ABSTRACT. In [1], W. Wolibner proved the following theorem:

Let a ≤ t_{1} < t_{2} < ... < t_{n} ≤ b be real numbers and let P be the set of all real polynomials on [a, b]. For each f in C([a, b]) and each ε > 0, there exists p in P such that (relative to the uniform norm)

a) ||f − p|| < ε, b) p(t_{i}) = f(t_{i}), i = 1, 2, ..., n and c) ||p|| = ||f||.

We were led by this result to make the following definition.

Definition. Let X be a real normed vector space, let M be a dense subspace of X, and let {x_{1}^{∗}, x_{2}^{∗}, ..., x_{n}^{∗}} be a finite subset of the dual space X^{∗}. Then (X, M, {x_{1}^{∗}, ..., x_{n}^{∗}}) has property SAIN (simultaneous approximation and interpolation with norm-preservation) if for each x ∈ X and ε > 0, there exists m in M with (i) ||x − m|| < ε, (ii) x_{i}^{∗}(m) = x_{i}^{∗}(x), i = 1, 2, ..., n, and (iii) ||m|| = ||x||.

Thus Wolibner’s Theorem asserts that (C[a, b]), P, {e_{t_{1}}, ..., e_{t_{n}}) has SAIN for any choice of points t_{1}, t_{2}, ..., t_{n} in [a, b]. (Here e_{t} in C([a, b])^{∗} denotes the point-evaluation functional at t in [a, b]).

We show, more generally, that if T is a compact Hausdorff space, A a dense subalgebra in C(T), and t_{1}, ..., t_{n} points in T, then (C(T), A, {e_{t_{1}}, ..., e_{t_{n}}}) has property SAIN. In a Hilbert space, property SAIN can be completely characterized as follows. Let M be a dense subspace of the real Hilbert space X and {x_{1}^{∗}, ..., x_{n}^{∗}} ⊂ X^{∗}. Then (X, M, {x_{1}^{∗}, ..., x_{n}^{∗}}) has property SAIN if and only if the representer of each x_{i}^{∗} is in M.