  This habilitation thesis centres on linearisation of vector-valued functions
which means that vector-valued functions are represented by continuous linear
operators. The first question we face is which vector-valued functions may be
represented by continuous linear operators where vector-valued means that the
functions have values in a locally convex Hausdorff space $E$. We study this
problem in the framework of $\varepsilon$-products and give sufficient
conditions when a space of $E$-valued functions coincides (up to an
isomorphism) with the $\varepsilon$-product of a corresponding space of
scalar-valued functions and the codomain $E$. We apply our linearisation
results to lift results that are known for the scalar-valued case to the
vector-valued case. We transfer the solvability of a linear partial
differential equation in certain function spaces from the scalar-valued case to
the vector-valued case, which also gives an affirmative answer to the question
of (continuous, smooth, holomorphic, distributional, etc.) parameter dependence
of solutions in the scalar-valued case. Further, we give a unified approach to
handle the problem of extending vector-valued functions via the existence of
weak extensions under the constraint of preserving the properties, like
holomorphy, of the scalar-valued extensions. Our approach also covers
weak-strong principles. In particular, we study weak-strong principles for
continuously partially differentiable functions of finite order and improve the
well-known weak-strong principles of Grothendieck and Schwartz. We use our
results to derive Blaschke's convergence theorem for several spaces of
vector-valued functions and Wolff's theorem for the description of dual spaces
of several function spaces of scalar-valued functions. Moreover, we transfer
known series expansions and sequence space representations from scalar-valued
to vector-valued functions.
