Projections, extendability of operators and the Gateaux derivative of the norm

L. Gajek, J. Jachymski and D. Zagrodny

Abstract: The Hahn--Banach extension theorem is generalized to the case of continuous linear operators mapping a subspace $Y$ of a normed space $X$ into a normed space $V$. In contrast with known results of this kind, we do not equip $V$ with a partial ordering neither impose any restrictions on $V$. The extension property is fully characterized by the sign of the one sided Gateaux derivative of the norm $\| \cdot \|_{X}$. Other characterizations, involving e.g. Birkhoff's orthogonality, are also provided.

Keywords: Hahn-Banach theorem, linear operators, Gateaux derivative,projections, Birkhoff's orthogonality

Classification (MSC2000): 47A20, 47A30

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