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.