Most of the material connecting analysis and category theory seems to be written by specialists in category theory who have observed some of the ways that insights from category theory can be brought to bear (for example look on Math Sci Net at the many papers (co)authored by Joan Wick Pelletier.) But here's an example which is a book on functional analysis that has a strong use of categories: @book {MR0296671, AUTHOR = {Semadeni, Zbigniew}, TITLE = {Banach spaces of continuous functions. {V}ol. {I}}, NOTE = {Monografie Matematyczne, Tom 55}, PUBLISHER = {PWN---Polish Scientific Publishers}, ADDRESS = {Warsaw}, YEAR = {1971}, PAGES = {584 pp. (errata insert)}, MRCLASS = {46E15 (46M99)}, MRNUMBER = {MR0296671 (45 \#5730)}, MRREVIEWER = {H. E. Lacey}, } -- Bob -- Robert L. Knighten RLK@knighten.org