# Structure of an intersection of $L^p$-spaces

In what follows, $$L^p$$ denotes the space of functions from $$\mathbb{R}$$ to $$\mathbb{R}$$ such that $$\int_{\mathbb{R}} |f(x)|^p\mathrm{d}x < \infty$$. I am interested to understand the structure we can put on intersection of $$L^p$$-spaces.

For $$I$$ an interval of $$[1,+\infty]$$, we define $$L^I = \bigcap_{p\in I} L^p.$$ Obviously, because $$L^a \cap L^b \subset L^c$$ for $$1\leq a\leq c\leq b \leq \infty$$, we have that $$L^{[a,b]} = L^a \cap L^b$$. Thus, $$L^I$$ is a Banach space for the norm $$\lVert \cdot \lVert_a + \lVert \cdot \lVert_b$$.

Questions: What can we say of the structure of $$L^I$$ if $$I$$ is an open (or semi-open) interval of $$[1,\infty]$$? What is the more natural topology on it that make it complete? Are these spaces studied somewhere, for instance as a subpart of a more general theory?

Extensions:

• I am also interested in spaces of the form $$L^p_+ = \bigcup_{\epsilon >0} \bigcap_{0 with similar questions as previously.

• What happens if $$I \subset (0,\infty]$$, knowing that $$L^p$$-spaces for $$p<1$$ are quasi-Banach spaces?

• The following article might be of interest, if you are still investigating about your question. It's rather about $L^p$ functions which avoid being $L^q$, even up to a local level, but still looked at within the $L^p$ space structure: arxiv.org/pdf/1207.3818.pdf Nov 29, 2019 at 18:21
• The following article is probably relevant: “An arbitrary intersection of $L_p$-spaces” by F. Abtahi, H. G. Amini, H. A. Lotfi and A. Rejali, Bull. Aust. Math. Soc., 85 (2012) 433–445, doi:10.1017/S0004972711003510. Nov 30, 2019 at 21:39

## 3 Answers

The spaces you describe in the first part are the intersection of a sequence of Banach spaces (in your case, it is natural to regard them as subspaces of the Fréchet space---in the sense of complete, metrizable, but not locally convex, topological vector space---of (equivalence classes of) measurable functions. As a consequence, they have a natural structure of a locally convex Fréchet space.

The concept of such an intersection of Banach spaces was used frequently in the early theory of functional analysis. There are various extensions---the dual one of the union of a sequence of Banach spaces and to the non-locally convex space of quasi-Banach that you mention.

Spaces of this type were introduced as ad hoc extensions of Banach spaces, in particular for the needs of the non-normable spaces of test functions and distribution which arose in PDE theory but soon were subsumed as special examples in the general theory of locally convex spaces.

However, the particular properties of the projective or inductive limit of countable spectra of Banach spaces have always been of interest in general functional analysis.

• I totally agree with you. Especially, the more natural structure is probably the one of Fréchet spaces for the locally convex case. I can define, in intersection or union of $L^p$-spaces that are not nicely normable, a topology using the projective or inductive limit. Also, I would like to know if these spaces I proposed were studied for themselves somewhere, and not only as simple example of non-normable spaces. Apr 16, 2014 at 9:10
• I have seen articles where these spaces occur explicitly but I can't trace any at the moment. You might try looking at Pietsch' work on operator ideals. Apr 16, 2014 at 9:19

I want only to mention you that the paper

Castillo, Díaz & Motos, On the structure of the Fréchet spaces $L_{p-}$, Manuscripta Math. 1998

could be of interest in this regard.

• That's definitely relevant! Thank you. Jan 30, 2015 at 17:28

First off I'd like to apologize that this is not exactly an answer to your original question, but I would nontheless like to point you to the theory of interpolation spaces, which provides a general framework for similar questions and constructions. Typically, the "Riesz-Thorin convexity theorem" is referred to as the starting point: Roughly speaking, if you have a continuous linear map between different $L^p$-spaces, then this also defines a continuous linear map for some $L^q$ "in between". This point of view is slightly different from the one posed in the original question, as it does not view the space just by itself but rather in its surrounding category of Banach spaces and linear maps, but it is rather similar and probably more fruitful.

The article on wikipedia provides a short overview of the topic, beyond that I found the books

"Interpolation spaces" by Bergh, Löfström "An Introduction to Sobolev Spaces and Interpolation Spaces" by Tartar

very helpful, but there are numerous other books that explore the subject in greater generality, such as Triebel's "Interpolation Theory - Function Spaces - Differential Operators".