Perhaps unsurprisingly, I'd tend to mildly endorse my own notes on functional analysis, which aim to emphasize practical/useful examples of spaces of functions... with natural/meaningful topologies/metrics ...
One idea is that, given a vector space of functions we find useful, give it a metric (or fancier topology...) so that it is complete, so then we know we can take certain limits and stay in the space. For example, spaces $C^k[a,b]$ with norm given by sup of sup-norms of derivatives up to order $k$.
An idea from the opposite side is to give a "good" structure (e.g., Hilbert-space) to a space of very-nice functions, which should/may preserve some good properties under taking the completion. A prototype for this is (Levi 1906!) $L^2$ Sobolev spaces, such as $H^k[a,b]$, which can be defined to be the completion of $C^\infty[a,b]$ under the Hilbert-space norm-squared $\sum_{0\le j\le k}|f^{(j)}|_{L^2}^2$. It is not elementary to understand exactly what this space is, but it has two good features: it is a Hilbert space, so has a genuine minimum principle (rather than a "false minimum principle"), and there is the Rellich lemma that $H^k[a,b]\subset C^{k-1}[a,b]$. That is, we can impose $L^2$-differentiability conditions to assure classical differentiability, with some loss of index.
So: doing analysis a little bit more conceptually than just explicit, perhaps-needlessly-detailed, brute-force estimates? :)
