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Urs, while it is good that spectral theorem is included into functional analysis table of contents, and it has functional analysis toc bar, I do not like that spectral theory is also included and also has this toc bar. My understanding is that spectral theory is much wider subject on the relation between the possibly categorified and possibly noncommutative function spaces (sheaf categories, noncommutative analogues) and the specifical “singular” features of those like prime ideals, like certain special objects in abelian categories, points of spectra in operator framework etc. In any case, in $n$POV, it is NOT a part of functional analysis, though some manifestations are. Like the concept of a space is not a subject of functional analysis, though some spaces are defined in the language of operator algebras. I find spectral theory on equal footing like space, “quantity” etc. Of course, the entry currently does not reflect this much (though it has a section on spectra in algebraic geometry), but it eventually will! Thus I will remove it from functional analysis contents.
One should also point out that using generators in the proof of Giraud’s reconstruction theorem of a site out of a topos is a variant of spectral idea: like points form certain spaces, so the generators of various kind generate or form a category. This is behind many spectral constructions (including recent Orlov’s spectrum which is very laconic but stems from that) and reconstruction theorems and if the category corresponds to coherent sheaves over a variety than often the geometric features of the variety give certain contributions to the spectrum.
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