# $L^p$ spaces with negative $p$

First of all: I have seen the question $L^p$ with negative $p$ and this is not intended to be a duplicate of it. (Despite the similar title.)

Let $p$ be a negative real number and $X$ a measure space. Define $L^p(X)$ to be the vector space of all measurable functions $X \to \mathbb{C}$. Then define a "norm" on this space as is usual in the non-negative case: $\| f\| = (\int|f|^{p}dx)^{1/p}$. If $f$ is equal to zero on a set of positive measure it makes sense to interpret the integral as having an infinite value, which would make the "norm" equal zero. The same interpretation should apply in the case where the integral diverges.

I realize this isn't a norm in the usual sense, but it still seems to retain some interesting properties, for example, it has the property of positive homogeneity. So I've been wondering:

Are such spaces being studied? Is there any good literature availible where one might learn about them?

I have tried searching google but couldn't find anything useful. Thanks in advance.

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Given the extensive development of the $p\ge 1$ norms, it might be worthwhile to view $|f|_{-p}=1/|1/f|_p$, that is, expressing the negative-$p$ thing in terms of the positive, at least for $p\ge 1$. (Note that for $0<p<1$, a slightly different set-up gives a non-convex metric topology, so there are extra complications there...) – paul garrett Dec 10 '11 at 23:41