Re: Functional analysis: Riesz representation theorem


<fjblurt@xxxxxxxxx> wrote in message
news:d9e66afa-2fbb-423b-9f3d-9e0d83483a32@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Hi folks,

Let X be a compact Hausdorff space, C(X) the Banach space of
continuous functions from X to R under the uniform norm. Let L be a
bounded linear functional on C(X). The Riesz representation theorem
says that L(f) = \int f dm for some finite signed measure m. Suppose
f_n is a bounded sequence in C(X) such that f_n(x) -> 0 pointwise (not
necessarily uniformly!).

*************
The dominated convergence theorem says that
L(f_n) -> 0.
***************************

I might suggest that you write down what this says, as precisely as you can,
in words. It's possible that they will surprise you.

-Bill



This is a surprising fact, in isolation. Just knowing that L is
bounded, you would think that you would need uniform convergence of
the f_n. I am wondering if there is a proof of this fact that does
not rely on the Riesz representation theorem. Somehow it seems like
there should be, since you can state it without any measure theory. I
thought about it for a while but didn't come up with anything. I'm
curious if anyone else has ideas.

Thanks!


.

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