Spectrum in functional-analysis and algebraic geometry Why do we use the notion "spectrum" both in functional-analysis and in algebraic geometry? Are there any analogies?
 A: The main difference is that in Functional Analysis the "Spectrum" is the family of maximal ideals of a ring, while in Algebraic Geometry, as Grothendieck defined it, the Spectrum $Spec(A)$ of a commutative ring with unit, is defined as the space (topoogical space with the natural Zariski topology) whose points are the prime ideals of the ring. In particular, since all maximal ideals are prime, but not viceversa, there are in the $Spec(A)$ (a la Grothendieck), points that are not closed, i.e., all points represented by those ideal which are primes, but not maximal. Whereas the spectrum in Functional Analysis, all points are closed, making the topology of Gorthendieck's Spectrum $Spec(A)$ much more interesting than the topology of the Gelfand spectrum in Functional Analysis.
It was actually one of the greatest insights of Alexander Grothendieck, to realize that a good definition of the Spectrum in Algebraic Geometry had to enclose all the prime ideals and not just maximal ideals as in Functional Analysis, and so it would have been quite more general. It is important to understand that Grothendieck started his career precisely in Functional Analysis.
