On simultaneous approximation and interpolation which preserves the norm

Abstract

Motivated by Wolibner's theorem, we consider the following more general problem. Let M bes. dense subspace of the real normed linear space X, and let {x*, • • • , #*} be a finite subset of the dual space X*. The triple (X, M, {xf, • • • , x*}) will be said to have property SAIN (simultaneous approximation and interpolation which is norm-preserving) provided that the following condition is satisfied: For each x £ X and each €>0 there exists a y£ilf such that ||#—;y|| <e, x*(y)=x*(x) (i = l, • • • , w), and \\y\\= \\x\\. In this note we shall outline some of the main results we have obtained regarding property SAIN. Detailed proofs and related matter will appear elsewhere.