Application of Integrals of Scalar Functions with Respect to a Vector Measure to Some Problems of Functional Analysis and the Theory of Linear Integral Equations

Abstract

We prove some statements on the decomposition of indefinite integrals of scalar functions with respect to a vector measure. We also consider continuous linear operators acting from the fundamental Banach space $X\left( {T,{\sigma ,\mu }} \right)$ to a Hilbert space H. This gives a representation theorem for continuous linear operators from X to H. These results are applied to most general linear integral equations of the form $\int\limits_T {x\left( t \right)dv = \varphi ,v:{\sigma } \to H,v \ll {\mu }}$. Such equations are equivalent to certain infinite systems of scalar integral equations and to infinite systems of linear algebraic equations. Bibliography: 11 titles.