## Some Banach Spaces

To complete what we were saying about the spaces, we need to show that they’re complete. As it turns out, we can adapt the proof that mean convergence is complete, but we will take a somewhat different approach. It suffices to show that for any sequence of functions in so that the series of -norms converges

the series of functions converges to some function .

For finite , Minkowski’s inequality allows us to conclude that

The monotone convergence theorem now tells us that the limiting function

is defined a.e., and that . The dominated convergence theorem can now verify that the partial sums of the series are -convergent to :

In the case , we can write . Then except on some set of measure zero. The union of all the must also be negligible, and so we can throw it all out and just have . Now the series of the converges by assumption, and thus the series of the must converge to some function bounded by the sum of the (except on the union of the ).

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