On the extension of linear operators
Let \(X\) be a Banach space and \(M\) a closed subspace of \(X.\) One says that \(M\) has the property \(\left( \mathcal{E}\right) \) if for every Banach space \(Y\), every bounded linear operator \(b:M\rightarrow Y\) has an extension \(B:X\rightarrow Y\) with \(\left\|B\right\|=\left\|b\right\|\). A Banach space \(X\) has the extension property if every closed subspace of \(X\) has the property \(\left( \mathcal{E}\right) \). By a result proved by \textit{S. Kakutani} in 1939 [Jap. J. Math. 16, 93-97 (1939; Zbl 0022.15001)], a Banach space \(X\) has the extension property if and only if \(X\) is a unitary space. The author constructs an example of a Banach space of dimension 2, whose subspaces of dimension \(1\) have all the property \(\left( \mathcal{E}\right) \), and which is not a unitary space. This example shows that Kakutani's theorem holds only for Banach spaces of dimension at least 3.
scientific article; zbMATH DE number 1780300