On a Riemannian manifold $(M,g)$, let $\mathcal L^p(M,k)$ denote the space of measurable $k$-tensors $T$ (i.e., the coordinate components in any chart are Lebesgue measurable) for which the norm $$||T||_p=\int_M |T|^p\,d\mu_g,\quad |T|=\sqrt{T_{i_1\cdots i_k}T^{i_1\cdots i_k} }$$ is finite. Is this space complete?

In compact case, I obtained an affirmative answer on Math Stack Exchange, but the answer does not seem to carry over to the general case. The book by Chavel, Eigenvalues in Riemannian Geometry, seems to claim this is true. He offers no proof.

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    $\begingroup$ Does any step of the standard proof that $L^p$ is complete fail here? $\endgroup$ – Will Sawin Jan 12 '17 at 13:11
  • $\begingroup$ @WillSawin Maybe nothing fails. I thought surely one couldn't use monotone/dominated convergence, but perhaps I didn't understand the proof for $L^p$ as well as I should. I have to carefully check this now. $\endgroup$ – Ryan Unger Jan 12 '17 at 14:00

There is a standard way to approach such questions using Grothendieck's completeness. Suppose that you have a complete topological vector space and an absolutely convex bounded closed subset thereof. Then the span of the latter is a Banach space with its Minkowski functional as norm. In your case, you can take the space of (equivalence classes of) measurable functions with its natural structure as a complete, metrisable tvs and the unit ball of your space of $L^p$-type tensors. A suitable reference for the Grothendieck result would be his notes on topological vector spaces which is available in english translation or the monograph of Schaefer.


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