetric basic sequences in some classes of Banach spaces with symmetric bases. We show that the Lorentz sequence spaced(a,p) has a unique symmetric basis and every infinite dimensional subspace ofd(a,p) contains a subspace isomorphic tolp. The symmetric basic sequences ind(a,p) are identified and a necessary and sufficient condition for a Lorents sequence space with exactly two nonequivalent symmetric basic sequences in given. We conclude by exhibiting an example of a Lorentz sequence space having a subspace with symmetric basis which is not isomorphic either to a Lorentz sequence space or to anlp-space.
Banach Space Sequence Space Basic Sequence Nonnegative Number Unit Vector Basis
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