On shrinking and boundedly complete Schauder bases of subspaces for Banach Spaces

Abstract

The main results in this thesis are concerned with Banach spaces each of which has a Schauder basis of subspaces that satisfies one or more of the following conditions: (1) it is shrinking, (2) it is boundedly complete, (3) each subspace in the basis is reflexive. It is proved that a Schauder basis of subspaces, {M_i; P_i.}, for a Banach space X is shrinking if and only if {R(P_i*)} is a boundedly complete Schauder basis of subspaces for csp U from 1 to infinity R(P_i*). It is also proved that if {M_i; P_i} is a weak* -Schauder basis of subspaces for X** then csp U M_i = J(X), the canonical image of X, if and only if {J^-1 (R(P_i'))}, where P_i' is the restriction of P_i to J(X), is a shrinking Schauder basis of subspaces for X and each M_i is reflexive. The following result is also proved: a Schauder basis of subspaces {M_i; P_i} for a Banach space X is shrinking and each Mi is reflexive if and only if {R(P_i**)} is a shrinking Schauder basis of subspaces for J(X) and a weak* -Schauder basis of subspaces for X**. It is shown that a Schauder basis of subspaces {M_i; P_i} for a Banach space X is boundedly complete if {R(P_i*)} is a shrinking Schauder basis of subspaces for csp U from 1 to infinity R(P_i*). If {M_i; P_i} is a Schauder basis of subspaces for a Banach space X and Y denotes csp U from 1 to infinity R(P_i*), then it is proved that {M_i; P_i} is boundedly complete and each M_i is reflexive if and only if the Y-canonical mapping is an isomorphism (a linear, 1-1, bicontinuous mapping) of X onto Y*. An example is given of a Banach space which has a boundedly complete Schauder basis of subspaces, but which is not isomorphic with any conjugate space. An example is also given of a Banach space X which has a boundedly complete Schauder basis of subspaces {M_i; P_i}, but X is not isomorphic with Y*, where Y = csp U from 1 to infinity R(P_i*). A Banach space B is exhibited which has the following properties: (1) B is non-reflexive, in fact, B is not quasi-reflexive; (2) B has a Schauder basis of subspaces which is both shrinking and boundedly complete; (3) B is isometric with B**.