I am aware of the meaning of the subspace of Lp:
$$ l^{p} = \sum_{i}^{\infty}|x_{i}|^{p} < \infty $$
in terms of an infinite dimensional vector/tensor space;
$$ \bar{V} = \sum_{p=1}^{\infty} \sum_{i=1}^{\infty}\hat{e}^{p}_{i}x^{p}_{i} $$
I am asking for a clarification of p-integral functions in abstract measure spaces and what the p index denotes in:
$$ \mathcal{L}^{p} (M, \sum, \mu) = \int_{M} |f|^{p} .d \mu < \infty $$
Also how normalization of a vector:
$$ ||V|| = \frac{V}{|V|} $$
is related to the definition in Banach spaces (normed) with the pth root:
for
$$ ||f||_{p} := (\int_{M}|f|^{p} . d \mu )^{\frac{1}{p}} $$
I am having trouble visualizing p-integrals and p-normalisation because the notation in my book is so compact.