  This paper is concerned with complex Banach-space valued functions of the
form $$ \hat{f}_k(r\cos\theta,r\sin\theta,z)=\mathrm{e}^{\mathrm{i} k
\theta}f_k(r,z), \qquad r \in [0,\infty), \theta \in \mathbb{T}^1, z \in
\mathbb{R}, $$ for some $k \in \mathbb{Z}$. It is demonstrated how classical
and Sobolev spaces for the radial function $f_k$ can be constructed in a
natural fashion from the corresponding standard function spaces for
$\hat{f}_k$. A theory of radial distributions is derived in the same spirit.
Finally, a new class of \textit{Hankel spaces} for the case $f_k=f_k(r)$ is
introduced. These spaces are the radial counterparts of the familiar
Bessel-potential spaces for functions defined on $\mathbb{R}^d$. The paper
concludes with an application of the theory to the Dirichlet boundary-value
problem for Poisson's equation in a cylindrical domain.
