Mathematical analysis, a branch of mathematics that deals with the establishment and techniques of differential and integral calculus, and other examples of using the limit (or threshold crossings) as the theory of (infinite) lines, infinite products, development of analytical functions in order, analytic extension, variation account and the like.
Historical Development
Differential and integral calculus founded by Newton and Leibniz were motivated by the geometry (tangent to the curve, the area under the curve) and mechanical (primarily relationship-speed particles). Their calculus, however, is not rigorous, and only approaches the intuitive notion of limit (limit) over the introduction of mathematical unsubstantiated infinitely small size.
Development of the concept of limit values
In clumsy hands, handy with an infinitely small sizes easily leads to a contradiction. Only Cauchy in the 19th century comes to a rigorous notion of limit ("epsilon-delta" definition), which ultimately based logic and calculus and thus provides the foundation for modern (mathematical) analysis. This relieves the need to account for the infinitely small quantities and handled with the limits calculated from the final small sizes. The advantage of working with finite sizes is also a disadvantage, because the idea of infinitely small size and very intuitively easily be transferred to engineers, but access via epsilon-delta technique. To this end, the middle 20th century, logician Abraham Robinson founded a new approach, called. non-standard analysis in which the infinitely small and infinitely large size rigorously imposed, and the rules of such accounts, which leads to a contradiction. In nonstandard analysis with simple, "standard" elements of the set of real numbers are introduced and the so-called. nonstandard elements. In addition to statements which speak only on the standard elements of nonstandard and introduced testimony in which they occur and non-standard elements.
Robinson introduced the principle of transfer or transfer that allows one to extend the standard expressions derived non-standard statements.
Development of concepts and measures of integrals
Integral calculus of functions of real variables in the interval axis, based in the 19th century Riemann, based on Cauchy term limits applied to the so-called. partial sums. Limes are taken by divisions (subdivision) interval of the argument function on sub-intervals whose maximum length tends to zero. Such an integral is applied to a class of functions for which this limit exists, example: integrable functions on Riemann or Riemann-integrable functions. In early 20th century, Lebesgue watching another kind of limit, where the subdivision instead of the argument, looking at the subdivision value, leading to a successive approximation of the so-called integral sum. steep functions. This process leads to a more flexible notion of integrable which leads to the so-called. Lebesgue-integrable functions, and the development of the modern theory of abstract generalizations of length and area of geometric sets, which is today called the theory of action.
Functional Analysis
Functional analysis is an examination of the infinite-dimensional spaces, a typical function spaces. Functional is a function defined on (typically) infinite dimensional spaces of mappings (these areas alone are often not linear spaces). The most frequently used locally convex topological vector spaces, and among them were historically important Hilbert, Banach and Frechet spaces.
Calculus of variations deals with finding the local extremes (minimums and maximums) functions.
The theory of functions
The theory of functions deals with the properties and relationships of various classes of functions of one or more real or complex variables. This refers to those properties that are typical of mathematical analysis: measurability, there extremum, different types of integrable, analytical, finite variation, continuity, Lipschitz continuity, the existence of statements (derivation), the maximum principle, the existence and properties of limits on the edge of the area definitions, the properties with respect to the various integral transformations (Mellina, Fourier, Hilbert, Laplace...), asymptotic properties, analytic extension, zeros, etc. How often do the class of functions linear spaces interesting from the standpoint of functional analysis, this area is intertwined with the functional analysis.
Harmonic analysis
Harmonic analysis came from studies of normal modes and waves oscillating system. It is very important in physics and engineering applications is important when the linear regime, for example in linear systems theory and linear response of nonlinear systems. Mathematically, the harmonic analysis deals with finding a certain group of dual symmetry, which allows, with some structures in the theory of action, development of a linear function of the corresponding bases in function spaces. This includes, for example, Fourier series and Fourier integrals. Non-commutative harmonic analysis deals with the decomposition of functional space is a key representation of continuous groups of symmetries, when these symmetry sufficiently many (Frechet, when the so-called valid. Plancherel theorem). Representations are analogues of one-dimensional functional subspace of commutative case.
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