Analysis is that branch of mathematics which deals with the real numbers and complex numbers and their functions. It has its beginnings in the rigorous formulation of calculus and studies concepts such as continuity, integration and differentiability in general settings.
Historically, analysis originated in the 17th century, with Newton's invention of calculus. In the 17th and 18th centuries, analysis topics such as the calculus of variations, differential and partial differential equations, Fourier analysis and generating functions were developed mostly in applied work. Calculus techniques were applied successfully to approximate discrete problems by continuous ones. All through the 18th century the definition of the concept function was a subject of debate among mathematicians. In the 19th century, Cauchy was the first to put calculus on a firm logical foundation by introducing the concept of Cauchy sequence. He also started the formal theory of complex analysis. Poisson, Liouville, Fourier and others studied partial differential equations and harmonic analysis. In the middle of the century Riemann introduced his theory of integration. The last third of the 19th century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, and introduced the ε-δ definition of limit. Then, mathematicians started worrying that they were assuming the existence of a continuum of real numbers without proof. Dedekind then constructed the real numbers by Dedekind cuts. Around that time, the attemps to refine the theorems of Riemann integration led to the study of the "size" of the discontinuity sets of real functions. Also, "monsters" (nowhere continuous functions, continuous but nowhere differentiable functions, space-filling curves) began to be created. In this context, Jordan developed his theory of measure, Cantor developed what is now called naïve set theory, and Baire proved the Baire category theorem. In the early 20th century, calculus was formalized using set theory. Lebesgue solved the problem of measure, and Hilbert introduced Hilbert space to solve integral equations. The idea of normed vector space was in the air, and in the 1920's Banach created functional analysis.
To a certain extent, the pedagogy of the subject follows the historical development and not the logically rigorous presentation of advanced texts. Still, the following presentation follows the logical development.
The study of real analysis usually starts with simple proofs in elementary set theory, a clean definition of the concept of function, and an introduction to the natural numbers and the important proof technique of mathematical induction. Then the real numbers are either introduced axiomatically, or they are constructed from sequences of rational numbers. Initial consequences are derived, most importantly the properties of the absolute value such as the triangle inequality and the Bernoulli inequality.
The concept of convergence, central to analysis, is introduced via limits of sequences. Several laws governing the limiting process can be derived, and several limits can be computed. Infinite series, which are special sequences, are also studied at this point. Power series serve to cleanly define several central functions, such as the exponential function and the trigonometric functions. Various important types of subsets of the real numbers, such as open sets, closed sets, compact sets and their properties are introduced next.
The concept of continuity may now be defined via limits, one can show that the sum, product, composition and quotient of continuous functions is continuous, and the important intermediate value theorem is proven. The notion of derivative may be introduced as a particular limiting process, and the familiar differentiation rules from calculus can be proven rigorously. A central theorem here is the mean value theorem.
Then one can do integration (Riemann and Lebesgue) and prove the Fundamental Theorem of Calculus, typically using the mean value theorem.
At this point, it is useful to study the notions of continuity and convergence in a more abstract setting, in order to later consider spaces of functions. This is done in point set topology and using metric spaces. Concepts such as compactness, completeness, connectedness, uniform continuity, separability, Lipschitz maps, contractive maps are defined and investigated.
We can take limits of functions and attempt to change the orders of integrals, derivatives and limits. The notion of uniform convergence is important in this context. Here, it is useful to have a rudimentary knowledge of normed vector spaces and inner product spaces. Taylor series can also be introduced here.
Functional analysis deals with spaces of functions. If U and V are normed vector spaces, then we can try to look for linear maps from U to V that are also continuous. If V is the field of scalars (either the real numbers or the complex numbers) then such a linear map is called a functional. If U=V then such a linear map is called an operator. The space of all functionals, itself a normed vector space, is called the dual space of U and denoted by U'. The space of all continuous operators is denoted L(U). We usually require some more structure of U and V, perhaps that they be complete and hence Banach spaces such as the Lp space and Hardy spaces or Hp spaces. Hilbert spaces, which are complete spaces with an inner product, are important in the mathematical treatment of quantum mechanics and also serve as a general framework for Fourier analysis. The notion of derivative is extended to arbitrary functions between Banach spaces; it turns out that the derivative of a function at a certain point is really a continuous linear map. Here we list some important results of functional analysis: If U is a Hilbert space, then U' is isomorphic (in the case of real numbers) or anti-isomorphic (in the case of complex numbers) to U (Riesz representation theorem). For Banach spaces, the double dual (U')' contains U and the triple dual is isomorphic to the dual space. The uniform boundedness principle is a result on sets of operators with tight bounds. The spectral theorem gives an integral formula for normal operators on a Hilbert space. It is of central importance in the mathematical formulation of quantum mechanics. The Hahn-Banach theorem is about extending functionals from a subspace to the full space, in a norm-preserving fashion. One of the triumphs of functional analysis was to show that the hydrogen atom was stable.
Harmonic analysis deals with Fourier series and their abstractions. Jean Baptiste Joseph Fourier, in his work on the heat equation, argued that any function could be written as a possibly infinite sum of sines and cosines. From his proposition, it can be argued that most of modern mathematics originated. A modern but simple introductions is as follows. If H is a Hilbert space, then a set {ek} in H is said to be an orthonormal basis if:
- <ej,ek> = 0 if j ≠ k and = 1 if j = k
- the linear span of {ek} is dense in H
In this case, it is easy to show that any arbitrary vector v in H can be written as
- v=∑k <ek,v>ek
This expression on the right is called the Fourier series of v. This reduces to Fourier's version, by taking H to be a suitable space of functions, and ek to be a suitable set of trigonometric functions. There are also other generalizations -- it turns out that there is a reconstruction formula of sorts for certain Lp spaces. In addition, if the domain is not the interval, but perhaps some strange and interesting group, a form of Fourier decomposition is possible with basis functions chosen from the group structure of the domain. See the Peter-Weyl theorem, representation theory, Lie group and Lie algebra.
Complex analysis deals with analysis in the plane of complex numbers. Functions from complex numbers to complex numbers have a special definition for the derivative, tailored to look identical to the definition for the real case. It turns out however that the requirements imposed by this definition are much more stringent. In particular, for a function to be once differentiable, it has to be infinitely differentiable. These functions are called holomorphic. Path integrals are central tool of complex analysis. A version of the mean value theorem for integrals called the Cauchy integral formula is a crucial stepping stone. Taylor series or power series expansion formulae are given for analytic functions. The Fundamental Theorem of Algebra can be proven at this point. Isolated singularities occur when a function is differentiable in a punctured neighborhood of a point; such functions are called meromorphic. These can be categorized into removable singuliarities, poles and essential singularities. An essential theorem is that an integral along a closed path in a simply connected domain for a holomorphic function is always zero. However, the presence of a pole inside a domain will yield a nonzero integral, and a formula involving residues can be given, which turns out to be very useful in evaluating various difficult integrals. Laurent series generalize power series. One also studies analytic continuation, a technique to extend the domain of definition of a given meromorphic function.