Saccoman, John J. On the extension of linear operators. (English) Zbl 1010.46003 Int. J. Math. Math. Sci. 28, No. 10, 621-623 (2001). 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 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. Reviewer: Costică Mustăţa (Cluj-Napoca) MSC: 46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators 47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.) Keywords:extension property; Banach space Citations:Zbl 0022.15001 PDFBibTeX XMLCite \textit{J. J. Saccoman}, Int. J. Math. Math. Sci. 28, No. 10, 621--623 (2001; Zbl 1010.46003) Full Text: DOI EuDML Link