# 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, 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}

Thank you.