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It seems that when a mathematical theory was newly invented, or a particular phenomenon was discovered, it is often while tackling a specific hard problem, but as more of the theory was developed it found use in a variety of problems, many of which are "easier", as in not assuming as much prerequisite. Thus, when the textbook is written it is often the case that the author chooses to motivate with, or to focus entirely on, the easy cases.

To narrow down the question, I'd like to ask for examples in which the distinction can be phrased as finite- vs infinite-dimensional, at some level. To give an old example, the calculus of variations is almost as old as calculus itself, and it's basically optimization over an infinite-dimensional space. Its finite-dimensional analogue is certainly very useful, and is "easy" with differential calculus. It is said that the Lagrange multiplier was first discovered in the infinite-dimensional case—see epi163sqrt's answer to History of Lagrange Multipliers—yet it's now taught to all multi-variable calculus students.

Lie theory, Morse theory, Lax pair all seem to follow a similar history, though I'd like to read what the experts have to say. I hope it's of interest here beyond the purely historical, as it may be a better motivation for the theory when we are not constrained by the textbook-level detail.

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    $\begingroup$ I think Morse theory is a good example here; it is a powerful tool for studying finite-dimensional manifolds, but as you say the original motivation was in the study of loops on a manifold, which form an infinite-dimensional space. Bott's "Morse Theory Indomitable" is a good read, I don't recall how much of the early history he develops... $\endgroup$
    – Ryan
    Feb 6, 2022 at 17:56
  • $\begingroup$ Lie theory certainly started off as the study of what seemed potentially to be an infinite collection of objects but turned out to be a collection of finitely many infinite series with finitely many exceptional groups—but in what sense did it start off as an infinite-dimensional theory? \\ This seems like a potentially better fit for HSMSE. $\endgroup$
    – LSpice
    Feb 6, 2022 at 19:20
  • $\begingroup$ @LSpice, Lie groups started out as (concrete) transformation groups to study differential equations, with the Lie algebra as infinitesimal transformations, so however you want to phrase it, the transformations (say, as diffeomorphisms) are on a higher order than matrices. $\endgroup$
    – liuyao
    Feb 7, 2022 at 2:08

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If I'm not mistaken, operator systems were introduced by Choi and Effros in 1977, and I believe they were regarded for many years as primarily being of interest in the infinite dimensional case. It wasn't until 2010 that Duan, Severini, and Winter proposed that finite dimensional operator systems could be thought of as a "quantum" version of finite simple graphs, and this led to the development of a rich theory which now includes, for instance, a quantum Ramsey theorem for finite dimensional operator systems.

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