91Ó°ÊÓ

Let \(Y\) and \(Z\) be closed subspaces of a Banach space \(X\) such that \(Y \cap Z=\\{0\\}\). Define a norm \(\|\cdot\|\) on \(Y \oplus Z\) by \(\|y+z\|=\|y\|+\|z\|\). (i) Show that \(\|\cdot\|\) is a complete norm on \(Y \oplus Z\). (ii) Show that the following are equivalent: (1) \(\|\cdot\|\) is equivalent to the original norm on \(Y \oplus Z\). (2) \(Y+Z\) is closed. (3) \(Y\) is complemented in \(Y+Z\); that is, \(Y \oplus Z\) is a topological sum. (4) There is \(k>0\) such that \(\|y\| \leq k\|y+z\|\) for every \(y \in Y\).

Show that all closed hyperplanes of \(C[0,1]\) are isomorphic to \(C[0,1]\) and \(C[0,1] \oplus \mathbf{R}\) is isomorphic to \(C[0,1]\). Consider the subspace \((C[0,1])_{0}\) formed by all functions in \(C[0,1]\) that vanish at 0 . Use it to show directly that \(C[0,1] \oplus \mathbf{R}\) is isomorphic to \(C[0,1]\).

Let \(f, f_{1}, f_{2}, \ldots \in L_{1}[0,1] .\) Show that if \(f_{n} \rightarrow f\) almost everywhere and \(\left\|f_{n}\right\|_{1} \rightarrow\|f\|_{1}\), then \(f_{n} \rightarrow f\) in \(L_{1}[0,1]\) (Vitali).