Three space properties and basis extensions (Q1275682)

If \(Y\) is a separable Banach space, \(X\) a closed subspace and if two of the spaces \(X\), \(Y\) and \(Y/X\) have bases then it does not necessarily follow that the third has a basis. A related question is whether a basis for \(X\) can necessarily be extended to one for \(Y\). As a consequence of the theorem proved here, the author gives the following corollaries: 1. If \(Y\) is a separable \(C(K)\) space and \(X\) a closed subspace with basis such that \(Y/X\) is not reflexive, then \(Y/X\) has a basis. 2. With the same conditions on \(X\) and \(Y\), if \(\{y_j\}\) is a basis for \(Y\) then any basis for \(X\) can be extended to a basis for \(Y\) that contains another subsequence equivalent to \(\{y_j\}\). 3. If \(Y\) is a separable \(L_1\) space and \(X\) a non-reflexive subspace such that \(Y/X\) has a basis then \(X\) has a basis. 4. With the same conditions on \(X\) and \(Y\), let \(\Omega_0\) be a basis for \(Y/X\) then there is a basis \(\Omega\) for \(Y\) such that the quotient map sends \(\Omega\) to \(\Omega_0\).