Functional Analysis book covering some of the oddities of infinite dimensional spaces I have been studying algebraic topology and came across the fact that the (algebraic) dual of an infinite dimensional space is not isomorphic to the space we started with.
In particular, I found that this answer
This then left me wondering if there might be an introductory book on functional analysis which takes the time to go over the complications involved in dealing with infinite dimensional vector spaces?
Most introductions I've seen seem to side-step them, which I think misses an oppurtunity to really motivate the need for topological tools in functional analysis.
 A: 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? :)
