Functional analysis

Functional analysis is the branch of mathematics that deals with the study of infinite-dimensional topological vector spaces and figures on such. In this analysis, topology and algebra are linked. Objective of these studies is to find abstract statements that can be applied to various practical problems. The functional analysis is the appropriate framework for the mathematical formulation of quantum mechanics and the investigation of partial differential equations.

Basic concepts

Of central importance are the terms

  • A functional illustration of vectors (e.g., functions) to scalar quantities and
  • Operator for a mapping from vectors to vectors. The term of the operator is actually much more general. It makes sense considering it but on algebraic and topological structured spaces such as topological, metric and normed vector spaces of all kinds

Examples of functionals are the terms episode limit, Standard, definite integral or distribution, Examples of operators are about differentiation, indefinite integral, quantum mechanical observables or shift operators for the consequences.

Basic concepts of analysis, such as continuity, derivatives, etc., are expanded in the functional analysis to functionals and operators. At the same time expands to the results of linear algebra (eg, the spectral theorem ) on topological linear spaces (for example, Hilbert spaces ), which is associated with very significant results.

The historical roots of functional analysis lie in the study of Fourier transform and related transformations, and the study of differential and integral equations. The word ' functional' goes back to the calculus of variations. As the founder of modern functional analysis Stefan Banach, Frigyes Riesz and Maurice René Fréchet apply.

Topological vector spaces

Basis of functional analysis are vector spaces over the real or complex numbers. The fundamental concept here is the topological vector space, which is characterized in that the vector space links are continuous, locally convex topological vector also more specific spaces and Fréchet spaces are investigated. Important statements are the Hahn- Banach theorem, the set of Baire and the Banach - Steinhaus. In particular, in the solution theory of partial differential equations play an important role in this, moreover, in the Fredholm theory.

Normed spaces, Banach spaces

The most important special case of locally convex topological vector spaces are normed vector spaces. Are these in addition completely, then they are called Banach spaces. More specifically considered to Hilbert spaces, in which the standard is produced by a dot. These spaces are essential for the mathematical formulation of the quantum mechanics. An important object of investigation are continuous linear operators on Banach or Hilbert spaces.

Hilbert spaces can be completely classified: exists for each cardinality of an orthonormal basis exactly ( up to isomorphism ) a Hilbert space into one body. Since finite-dimensional Hilbert spaces are detected by linear algebra and every morphism can be decomposed between Hilbert spaces in morphisms of Hilbert spaces with a countable orthonormal basis, we consider the functional analysis mainly Hilbert spaces with a countable orthonormal basis and their morphisms. These are isomorphic to the sequence space of all sequences with the property that the sum of the squares of all sequence elements is finite.

Banach spaces, however, are much more complex. For example, there is no practically useful general definition of a base, so can be bases from under base ( vector space ) described type (also called a Hamel basis) in the infinite-dimensional case, do not specify a constructive and are always uncountable ( see Theorem of Baire ). Generalizations of the Hilbert space orthonormal bases lead to the concept of Schauder basis, but not everyone has such a Banach space.

For every real number there is the Banach space "of all Lebesgue measurable functions whose -th power of the sum is a finite integral has " (see Lp space), this is just for a Hilbert space.

In the study of normed spaces, the investigation of the dual space is important. The dual space consists of all continuous linear functions from the normed space in his Skalarkörper, so in the real or complex numbers. The Bidual, ie the dual space of the dual space, need not be isomorphic to the original space, but there is always a natural monomorphism from a room in his Bidual. Is this special also surjective monomorphism, then one speaks of a reflexive Banach space.

The concept of derivation can be generalized to functions between Banach spaces for the so-called Fréchet derivative, so that the derivative is a continuous linear map at one point.

Operators, Banach algebras

While the Banach spaces and Hilbert spaces represent generalizations of finite-dimensional vector spaces of linear algebra, generalizing the continuous linear operators between them, the matrices of linear algebra. The diagonalization of matrices, which attempts to represent a matrix as a direct sum of dilates of so-called eigenvectors, spectral theorem for self-adjoint expands to or normal operators on Hilbert spaces, which leads to the mathematical formulation of quantum mechanics. The eigenvectors form the quantum mechanical states, the operators, the quantum mechanical observables.

As products of operators are operators again, we obtain algebras of operators, which are Banach spaces with the operator norm, so that for two operators and the multiplicative triangle inequality applies. This leads to the concept of Banach algebra whose most accessible representative of the C *-algebras and von Neumann algebras.

For the investigation of locally compact groups you shall use the Banach space of dimension integrable functions with respect to the hair, which is the convolution as multiplication in a Banach algebra. This establishes the harmonic analysis as a functional analytic approach to the theory of locally compact groups; the Fourier transform arises in this view, as a special case of Banach algebras studied in the theory of Gelfand transform.

Partial Differential Equations

The functional analysis provides an appropriate framework for solving theory of partial differential equations. Such equations have often the shape, wherein the desired function, and the right side functions in a field and a differential expression. There are also so-called boundary conditions that prescribe the behavior of this function on the boundary of. An example of such differential expression is about the Laplace operator, other important examples arise from the wave equation or the heat equation.

The differential expression is now viewed as an operator between spaces of differentiable functions, in the example of the Laplace operator about as operator between the space of twice continuously differentiable functions and the space of continuous functions on. Such spaces of differentiable in the classical sense function spaces, however, prove to be an exhaustive solution theory as inappropriate. By moving to a more general Differenzierbarkeitsbegriff (weak dissipation, distribution theory ) can be the differential expression, regarded as operator between Hilbert spaces, so-called Sobolev spaces, which consist of appropriate L2 functions. In this framework can be used in important cases satisfying theorems on existence and uniqueness of solutions prove. These questions will be such as the dependence of the right side, as well as questions of regularity, ie smoothness properties of the solution depending on smoothness properties of the right-hand side, studied with functional analytic methods. This can be further to more general classes of area, such as spaces of distributions generalize. If the right side is equal to the delta function and one has found a solution for this case, a so-called fundamental solution, one can construct in some cases, solutions for arbitrary right-hand sides by folding.

In practice, numerical methods for the approximate determination of solutions of such differential equations are used, such as the finite element method, especially if no solution can be given in closed form. Even with the construction of such approximations and the determination of the approximation functional analytic methods play an essential role.