Anyone who studies advanced calculus must be strongly motivated by a genuine love of mathematics. By its very nature the subject is hard, and perhaps not so glamorous. To spend a year at it requires dedication.
Therefore the author has made a real effort to "break the monotony," so to speak, by shifting abruptly from one topic to another. Of course, the topics are related; topology, linear algebra, and inequalities all fit into the grand scheme of advanced calculus. To the student, however, they will probably be as distinct as night and day. If the author's timing has been good, just when one has had all the topology one can stand, inner-product spaces present themselves in all their pristine glory, Fourier series for the most striking
curves unfold, and the surprising secret of Pythagoras is revealed.