$L^p$-spaces for locally convex spaces

Question

Let $(X,\sigma)$ be a locally convex space, say generated by a family of seminorms $\mathfrak{P}$. I know that there is the notion of the space of integrable functions $f:\Omega\rightarrow(X,\sigma)$ for $(\Omega,\Sigma,\mu)$ a probability measure space, i.e. $p(f)$ has to be integrable for each $p\in\mathfrak{P}$ (see also PhD Thesis by Edward Reinke). My question is if there exists a notion of $L^p$ spaces for function with values in locally convex spaces and if the corresponding $L^p$ space has nice properties. Can someone give help or references to this topic?

Thank you and kind regards.

Answer 1

There is a notion of vector-valued $L^p$-spaces. The typical definition is the following: Denote by $S(F)$ the space of (up to equality almost everywhere) the functions of the form $$ s(t) = \sum_{i=1}^{n} \chi_{A_i}(t) y_i$$ where the $A_i$ are pairwise disjoint measurable sets and such that for every continuous seminorm $q$ the sum $\sum_{i=1}^{m} \mu(A_i)^p q(y_i)^p$ is finite. We equip this space with the topology generated by the seminorms $$\pi_q(s) = \Big(\int_{X} q(s(t))^p\,\mathrm{d}\mu\Big)^{1/p} = \Big(\sum_{i=1}^{m} \mu(A_i)^p q(y_i)^p\Big)^{1/p}.$$ The space $L_{X,\mu}^{p}\{F\}$ is defined as the completion of $S(F)$.

This definition can be found in the literature at the following places:

• Chapter 46.1 in F. Trèves: Topological Vector Spaces, Distributions and Kernels
• These spaces are also mentioned in Grothendieck's thesis "Produits tensoriels topologiques et espaces nucléaires", see also Exemple 1 in his "Résumé des résultats essentiels dans la théorie des produits tensoriels topologiques et des espaces nucléaires" Here he refers to the Bourbaki book on integration but I could not find the general case there.
• The spaces $L^{p}(X,F)$ are also studied in Chapter 8.18 of R. E. Edwards: Functional Analysis. Here the focus lies on the case where $F$ is a Banach space.

Unfortunately, it seems not much is known about these spaces. In the general case, it might even happen that elements of these spaces need not be functions with values in $F$ (see the quote in my question).