An inner product and orthonormalization inl p -spaces and solution of infinite systems of infinite linear equations

Abstract

Let the rows of an infinite square matrixM be elements ofl p -space (p>1) andX be an infinite column vector of unknowns andC an infinite column vector of real numbers. To our knowledge the solvability ofMX=C has nowhere been satisfactorily studied in the literature. This is also true of Riesz’classical work [2]. A reason for this is that not until recently [1] an appropriate inner product and the corresponding orthonormalization forp≠2 has been introduced. In this paper, based on [1], it is shown thatMX=C has a solution which is an element ofl q if and only if upon our process of orthonormalization of the rows ofM the system yields an equivalent systemAX=K where the rows ofA form an orthonormal sequence (in our sense) of elements ofl p andK becomes an element ofl q withp −1+q −1=1. A solution is then given byX=(A (q) (AA (q) )−1)K whereA (q) is ourq-transpose ofA. This solution is the solution of the minimall q -norm. Otherwise, the obvious dual concept of the best approximating solution inl q -norm is introduced and obtained.