  The odd dimensional quantum sphere $S_q^{2\ell+1}$ is a homogeneous space for
the quantum group $SU_q(\ell+1)$. A generic equivariant spectral triple for
$S_q^{2\ell+1}$ on its $L_2$ space was constructed by Chakraborty & Pal. We
prove regularity for that spectral triple here. We also compute its dimension
spectrum and show that it is simple. We give detailed construction of its
smooth function algebra and some related algebras that help proving regularity
and in the computation of the dimension spectrum. Following the idea of Connes
for $SU_q(2)$, we first study another spectral triple for $S_q^{2\ell+1}$
equivariant under torus group action constructed by Chakraborty & Pal. We then
derive the results for the $SU_q(\ell+1)$-equivariant triple in the $q=0$ case
from those for the torus equivariant triple. For the $q\neq 0$ case, we deduce
regularity and dimension spectrum from the $q=0$ case.
