It is probably dangerous to answer without reading Landsman’s whole paper (and the question seems likely to be closed as “opinion-based”), but I’ll record my first reaction as much like [lcv](//mathoverflow.net/questions/348126/any-real-contribution-of-functional-analysis-to-quantum-theory-as-a-branch-of-ph#comment871820_348126)’s (<sup>a</sup>), namely, it sounds a little bit strawman-ish to separate the two and then pit one (FA) against the other (QM).

Footnotes (by Born) on pp. 583, 585 of the famous [1926 Dreimännerarbeit](//doi.org/10.1007/BF01379806) (see commented [translation](//archive.org/details/SourcesOfQuantumMechanics) pp. 351, 352) immediately identified [Hilbert’s work on linear operators](//archive.org/details/grundzugeallg00hilbrich) as the correct framework for QM, and are a literal blueprint for its extension to unbounded operators by von Neumann. If this, and Weyl’s group reformulation of $[P,Q]=I$, are not contributions of functional analysis, then I don’t know what could be!

Consider also that, as far as could be determined at [this question](//mathoverflow.net/questions/313967/reference-request-oldest-linear-algebra-books-with-exercises), the first two times our very *phrase* “linear algebra” (I don’t mean the *thing*) appears in the literature are (<sup>b,c</sup>)

1. In Hermann Weyl’s [1919 book](//archive.org/details/raumzeitmateriev00weyl/page/20) on general relativity; the phrase didn’t catch on then.

2. In Hermann Weyl’s [1928 book](//books.google.com/books?id=-VReAAAAIAAJ&q=%22lineare+Algebra%22) on quantum mechanics: ***then*** it caught on.



So QM and FA *both* played a role in establishing the other as a field of study.

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<sup>a. Which I would qualify to: FA (some say linear algebra) “is essentially the same thing” as *those parts of QM we understand*.</sup>

<sup>b. The phrase also appears, just once fleetingly, in Courant and Hilbert’s [1924 book](//books.google.com/books?id=ikOHBwAAQBAJ&ots=V1OHN8hXMp&pg=PA100&dq=%22Verallgemeinerung+des+Problems+der+linearen+Algebra%22), of which H. Hameka writes in his nice informed [account](//books.google.com/books/?id=wtYcXH5OAS4C), p. 11: “by a fortunate coincidence, linear algebra was the subject of the first chapter in the newly published book *Methods of Mathematical Physics* by Courant and Hilbert.” Born and Jordan cited it in 1925 ([translation](//archive.org/details/SourcesOfQuantumMechanics) p. 279); like Landsman, I think the idea that physicists needed no one’s help came largely from *Dirac*’s failure to cite almost anyone.</sup>

<sup>c. Correction: I have since found the expression defined in Hellinger-Toeplitz ([1910](http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002263432), p. 292): “diejenigen Partien der Algebra, die man etwa unter dem Sammelnamen einer *linearen Algebra* vereinigen könnte: bilineare Formen (Rangverhältnisse), Trägheitsgesetz der quadratischen Formen, Formenscharen (Elementarteilertheorie von Weierstraß, Kronecker, Frobenius usw.).” There remains that it only caught on after 1928. (The *Dreimännerarbeit* cites Hellinger along with Hilbert; Hellinger and Toeplitz were Born’s classmates, and after 1904 all 3 and Courant reunited in Göttingen as the “[group from Breslau](//books.google.com/books?id=7gYAAAAAMBAJ&pg=PA3)”.)</sup>