[llamaz]

To add to what you said:

In Australia the follow on course after second year analysis is a course on topology, metric spaces, and basic functional analysis. Here you learn about norms from an analysis point of view and its relationship to topology (e.g. the euclidean norm induces the euclidean topology, which is a set of of open balls satisfying some properties).

One specific topic in the blog post, "best approximation" [1], is used to add some amount of rigour to the engineer's version of fourier analysis.

[1]: this is the first ref I could find on google: http://people.math.gatech.edu/~meyer/MA6701/module5.pdf