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Some simple applications of the closed graph theorem

1964
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Proceedings of the American Mathematical Society
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Our discussion is based on the following two simple lemmas. Our field of scalars can be either the reals or the complex numbers. Lemma 1. Let Ebea Banach space under a norm || || and Fa normed space under a norm || ||i. Suppose that || H2 is a second norm of F with respect to which F is complete and such that || ||2^^|| \\ifor some k>0. Assume finally that T: E-+Fis a linear transformation which is bounded with respect toll II and II IK. Then T is bounded with respect to\\ \\ and Proof. The

doi:10.1090/s0002-9939-1964-0164239-3
fatcat:4dl3bo7luvgvpdve7d4awhcmaa