Real analysis, Royden-fitzpatrick

Real analysis, Royden-fitzpatrick

        The book is divided the three parts: Part I treats Lebesgue measure and Lebesgue integration for functions of a single real variable; Part II treats abstract spaces-topological spaces, metric spaces, Banach spaces, and Hilbert spaces; Part III treats integration over general measure spaces, together with the enrichments possessed by the general theory in the presence of topological, algebraic, or dynamical structure. The material in Parts II and III does not formally depend on Part I. However, a careful treatment of Part I provides the student with the opportunity to encounter new concepts in a familiar setting, which provides a foundation and motivation for the more abstract concepts developed in the second and third parts. Moreover, the Banach spaces created in Part I, the LP spaces, are one of the most important classes of Banach spaces. The principal reason for establishing the completeness of the LP spaces and the characterization of their dual spaces is to be able to apply the standard tools of functional analysis in the study of functionals and operators on these spaces. The creation of these tools is the goal of Part II. 

Part I 

       The concept of uniform integrability and the Vitali Convergence Theorem are now presented and make the centerpiece of the proof of the fundamental theorem of integral calculus for the Lebesgue integral. 

Part II 

     General structural properties of metric and topological spaces are now separated into two brief chapters in which the principal theorems are proven. • In the treatment of Banach spaces, beyond the basic results on bounded linear operators, compactness for weak topologies induced by the duality between a Banach space and its dual is now examined in detail. • There is a new chapter on operators in Hilbert spaces, in which weak sequential com-pactness is the basis of the proofs of the Hilbert-Schmidt theorem on the eigenvectors of a compact symmetric operator and the characterization by Riesz and Schuader of linear Fredholm operators of index zero acting in a Hilbert space. 

 Part III 

      General measure theory and general integration theory are developed, including the completeness, and the representation of the dual spaces,

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