91Ó°ÊÓ

Use the notion of basic sequence to prove that a Hamel basis of an infinite- dimensional Banach space has cardinality at least continuum.

Let \(\left\\{e_{i}\right\\}\) be a Schauder basis. For \(n \leq m \in \mathbf{N}\), define \(T_{n, m}\left(\sum_{i=1}^{\infty} a_{i} e_{i}\right)=\) \(\sum_{i=n}^{m} a_{i} e_{i} .\) We say that \(\left\\{e_{i}\right\\}\) is bimonotone if \(\left\|T_{n, m}\right\|=1\) for all \(n, m .\) 192 6. Schauder Bases (i) Show that \(\left\|T_{n, m}\right\| \leq 2 \mathrm{bc}\left\\{e_{i}\right\\}\). (ii) Show that there is an equivalent norm \(\|\cdot\|\) on \(X\) such that \(\left\\{e_{i}\right\\}\) is a bimonotone basis of \((X,\|\cdot\|)\)

A Banach space \(X\) is called weakly sequentially complete if every weakly Cauchy sequence is weakly convergent in \(X .\) An example of a weakly sequentially complete space is any reflexive space (show this) and any general \(L_{1}(\mu)\) space (this is a classical Steinhaus theorem). Show that if \(X\) is weakly sequentially complete and nonreflexive, then \(X\) must contain an isomorphic copy of \(\ell_{1}\).