# Functional analysis

The functional analysis is the branch of mathematics that deals with the study of infinite-dimensional topological vector spaces involved and illustrations on such. Here analysis , topology and algebra are linked. The aim of these investigations is to find abstract statements that can be applied to various concrete problems. Functional analysis is the suitable framework for the mathematical formulation of quantum mechanics and for the investigation of partial differential equations .

## Basic terms

The terms are of central importance

• Functional for mapping vectors (e.g. functions ) to scalar sizes and
• Operator for mapping vectors to vectors. The concept of the operator is actually much more general. However, it makes sense to look at them in algebraically and topologically structured spaces, such as B. topological, metric or standardized vector spaces of all kinds.

Examples of functional are the terms sequence limit value , norm , definite integral or distribution , examples of operators are differentiation , indefinite integral , quantum mechanical observable or shift operators for sequences.

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

The historical roots of functional analysis lie in the study of the Fourier transformation and similar transformations and the investigation of differential and integral equations . The word component “functional” goes back to the calculus of variations . Stefan Banach , Frigyes Riesz and Maurice René Fréchet are considered the founders of modern functional analysis .

## Topological vector spaces

Functional analysis is based on vector spaces over real or complex numbers. The basic concept here is the topological vector space, which is characterized by the fact that the vector space links are continuous; locally convex topological vector spaces and Fréchet spaces are also examined more specifically . Important statements are the Hahn-Banach theorem , the Baire theorem and the Banach-Steinhaus theorem . In particular in the solution theory of partial differential equations , these play an important role, moreover in the Fredholm theory .

## Standardized spaces, Banach spaces

The most important special case of locally convex topological vector spaces are normalized vector spaces . If these are also complete , then they are called Banach spaces . Hilbert spaces are considered even more specifically , in which the norm is generated by a scalar product . These spaces are fundamental to the mathematical formulation of quantum mechanics . An important subject of investigation are continuous linear operators on Banach or Hilbert spaces.

Hilbert spaces can be fully classified: for every thickness of an orthonormal basis there is exactly one Hilbert space for a body (except for isomorphism ) . Since finite-dimensional Hilbert spaces are covered by linear algebra and every morphism between Hilbert spaces can be decomposed into morphisms of Hilbert spaces with a countable orthonormal basis, one considers mainly Hilbert spaces with a countable orthonormal basis and their morphisms in functional analysis. These are isomorphic to the sequence space of all sequences with the property that the sum of the squares of all sequence members is finite. ${\ displaystyle \ ell ^ {2}}$

Banach spaces, on the other hand, are much more complex. For example, there is no general definition of a basis that can be used in practice, so bases of the type described under basis (vector space) (also called Hamel basis ) cannot be given constructively in the infinite-dimensional case and are always uncountable (see Baire's theorem ). Generalizations of the Hilbert space orthonormal bases lead to the concept of the shudder base , but not every Banach space has one.

For every real number there is the Banach space "of all Lebesgue-measurable functions whose -th power of the amount has a finite integral" (see L p -space ), this is exactly for a Hilbert space. ${\ displaystyle p \ geq 1}$${\ displaystyle p}$${\ displaystyle p = 2}$

When studying standardized spaces, it is important to examine the dual space . The dual space consists of all continuous linear functions from the normalized space into its scalar body , i.e. into the real or complex numbers. The bidual , i.e. the dual space of the dual space, does not have to be isomorphic to the original space, but there is always a natural monomorphism of a space in its bidual. If this special monomorphism is also surjective , then one speaks of a reflexive Banach space .

The term derivation can be generalized to functions between Banach spaces to the so-called Fréchet derivation , so that the derivation at one point is a continuous linear mapping.

## Operators, Banach algebras

While the Banach spaces or Hilbert spaces represent generalizations of the finite-dimensional vector spaces of linear algebra, the continuous, linear operators between them generalize the matrices of linear algebra. The diagonalization of matrices, which a matrix tries to represent as a direct sum of stretchings of so-called eigenvectors , expands to the spectral theorem for self-adjoint 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 with the operator norm are Banach spaces, allowing for two operators and the multiplicative triangle inequality holds. This leads to the concept of Banach algebra , the most accessible representatives of which are the C * algebras and Von Neumann algebras . ${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle \ | A \ circ B \ | \ leq \ | A \ | \ | B \ |}$

For the investigation of locally compact groups one uses the Banach space of the functions integrable with respect to the hair measure , which becomes a Banach algebra with the convolution as multiplication. This justifies the harmonic analysis as a functional analytical approach to the theory of locally compact groups; The Fourier transformation results from this point of view as a special case of the Gelfand transformation examined in the Banach algebra theory . ${\ displaystyle G}$${\ displaystyle L ^ {1} (G)}$

## Partial differential equations

Functional analysis offers a suitable framework for the solution theory of partial differential equations. Such equations often have the form where the function sought and the right hand side are functions in a domain and is a differential expression. In addition, there are so-called boundary conditions that prescribe the behavior of the function sought on the boundary of . An example of such a differential expression is the Laplace operator ; other important examples result from the wave equation or the heat conduction equation . ${\ displaystyle you \, = f}$${\ displaystyle u}$${\ displaystyle f}$${\ displaystyle \ Omega \ subset \ mathbb {R} ^ {n}}$${\ displaystyle D}$${\ displaystyle u}$${\ displaystyle \ partial \ Omega}$${\ displaystyle \ Omega}$ ${\ displaystyle D = {\ frac {\ partial ^ {2}} {\ partial x_ {1} ^ {2}}} + \ dotsb + {\ frac {\ partial ^ {2}} {\ partial x_ {n } ^ {2}}}}$

The differential expression is now viewed as an operator between spaces of differentiable functions, in the example of the Laplace operator, for example, as an operator between the space of twice continuously differentiable functions and the space of continuous functions . Such spaces of function spaces that can be differentiated in the classical sense turn out to be unsuitable for an exhaustive solution theory. By moving to a more general concept of differentiability ( weak derivative , distribution theory ), one can view the differential expression as an operator between Hilbert spaces, so-called Sobolew spaces , which consist of suitable L 2 functions. In this context, in important cases, satisfactory theorems about the existence and uniqueness of solutions can be proven. For this purpose, questions such as the dependence on the right side , as well as questions about the regularity, i.e. smoothness properties of the solution depending on the smoothness properties of the right side , are investigated using functional analytical methods. This can be further generalized to more general room classes, such as rooms of distributions. If the right-hand side is the same as the delta distribution and a solution has been found for this case, a so-called fundamental solution, in some cases solutions for any right-hand side can be constructed using convolution . ${\ displaystyle \ Omega}$${\ displaystyle f}$${\ displaystyle u}$${\ displaystyle f}$${\ displaystyle f}$

In practice, numerical methods are used to approximate solutions of such differential equations, such as the finite element method , especially when no solution can be given in closed form. Functional analytical methods also play an essential role in the construction of such approximations and the determination of the approximation quality .

## literature

The books Alt (2006) and Heuser (1992) offer an introduction and a first overview of “classical” sentences of functional analysis. In doing so, physical applications are repeatedly discussed as a common thread. Heuser has exercises for each chapter, most of which are outlined in the appendix. The last chapter “A look at the emerging analysis” describes the most important steps in the historical development towards today's functional analysis.