I'm a Ph.D. student in functional analysis, and struggling to understand why most research-level math in FA is useful. I talked to my advisor about this, and his response was, quite literally, to shrug his shoulders. In so many words, he said that he agrees with me that FA research is mostly useless, and suggests that I ignore the problem until I'm sufficiently well-established in the FA community.
Please don't misunderstand this question as suggesting that FA itself is useless. Clearly it is extremely useful, and there are even examples, albeit uncommon, of very useful cutting-edge FA.
Instead, let me give a specific example of what I am talking about. In a presentation about Ted Odell's work on spreading models, it was asked (my paraphrase):
Assume $\ell_1$ does not isomorphically embed into a Banach space $X$, and all asymptotic models generated by weakly null arrays in $X$ are equivalent to the $c_0$ basis. Is it necessary that $X^*$ is both separable and asymptotically-$c_0$?
This is a difficult problem, and anyone who solves it will be rightly praised as an intelligent and clever mathematician. But are such "clever" mathematicians really contributing in a significant way to human knowledge?
In some of my published papers, I have solved some "problems" in FA which are roughly on par with the above example. This left me feeling very proud. And yet, despite the recognition it has earned me in the FA community, I feel like I have really just been playing a game---the game of pandering to higher-ups ransoming postdocs.
With all this in mind, I suggest the following questions.
(1) Should a math researcher prefer to investigate issues for which there are no immediate practical applications?
(2) Do higher-ups (i.e., people on hiring committees) care about math research which has no obvious practical application? Should they?