Banach space is the standard setting for the French Analysis courses of the second half of XX century, for example H. Cartan, Calcul differentiel. Formes differentielles, Herman, Paris, 1967. It does not have Lebesgue integral. The course of Schwartz has everything, including a careful discussion of differences between the finite-dimensional case and the general case. Laurent Schwartzm Analyse mathematique I, II. Herman, Paris, 1967. There is also a course of Dieudonne.

Banach space is the standard setting for the French Analysis courses of the second half of XX century, for example H. Cartan, Calcul differentiel. Formes differentielles, Herman, Paris, 1967. It does not have Lebesgue integral. The course of Schwartz has everything, including a careful discussion of differences between the finite-dimensional case and the general case. Laurent Schwartz Analyse mathematique I, II. Herman, Paris, 1967. There is also a course of Dieudonne.