\(L^p(l^q)\) space

Let \(p,q \in [1, \infty]\) and \(\{f_k\}_{k\in \mathbb{N}}\) a sequence of complex-values Borel-measurable functions on \(\mathbb{R}^n\). We define the norm

\begin{equation*} \lVert f_k\rVert_{L^p(l^q)}:=\lVert \lVert f_k(\cdot)\rVert_{l^q}\rVert_{L^p(\mathbb{R}^n)}. \end{equation*}

and

\begin{equation*} L^p(l^q):= \{\{f_k\}\mid \lVert f_k\rVert_{L^p(l^q)}<\infty\}=\biggl(\int_{R^n}\Bigl(\sum_{k\in \mathbb{N}} |f_k(x)|^q\Bigr)^{\frac{p}{q}} \mathrm{d} x\biggr)^{\frac{1}{p}}. \end{equation*}