# Riemann Integral of Banach space valued functions

Does anyone know of a good undergraduate or graduate text that gives a brief rundown of the Riemann integral on Banach space valued functions?

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You may also want to take a look at the answers to this question: mathoverflow.net/questions/47721/… –  Willie Wong Feb 5 '11 at 18:56
No, I don't know a reference. But, to sketch an answer for the implied question you didn't ask, it's relatively easy if the Banach space valued function $f:[0,1] \to E$ is continuous (the norm topology throughout). The convex hull of the compact set $f([0,1]) \subset E$ has compact closure; now directly use a sequence of approximating Riemann sums $\sum_j f(t_j) (t_{j+1}-t_j)$; each one of these is, of course, a convex combination. Filling in the details is a fun exercise...! (It was several years ago when I worked out these details, so apologies if I've misremembered them.) –  Zen Harper Feb 5 '11 at 23:58