I am starting to get familiar with PDEs theory. As far as I know is the purpose is to apply functional analysis methods to study theses complicated problems. and this requires to define certain function spaces in which our unknown "lives". I understand that these spaces typically defined via integrals, so there's a use of $L^{p}$ spaces of measurable functions for which the {\displaystyle p}p-th power of the absolute value is Lebesgue integrable.
I'm just curious weather there's any application in PDEs theory of $\ell^{p}(I)$ spaces over a general index set $I$ (and $1 \leq p<\infty)$, which defined as follows
$\ell^{p}(I)=\left\{\left(x_{i}\right)_{i \in I} \in \mathbb{K}^{I} ; \sum_{i \in I}\left|x_{i}\right|^{p}<\infty\right\}$
Thank you.