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It seems to me (correct me if I'm wrong) that Calculus of Variation is a subset of Analysis in Banach Spaces (see this post for an example).

Is there any text that approaches more general results on Banach Spaces while delving into the application of these results to the Calculus of Variations?

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    $\begingroup$ Maybe Differential Calculus (1986) by André Avez? (Book review) $\endgroup$ Commented Apr 3 at 1:37

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Jurgen Jost and Xianqing Li-Jost Calculus of Variations is an advanced monograph that might do what you want.

Part II of the book (about the last two-thirds) treats variational methods and function spaces in some depth

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