Categorical formulations of basic results and ideas from functional analysis? I'm taking a first (undergrad) course on functional analysis. Though the material is nice, the approach seems very ad hoc and in a sense, near-sighted (?).
I was wondering whether the/a big picture of (parts of) the elementary landscape of functional analysis admits some nice categorical descriptions.
What are some basic facts, theorems, and constructions in elementary functional analysis admit enlightening categorical formulations?
 A: I think Lectures and Exercises on Functional Analysis by A. Y. Helemskii might be exactly what you're looking for. Quoting from the introduction:

Perhaps the main idea is that our book is written from the categorical
  point of view. Everywhere we stress and comment on the categorical nature of the fundamental constructions and results (like the
  constructions of adjoint operators and completion, the Riesz-Fischer
  and the Schmidt theorems, and closer to the end of the book, the
  great Hilbert spectral theorem). This, as we believe, provides a new
  level of understanding of the topics discussed. We are sure that
  students (and even professors!) are ready for the perception of the
  very basic categorical notions (and only those are used) and, what is
  more important, for the unifying mathematical language of category
  theory. Functional analysis, with its synthetic algebraic and topological content, works very well for first acquaintance with
  categories, the same way as "Analysis III" did for the exposition of
  the foundations of set theory 50 years ago. (Of course, the exposition
  must be accompanied by a sufficient supply of examples and exercises;
  but we shall discuss this later.)

