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

$\begingroup$ You may also want to take a look at the answers to this question: mathoverflow.net/questions/47721/… $\endgroup$ – Willie Wong Feb 5 '11 at 18:56

$\begingroup$ 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.) $\endgroup$ – Zen Harper Feb 5 '11 at 23:58

$\begingroup$ As a slightly different elementary theory, you may like the Cauchy integral for Banach valued regulated functions (that is, functions that have left and right limit at any point). See Dieudonné's "Foundations of modern analysis". $\endgroup$ – Pietro Majer Jan 1 '16 at 20:19

$\begingroup$ See Sec. V.3 on Bochner integral in Yosida's book ``Functional Analysis''. $\endgroup$ – Liviu Nicolaescu Jan 1 '16 at 20:38
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: 9783764374723; 3764374723
I also think there should be something in Henri Cartan's book on Differentiable Calculus.

$\begingroup$ 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) $\endgroup$ – CStarWStar May 19 '14 at 19:06
If I remember correctly, Hans Triebel's, "Higher Analysis" provides some useful results.