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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 available where one might learn about them?

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

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    From the Sobolev space point of view, there are spaces $H^{-p}$ that are typically defined as the continuous dual of the space of the space $H^{p}$. When your underlying space isn't $L^2$ you can still define a dual space, and this leads to the $W$ spaces.2017-04-06

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