Journal of Applied Analysis Vol. 1, No. 1, pp. 2938 (1995) 

Projections, extendability of operators and the Gateaux derivative of the normL. Gajek, J. Jachymski and D. ZagrodnyInstitute of MathematicsTechnical University of Lodz al. Politechniki 11 90924 Lodz, Poland Abstract: The HahnBanach 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: HahnBanach theorem, linear operators, Gateaux derivative,projections, Birkhoff's orthogonality Classification (MSC2000): 47A20, 47A30 Full text of the article:
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