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, …, x_N0 in S and given directions x_ki ∈ E for all k ∈ {1, …, N}, i ∈ {1, …, m} such that x_kl ≠ 0, we prove that there exists an E-polynomial p:R → E with values in S such that p⁽ⁱ⁾(k) = x_ki ∀k ∈ {1, …, N}, i ∈ {0, …, m}, p(t) ∈ S, p′(t) ≠ 0 ∀t ∈ [l, N].