In math, what is sometimes meant by Analysis

[marcus]

in math, what is sometimes meant by "Analysis"

not that "Mathematical Atlas" is the greatest, it provides a rough indication of conventional university-level math-speak, and its welcome page lists these subheadings under "Analysis"

Functional Analysis
Real A.
Complex A.
Differential Equations
Numerical A.


At upperdivision or grad level, unless specified otherwise or modified, the term "Analysis" refers mostly to Functional A. To get more specific, exerpting the Math Atlas, Functional Analysis means things like:

"Function spaces, ...(infinite-dimensional) vector spaces with some kind of metric or other structure, including ring structures (Banach algebras and C-* algebras for example)...

Fourier analysis...

Abstract harmonic analysis...

Integral transforms ...The general theory of transformations between function spaces is part of Functional Analysis... Also includes convolution operators and operational calculi.

Operator theory studies transformations between the vector spaces studied in Functional Analysis, such as differential operators or self-adjoint operators..."

For some reason Math Atlas does not mention Measure Theory as part of Analysis----measures on other structures besides the familiar real line and such.

Math Atlas isn't perfect but it does give a quickanddirty idea of what many people mean by "Analysis"----i.e. Functional Analysis.
Wikipedia probably gives a similar perspective, I haven't looked.

 

[quartodeciman]

The Math Atlas has Measure Theory listed under Calculus and Real Analysis.

I suspect that the term Analysis came from the eighteenth century and originally referred to analysis of mechanical situations, both terrestrial and celestial. By the nineteenth century, Algebra and Geometry took off on their own separate developments and Analysis had a life of its own, but borrowed from the other two subjects. In the twentieth century there came the great triplet of subjects: Algebra, Topology and Analysis.

Functional Analysis means "analysis of functionals". Functionals are
real-values functions on vector spaces. It's a funny name, but there it is.

 

[marcus]

Originally posted by quartodeciman
The Math Atlas has Measure Theory listed under Calculus and Real Analysis...

some of the more interesting measures I can think of are on groups or on infinite-dimensional spaces of one sort or another,
you likewise?


the threefold division you mention agrees with how I've often pictured the terrain divided up (always allowing a few specialties that don't fit)

Algebra
Geometry/Topology
Analysis

indeed faculties tend to divide up this way to first approximation, as do sets of qualifying exams

so to a mathematician, "Analysis" ought to be a big word pointing to a broad range of subjects-----it wouldn't sound right to limit it to things like "functions of a real variable" "calculus" "complex variable" or whatever the historical (pre-20thC) meanings may have been.

WHOAH I just looked in Britannica----the article "Analysis" says "one of the three main divisions of mathematics, the other two being (1) geometry and topology, and (2) algebra and arithmetic

So Britannica generally agrees with what we were saying----Analysis is roughly (and in the subjective view of 20thC mathematicians) one third of the mathematical pie.

 

[quartodeciman]

The problem is that while it is convenient to engage in systematic study in neat separated categories, each of these borrows and steals from the others shamelessly. When significant results are attained, both areas (the taker and the taken) grow.

The big three become major toolchests for work on all the later research conquests. The Math Atlas symbolizes this fairly well with its diagram of a few big bubbles in a sea of little bubbles.

The measure idea is a rich one for various levels of application and for foundations. Unfortunately, one doesn't meet the idea until well into study of analysis. Everyone's slate is already full. Notice that one consequence was bringing the very-analysis-originated concept of integral back to algebraic/number-theoretic domains as applications.

 

[HallsofIvy]

Yep, mathematicians are shameless!

Actually, I would be inclined to say that the TWO main fields of mathematics are "Algebra" and "Geometry" (i.e. algebra and topology) and that analysis is the combination of those two (the "appropriate structure" for doing analysis (the theory of calculus) is the "topological vector space" or "topological group"- combining topology and algebra.

 

[ogb p]

Analysis is the process of breaking down a proposition or theory to its fundamental propositions/assumptions, known in math as axioms.

The reverse of analysis is synthesis. In a synthetic approach you start with basic assumptions/axioms and prove theories that follow from the given axioms. Eculidean Geormetry is a synthetic aproach.

Despite being inverses of each other, analysis and synthesis are not opposed to each other, they are complimentary. The goal of analysis is to discover the least number of a priori assumptions (axioms) needed to prove the origonally analyzed proposition synthetically.

 

[HallsofIvy]

Analysis is the process of breaking down a proposition or theory to its fundamental propositions/assumptions, known in math as axioms.

Yes, but that wasn't the question. The reference to "analysis" (and indeed in the title of this forum) is to "mathematical analysis" which very specifically means the "theory of (or behind) calculus".

 

[ogb p]

The title of this thread is, "in math, what is sometimes meant by 'Analysis'" - and the definition I provided is a direct answer to that question... IN TERMS OF MATHEMATICS!

IN MATH the term analysis means exactly what I described it as meaning. How many other disciplines try to break things down to 'fundamental' Axioms?

The method of Analysis was applied to the subject of Calculus in the late 19th century by mathematicans like Cauchy who realized that the subject did not have a rigorus foundation. The subjects of Real Analysis, Complex Analysis, Functional Anaylsis, etc., are just descriptive titles for Analytic methods being aplied to specific subjects like the Real Number Line, Complex Plane, Functions, etc.
Math is more than Calculus (although it is all related), and "Topological Analysis", or "Geometric Analysis" is just as much "mathematical Analysis" as the analysis of Calculus.