Reflexivity of $L(E,\,F)$

Let $ E$ be Banach spaces and denote by $ L(E,F)$ the space of all bounded linear operators (resp., all compact operators) from $ E$. In this note the following theorem is proved: If $ E$ are reflexive and one of $ E$ has the approximation property then the following are equivalent: (i) $ L(E,F)$is reflexive, (ii) $ L(E,F) = K(E,F)$ (iii) if $ T \ne 0 \in L(E,F)$ $ \vert\vert T\vert\vert = \vert\vert Tx\vert\vert$ $ x \in E,\vert\vert x\vert\vert = 1$ This result extends a recent result of Ruckle (Proc. Amer. Math. Soc. 34 (1972), 171-174) who showed (i) and (ii) are equivalent when both $ E$ have the approximation property. Moreover the proof suggests strongly that the assumption of the approximation property may be dropped.