This paper extends results of [1], [2], of Goldstein, and [3] of Vainberg concerning steepest descent and related topics. An example Is given taken from a simple rendezvous problem in control theory. The problem is one of minimizing a norm on an affine subspace. The problem here is solved in the primal. A solution in the dual is given by Neustadt [4]. I. GENERATION OF MINIMIZING SEQUENCES Let E...

Let T be any normed linear space [l, p. S3]. Then an inner product is defined in T if to each pair of elements x and y there is associated a real number (x, y) in such a way that (#, y) » (y, x), \\x\\ = (#, #), (x, y+z) = (#,y) + (x, 2), and (/#,y) = /(#, y) for all real numbers /and elements x and y. An inner product can be defined in T if and only if any two-dimensional subspace is equivalen...

Every continuous linear functional defined on a vector subspace of a real normed space can be extended to the whole space so as to remain linear and continuous, and with the same norm(2). The extension of continuous linear transformations between two real normed spaces has been studied by several authors and for a long time it has been recognized that this problem has a close connection with th...

1. Introduction. In addition to its well known role in analysis, based on measure theory and integration, the study of the Banach space B(X) of real bounded continuous functions on a topological space X seems to be motivated by two major objectives. The first of these is the general question as to relations between the topological properties of X and the properties (algebraic, topological, metr...

Let i, n be elements of N. The functor proj(i, n) yielding a function from Rn into R is defined by: (Def. 1) For every element x of Rn holds (proj(i, n))(x) = x(i). Next we state two propositions: (1) dom proj(1, 1) = R1 and rng proj(1, 1) = R and for every element x of R holds (proj(1, 1))(〈x〉) = x and (proj(1, 1))−1(x) = 〈x〉. (2)(i) (proj(1, 1))−1 is a function from R into R1, (ii) (proj(1, 1...