Operators and Locatedness

Abstract

We begin by introducing normed spaces on which the norm is differentiable in some fashion. In Section 2, with substantial help from the λ-technique, we provide criteria for the locatedness of certain convex sets in a normed space. This work is applied in Section 3 to proving that a bounded operator on a Hilbert space H has an adjoint if and only if it maps the unit ball to a located set. In the next section we construct approximate eigenvectors of a selfadjoint operator on H, and then show that a bounded positive operator has a unique positive square root. This result is applied in Section 5, in which we make further good use of the λ-technique to show that, for a so-called weak-sequentially open operator T on H, the range of T is located if and only if the range of its adjoint is located. The section ends with a proof of the closed range theorem for operators with an adjoint. The final section of the chapter presents a version of Baire’s theorem, which is then applied to the proofs of three of the pillars of functional analysis: the open mapping, inverse mapping, and closed graph theorems.