Peano's kernel theorem for vector-valued functions: a weak version in normed spaces

Improving a recent result by the authors, a vector-valued version of Peano's Kernel Theorem is proposed, which gives an integral estimate for a class of linear operators L: C^{n+1}([a,b],X)->X , with X general normed space, vanishing on all abstract polynomials of degree < n. Continuity and derivatives are intended in the weak sense. When lthe space is complete, the usual integral representation is retrieved. We show that all usual remainder estimates for polynomial, piecewise polynomial, and spline polynomial interpolation, numerical differentiation and numerical quadrature, can be readily transferred in the vector-valued setting.