Operators and Locatedness

Part of the Universitext book series (UTX)


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.


Hilbert Space Banach Space Normed Space Convex Subset Cauchy Sequence 
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Copyright information

© Springer Science+Business Media, LLC 2006

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