TITLE: Abstract treatment of multivariate calculus relevant for optimization
QUESTION [8 upvotes]: After studying the basics of (convex) optimization, I've become convinced there's sometimes a conceptual benefit in thinking of quantities like gradients etc. in a coordinate-free way, and keeping track of the spaces they naturally come from (e.g., while oftentimes we pick a basis $\mathbb{R}^n$ and think of the gradient as a vector also in $\mathbb{R}^n$, one can define gradients abstractly and see they naturally come from the dual $V^*$; another example would be optimizational duality).
I'm looking for a text that emphasizes such abstract points of view on the aspects of multivariate analysis relevant to (convex) optimization (no upper limit to how abstract (e.g. categories are fine)... I guess). Suggestions that are not obviously related to (convex) optimization are also welcome.

REPLY [3 votes]: I can even top Dirk's reference:

Constantin Zălinescu, MR1921556, Convex analysis in general vector spaces, World Scientific Publishing Co., Inc., River Edge, NJ, 2002.

(Although I draw the line at metric spaces...)