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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:… –  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

1 Answer 1

This one might be what you are looking for:

MR2419362 (2009e:00001) Amann, Herbert ; Escher, Joachim . Analysis. II. Translated from the 1999 German original by Silvio Levy and Matthew Cargo. Birkhäuser Verlag, Basel, 2008. xii+400 pp. ISBN: 978-3-7643-7472-3; 3-7643-7472-3

I also think there should be something in Henri Cartan's book on Differentiable Calculus.

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Amann and Escher do not offer the Riemann version but rather the Bochner version and these fail to agree even on the compact interval (see link in the comment above) –  Freeze_S May 19 '14 at 19:06

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