functional analysis – $l^{p}$ and $L^{p}$ applications in PDEs

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

$ell^{p}(I)=left{left(x_{i}right)_{i in I} in mathbb{K}^{I} ; sum_{i in I}left|x_{i}right|^{p}<inftyright}$

Thank you.