  We prove that the outer $L^p_\mu(\ell^r)$ spaces, introduced by Do and
Thiele, are isomorphic to Banach spaces, and we show the expected duality
properties between them for $1 < p \leq \infty, 1 \leq r < \infty$ or $p=r \in
\{ 1, \infty \}$ uniformly in the finite setting. In the case $p=1, 1 < r \leq
\infty$, we exhibit a counterexample to uniformity. We show that in the upper
half space setting these properties hold true in the full range $1 \leq p,r
\leq \infty$. These results are obtained via greedy decompositions of functions
in $L^p_\mu(\ell^r)$. As a consequence, we establish the equivalence between
the classical tent spaces $T^p_r$ and the outer $L^p_\mu(\ell^r)$ spaces in the
upper half space. Finally, we give a full classification of weak and strong
type estimates for a class of embedding maps to the upper half space with a
fractional scale factor for functions on $\mathbb{R}^d$.
