Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorzskoda
    • CommentTimeJul 20th 2011
    • (edited Jul 20th 2011)

    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 nnPOV, 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.