Euclidean Spaces
meurer.xyz
meurer.xyz
The problem is that you need to have mulled over the problem for months to years before you can develop insight.
These notes are contained in a chapter or two of any standard linear algebra textbook. This can serve as a background when studying analysis. Analysis starts with considering the real line first, then moves on to the metric spaces, then the normed spaces etc. That's when this stuff comes in handy. Typically, in linear algebra course one is introduced to norms and their properties; but analysis doesn't care about this stuff - it's just that LA ideas are used to further generalize analysis concepts. Fourier analysis (in mathematically rigorous sense) is introduced relatively late in ones analysis edjumacation. But the subject is important to engineers and physicists, so they get to be introduced to Fourier stuff as early as possible, but with much of the analytic rigor stripped.
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