  This work is dedicated to the development of the theory of Fourier
hyperfunctions in one variable with values in a complex non-necessarily
metrisable locally convex Hausdorff space $E$. Moreover, necessary and
sufficient conditions are described such that a reasonable theory of $E$-valued
Fourier hyperfunctions exists. In particular, if $E$ is an ultrabornological
PLS-space, such a theory is possible if and only if E satisfies the so-called
property $(PA)$. Furthermore, many examples of such spaces having $(PA)$ resp.
not having $(PA)$ are provided. We also prove that the vector-valued Fourier
hyperfunctions can be realized as the sheaf generated by equivalence classes of
certain compactly supported $E$-valued functionals and interpreted as boundary
values of slowly increasing holomorphic functions.
