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The theory of focal points and conjugate points is an important part of the study of problems in the calculus of variations and control theory. In previous works we gave a theory of focal points and of focal intervals for an elliptic form J(x) on a Hilbert space a?. These results were based upon inequalities dealing with the indices s(a) and n(a) of the elliptic form J(x; a) defined on the closed subspace d(a) of a?, where a belongs to the metric space (E:, p)In this paper we give an approximation theory for focal point and focal interval problems. Our results are based upon inequalities dealing with the indices s(u) and u(u), where ,u belongs to the metric space (M, d), M=E1 x E. For the usual focal point problems we show that Xj(a), the nth focal point, is a p continuous function of a. For the focal interval case we give sufficient hypotheses so that the number of focal intervals is a local minimum at a0 in E. Neither of these results seems to have been published before (under any setting) in the literature. For completeness an example is given for quadratic problems in a control theory setting. 1. Preliminaries. We now state the approximation hypothesis given in [1] and [2] and the focal point hypothesis given in [5]. The former is contained in conditions (1) and (2), the latter in (3). QW will denote a real Hilbert space with inner product (x,y) and norm 11x11=(x,x)'12. Strong convergence is denoted by Xq=>Xo and weak convergence by Xq*X0. Let E be a metric space with metric p. A sequence {ar} in E converges to ao in E, written or(>OX if limr,, p(or, coo)=O. For each a in E let !/(a) be a closed subspace of v/ such that (la) if ar-`O0, Xr int d(ar), xr-oyo then yo is in Q(ao); (lb) if x0 is in Q1(ao) and e>0 there exists 3>0 such that whenever p(a, oro)<6, there exists Xe in A(oa) satisfying lxOx-xeII <8. Received by the editors March 26, 1971. AMS 1969 subject classifications. Primary 4160, 4617; Secondary 49XX, 9340.

An approximation theory for focal points and focal intervals