Some simple applications of the closed graph theorem

Jes{ús Gil de Lamadrid
1964 Proceedings of the American Mathematical Society  
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
more » ... and Proof. The product topology oî EX F with respect to || || and ||i is weaker (coarser) than the product topology with respect to || and || 112. Since T is bounded with respect to j| || and || ||i, its graph G is (|| | , II ||i)-closed. Hence G is also (|| ||, || ||2)-closed. Therefore T is ( | [[, || ||2)-bounded by the closed graph theorem. Lemma 2. Assume that E is a Banach space under a norm || ||i and that Fis a vector subspace2 of E, which is a Banach space under a second norm || ||2=è&|| ||i, for some ft>0. Suppose that T: E->E is a bounded operator with respect to \\ \\i and \\ ||i smcA that T(F)EF. Then the restriction of T to F is bounded with respect to || H2 and || ||2. Proof. Since the || || 2-topology of F is stronger (finer) than its ||i-topology, T restricted to F is (|| H2, || ||i)-continuous. We now apply our previous lemma with E replaced by F to get that the restriction of T to F is (|| || 2, || 112)-continuous. The above two lemmas have some interesting applications to certain questions about simple Banach algebras and other questions about tensor products of Banach spaces. These applications will be discussed in a subsequent article. Here we only give some simple ap-
doi:10.1090/s0002-9939-1964-0164239-3 fatcat:4dl3bo7luvgvpdve7d4awhcmaa