  We introduce a "dual-space approach" to mixed
Nevanlinna-Pick/Carath\'eodory-Schur interpolation in Banach spaces X of
holomorphic functions on the disk. Our approach can be viewed as complementary
to the well-known commutant lifting approach of D. Sarason and B.
Nagy-C.Foia\c{s}. We compute the norm of the minimal interpolant in X by a
version of the Hahn-Banach theorem, which we use to extend functionals defined
on a subspace of kernels without increasing their norm. This Functional
extensions lemma plays a similar role as Sarason's Commutant lifting theorem
but it only involves the predual of X and no Hilbert space structure is needed.
As an example, we present the respective Pick-type interpolation theorems for
Beurling-Sobolev spaces.
