I was reading a post on Quora regarding the application of "$l_1$", "$l_2$" norms for convex linear programming when I became very confused at which $L$-norm the posters are actually referring to.
I am used to make a distinction between $l^p$ and $L^p$ spaces and for me it is convenient and logical to say (following the convention of $l^p$ and $L^p$ spaces) that:
little $l$, $l_p$-norm refers to $\|x\|_p = \sum\limits_{i = 1}^{\infty} |x_i^p|^\frac{1}{p}$, and
big $L$, $L_p$ refers to $\|f(x)\|_p = (\int\limits_{\mathbb{X}} |f(x)^p|dx)^\frac{1}{p} $
To me it makes a huge difference which L-norm you are referring to. But on Quora and as well as on mse (and another instance here on physics.se) I see people "seemingly" mixing up the little $l$ and big $L$ norms frequently to the point I have no idea which $L$-norm people are referring to. For example, I can say that "a system is BIBO stable if the L1 norm is bounded". Which L-norm do you think I am referring to if you had no idea what BIBO stability is?
But does this actually make that big of a difference? Since the intuition of the norms (energy, stability, etc.) is preserved regardless of dimension. What are some reasons why difference between the two norms should or should not be enforced?
The $L_p$ norm is more general, but you need to specify a measure space for it to make sense. You're integrating over $\mathbb X$ after all!
The $l_p$ norm can be seen as a particular case of the above, as @Stephen Montgomery-Smith noted, with the counting measure on positive integers.
So I don't think there really is any source of ambiguity: either I specify which measure space I'm in, and then I'm clearly talking about $L_p$ on that measure space, or I don't (and this usually means that it's clear from context what I mean).