Integral equations occur in a natural way in numerous physical problems and have attracted the attention of many mathematicians including Volterra, Fredholm, Hilbert, Schmidt, F. Riesz, and others. The works of Volterra and Fredholm on integral equations emphasized the usefulness of the techniques of the integral operators. Soon it was realized that many problems in analysis could be attacked with greater ease if placed under a suitably chosen axiomatic framework. Axioms closely related to those of a Banach space were introduced by Bennett.† Using the axioms of a Banach space, F. Riesz‡ extended much of the Fredholm theory of integral equations. In 1922 using similar sets of axioms for such spaces, Banach,§ Wiener,‖ and Hahn¶ all independently published papers. But it was Banach who continued making extensive and fundamental contributions in the development of the theory of these spaces, now well known as Banach spaces. Banach-space techniques are widely known now and applied in numerous physical and abstract problems. For example, using the Hahn—Banach Theorem (asserting the existence of nontrivial continuous linear real-valued functions on Banach spaces), one can show the existence of a translation-invariant, finitely additive measure on the class of all bounded subsets of the reals such that the measure of an interval is its length.
KeywordsBanach Space Compact Operator Bounded Linear Operator Closed Subspace Weak Topology
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