CONDITIONS FOR SIMULTANEOUS APPROXIMATION AND INTERPOLATION WITH NORM PRESERVATION IN C[a,b]

This paper gives necessary and sufficient conditions for a triple (X, M, Γ) to have property SAIN (simultaneous approx-imation and interpolation which preserves the norm), X being an arbitrary Banach space. The best previous result concerned X, a reflexive, rotund Banach space. The paper proceeds to use this result to yield geometric proofs of the work of D. J. Johnson concerning property SAIN and C[a, b]. The concept of simultaneous approximation and interpolation with norm preservation (SAIN) was introduced by F. Deutsch and P. D. Morris [2]. DEFINITION 0.1. The triple (X, M, Γ) satisfies the hypotheses of the SAIN problem if X is a normed linear space, M a dense subspace of X, and Γ a finite dimensional subspace of X*. The triple (X, Λf, Γ) has a SAIN solution at x in X if given e> 0, there exists y in M such that ||JC —y | |<e, | | * | | = ||y||, and y(x) = y(y) for every γ in Γ. The triple (X, M, Γ) is said to have property SAIN if (X, M, Γ) has a SAIN solution for every x in X. The papers [4] and [10] took a geometrical approach to the SAIN problem and along with [9] extended the concept by allowing M to be a dense convex subset of X. In [7,8], specific questions posed in [2,4] were resolved in the standard Banach spaces f u