Regular Hermite Interpolation in an Arbitrary Connected Open Subset of a Topological Vector Space

We establish the existence of regular Hermite interpolations in an arbitrary connected open subset S of a topological vector space E: given N points x10, …, xN0 in S and given directions xki ∈ E for all k ∈ {1, …, N}, i ∈ {1, …, m} such that xkl ≠ 0, we prove that there exists an E-polynomial p:R → E with values in S such that p(i)(k) = xki ∀k ∈ {1, …, N}, i ∈ {0, …, m},p(t) ∈ S, p′(t) ≠ 0 ∀t ∈ [l, N].