Funktionalanalysis

the Funktionalanalysis is the branch of mathematics, which with the study of function areas concerns itself. The Funktionalanalysis can be regarded as part or as extension of the analysis.

The historical roots of the Funktionalanalysis lie in the study of the Fourier transform (and similar transformations) and the investigationof differential and integral equations. The word component “functionally” goes back on the variation calculation: Functional ones are functions, whose arguments are again functions.

Of modern view the Funktionalanalysis from the study of complete vector spaces over the real or complex numbers consists. Such areas are called Banachräume.An important example are Hilberträume, with which the standard is produced by a dot product. These areas are of fundamental importance for the mathematical formulation of quantum mechanics. In the Funktionalanalysis also Fréchet areas and other topological vector spaces are somewhat more generally examined, which do not have a standard.

An important investigation article are constant linear operators on Banach or Hilbert spaces.

Hilberträume can be completely classified: For each power (up to isomorphism) exactly a Hilbertraum exists to a basis to a body. There finite-dimensional Hilberträume by linear algebra to be seized and each Morphismusbetween Hilberträumen to be divided aleph_0 /math can in Morphismen of Hilberträumen <>of the dimension math< \> (To Aleph zero, see counting barness), one regards the Hilbertraum of the dimension Aleph zero and its Morphismen in the Funktionalanalysis mainly. This is the area <math> \ ell^2< /math> all consequences with the characteristic that the sumthe squares of all consequence members is finite.

Banachräume are many more complex against it. There is for example no practically usable general definition of a basis (e.g. if a basis is here usually non-constructional type described by under basis (vector space ) (also Hamelbasis called), cannot be indicated thus explicitly).

Foreach real number of p ≥ 1 gives it to the Banachraum of “all Lebesgue measurable functions, their PSE power of the amount a finite integral has " (see LP area).

With the study of Banachräumen the investigation of the dual area is an important part. The dual area consists of all constant linear functionsof the Banachraum into the real (or complex) numbers. As in linear algebra the dual area of the dual area does not have to be isomorphic also here to the original area, but there is always a Monomorphismus of an area in dual of its dual area.

The term of the derivative leaveson functions between Banachräumen it generalizes itself in such a way that the derivative is in one point a constant linear illustration.

literature

  • Yosida, K.: Functional analysis, Springer publishing house, 6th edition, 1980
  • Schechter, M.: Principles OF Functional analysis, AMS, 2nd edition, 2001
  • Hutson, V., Pym, J.S., CloudM.J.: Applications OF Functional analysis and operator Theory, 2nd edition, Elsevier Science, 2005, ISBN 0444517901
  • Dunford, N. and Schwartz, J.T. : Linear operator, general Theory, and OTHER 3 volumes, includes visualization charts
  • Brezis, H.: Analysis Fonctionnelle, Dunod
  • Sobolev, S.L.: Applications OF Functional analysis in MathematicalPhysics, AMS, 1963
  • Lebedev, L.P. and Vorovich, I.I.: Functional Anlysis in mechanic, Springer publishing house, 2002
  • alto, H.W.: Linear Funktionalanalysis, Springer publishing house, 2002
 

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