The following book treats parts of Banach space theory and harmonic analysis from the point of view of category theory: Johann Cigler, Viktor Losert, Peter W. Michor: Banach modules and functors on categories of Banach spaces. Lecture Notes in Pure and Applied Mathematics 46, Marcel Dekker Inc., New York, Basel, (1979), MR 80j:46112, Zbl 411.46044. Review in Bull. AMS 3,2 (1980) xv+282 pp., Scanned pdf here.