Just as the topic says, how did the name "analysis" come to denote the specific mathematical branch dealing with limits and stuff? The term "analysis" seems very generic compared to the words for the other two main branches, "algebra" and "geometry", which do not seem to have other unrelated meanings.
[Math] Why is analysis called “analysis”
math-historyreal-analysissoft-questionterminology
Related Solutions
There is a lot of problems in functional analysis one can mention. For instance,
- The approximation problem. Is every compact operator approximated via finite rank operators? The answer is no in general Banach spaces and yes in Hilbert spaces. (Enflo, 1973)
- Kato conjecture. Are square roots of certain class of elliptic operators analytic? The answer was given is 2002 and you can compare this article.
- Existence and uniqueness of the solution of the Schroedinger equation. Does the Schroedinger equation admit a solution when there is a potential? And is such solution unique? The answer is yes for a large class of potentials, due to the Phillips-Lumer theorem. (1961) However, this is a particularly significative example of a more general problem.
- Existence of topologically complementary of closed sets in Banach spaces. When every closed subset of an infinite dimensional Banach $X$ space has topological complement? When $X$ is Hilbert, thanks to the Lindenstrauss-Tzrafiri theorem. (1970)
- When absolute convergence is equivalent to unconditional convergence? The answer is provided by the Dvrorestky-Rogers theorem (1953) and is: when we are on a finite-dimensional Banach space.
- Self-adjointness of hamiltonians in Quantum Mechanics. Are typical hamiltonian operators of quantum mechanics self-adjoint? Kato-Rellich theorem ensures they are, even for a certain class of singular potentials (such as coulombian ones). (1951)
A never closed (so far) problem is that of Navier-Stokes equations, in the precise statement of the 6-th Millennium Problem. The 5-th Millennium Problem, concerning Yang-Mills theories, would probably be strictly connected with functional analysis in its solution. Furthermore, I remember my lecturer said Perelman proved Poincaré conjecture (more precisely, Thurston' geometrization conjecture) making use of functional analysis methods (2001-2002).
Obviously, such a list is subjective, in the sense those problems are the ones has impressed me so far and surely is incomplete. Moreover, they reflect my own formation. I must point out I don't really know how all of those problems were popular at the time they were open, and that some of them are rather specific, but I think they should be mentioned at least in view of the importance of their applications.
Added. For the third part, I'd say:
- Operator algebras ($C^*$-algebras, von Neumann algebras, Banach algebras);
- Nonlinear functional analysis;
- Geometry in Banachs spaces (developed especially from 1970s by Isreaeli school);
- Hilbert manifolds (this topic also developed from 1970s, as far as I know).
Well of course this has historic reasons. I don't know the details, though. But I would like to explain why the notion of an algebra over a ring, suitably generalized, is fundamental.
There are various notions which look very similar:
- ring
- monoid
- algebra over a ring
- normed algebra
- Banach algebra
- sheaf of rings
- topological monoid
- topological ring
- ring spectrum
- ...
Category theory is the unique field of mathematics where "similar" things are united to "one" thing. And in fact, in the context of monoidal categories, the mentioned examples are actually instances of one single notion: Monoid object, often also called "algebra object". One just has to apply this notion to different monoidal categories. In the above examples, these are:
- abelian groups
- sets
- modules over a ring
- normed vector spaces
- Banach spaces
- sheaves of abelian groups
- topological spaces
- topological abelian groups
- symmetric spectra
Best Answer
There is a tradition on early modern mathematics regarding the usage of the term analysis :
François Viète, Isagoge in Artem Analyticem (Introduction to the Analytic Art), Tours, 1591 (several successive editions and translations);
Thomas Harriot, Artis Analyticae Praxis, London, 1631.
The background is the "rediscovery" of ancient Greek mathematics and, in particular of Pappus of Alexandria, (c.A.D. 290 – c.350) and his main work in eight books titled Synagoge or Collection, which Book VII explains the terms analysis and synthesis, and the distinction between theorem and problem.
See Henk Bos, Redefining Geometrical Exactness. Descartes' Transformation of the Early Modern Concept of Construction (2001), page :
Reference to Pappus' problems is also found into René Descartes' La Géométrie (1637).
The two main line to be understood are :
analysis as a "method" to solve problem
analysis as the technique of treating geometrical problems with algebraic methods.
Both, I think, are "involved" into the use of analysis to name the new method introduced by Newton and Leibniz.
You can see :
and :