Analysis

Initially, students are taught how to [mathematical proof]? simple theorems. Pythagoras was most distraught when one of his disciples showed him that the diagonal of a square of side 1 had a length that could not be expressed as the quotient of two natural numbers. This is one of the first proofs that any student of mathematics will see.

Quickly, the basic tools of analysis are built: natural numbers, integers and rational numbers have a mostly algebraic construction consistent with the [triangle inequality]?. Real numbers involve a kind of limit process so one could claim that analysis begins there. Sequences of numbers and limits are investigated simultaneously, with series inducing an early form of the CBS inequality. This leads to continuity and differentiation. An introduction to Cauchy sequences is also normally reviewed at this point.

Then we can do integration (Riemann? and Lebesgue) and prove the Fundamental Theorem of Calculus, typically using the mean value theorem. We can take limits of functions and attempt to change the orders of integrals, derivatives and limits. Here, it is useful to have a rudimentary knowledge of [normed vector spaces]? and [inner product spaces]?. Taylor series can also be introduced here.

Notions of point set topology and metric spaces such as compactness, completeness?, connectedness, [uniform continuity]?, separability, Lipschitz maps, [contractive maps]? and so on are also investigated. Armed with these tools, it is possible to approach more abstract analysis.

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 analytic?. 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. Functions that are differentiable on an open domain in the complex plane are also studied. [Isolated singularities]? occur when a function is differentiable in a [punctured neighborhood]? of a point. These can be categorized into [removable singuliarities]?, poles? and [essential singularities]?. Path integrals are also useful. An essential theorem is that an integral along a path in a simply connected domain for a differentiable 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. [Laurent series]? generalize power series. We can also study [analytic continuation]?.

Functional analysis deals with linear operations. If U,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. We usually require some more structure of U and V, perhaps that they be Banach spaces or Hilbert spaces. The space of all continuous linear maps from U to V is denoted by L(U,V). The space of functionals is denoted by U*. The space of all continuous operators is denoted L(U). Here we list some important results of gunctional analysis: If U is a Hilbert space, then U*=U ([Riesz representation theorem]?). For Banach spaces, U** contains U and U***=U*. 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. The Hahn-Banach theorem is about extending functionals from U to V when U is a subspace of V, in a norm-preserving fashion. One of the triumphs of functional analysis was to show that the hydrogen atom was stable. Also of interest: the [[L^p spaces]] and [Hardy spaces]? or H^p spaces.

[harmonic analysis]? deals with [Fourier series]?. [Jean Baptiste Joseph Fourier]?, in his work on the [heat equation]?, argued that any function could be written as a 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 {e_k} in H is said to be a basis if:

1) <e_j,e_k> = 0 if j \neq k and 1 if j \eq k 2) the [linear span]? of {e_k} is dense in H

In this case, it is easy to show that any arbitrary vector v in H can be written as

v=\sum_k <v,e_k>e_k

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 e_k 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 L^p 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 groups and lie algebras.

See also [analytic philosophy]?; [paradox of analysis]?.

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