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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 $L^p$. 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.