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?

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    $\begingroup$ If you make the measure space the positive integers with counting measure, then $L_p$ is the same as $\ell_p$. $\endgroup$ – Stephen Montgomery-Smith Aug 29 '15 at 2:08

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).


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