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.
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