Functional

This article covers the mathematical functional. For the adjective functional see functionality and function .

As a functional basis is known in mathematics is usually a function whose definition set as a subset of a vector space is included, while its target amount in the associated Skalarkörper is. ${\displaystyle V}$

The concept of functional is closely related to the mathematical branch of functional analysis , which derives its name from it, as it emerged from the study of such functionals. Here, the vector space examined is mostly a function space, i.e. a vector space, the elements of which are real or complex-valued functions, whereby these are assigned scalars by functional. The Lebesgue integral can be regarded as an important example of such a functional . ${\displaystyle V}$

This article deals with the (most studied) cases in which the scalar field is based on the field of real numbers or the field of complex numbers and the definition set of the respective functional coincides with the vector space . As a fundamental distinction, it makes sense to consider linear and nonlinear functionals separately, since these two types of functional are treated in very different ways in mathematics. ${\displaystyle \mathbb {K} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle V}$

definition

Be a - vector space with . A functional is an image ${\displaystyle V}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle \mathbb {K} \in \{\mathbb {R} ,\mathbb {C} \}}$ ${\displaystyle T}$ ${\displaystyle T\colon V\to \mathbb {K} .}$

Examples

A linear functional on the vector space of the functions on the real axis is the evaluation functional at zero ${\displaystyle {\mathcal {F}}(\mathbb {R} ,\mathbb {K} )}$

${\displaystyle \delta \colon {\mathcal {F}}(\mathbb {R} ,\mathbb {K} )\to \mathbb {K} ,\quad f\mapsto \delta [f]=f(0).}$

This functional is called delta distribution or Dirac delta.

A nonlinear functional on the vector space of the curves in space, especially here continuously differentiable functions from to , is the arc length functional ${\displaystyle \left[0,1\right]}$ ${\displaystyle \mathbb {R} ^{3}}$

${\displaystyle L\colon {\mathcal {C}}^{1}\left(\left[0,1\right],\mathbb {R} ^{3}\right)\to \mathbb {R} ,\quad c\mapsto L[c]=\int _{0}^{1}\left\Vert {\dot {c}}(t)\right\Vert \ \mathrm {d} t.}$

Linear functionals

In most areas of functional analysis, for example in the theory of topological vector spaces , the term functional (without any further addition) is used as a synonym for linear functionals. By definition, such a functional is a linear form , that is, a linear mapping of the vector space into its scalar body . The set of all these functionals is in turn in natural form a vector space over the same field by defining point-wise for two functionals and via addition and scalar multiplication , i.e. H. ${\displaystyle V}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle \mathbb {K} }$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle V}$

${\displaystyle (f+g)(x):=f(x)+g(x)\quad (\lambda f)(x):=\lambda (f(x)),x\in V.}$

The vector space of the linear functionals on the vector space is called the algebraic dual space and is often referred to as. ${\displaystyle V}$ ${\displaystyle V^{*}}$

Examples of dual spaces

For the vector space , the dual space is canonically isomorphic to the vector space itself, ie . The canonical isomorphism is conveyed via the standard scalar product: ${\displaystyle V=\mathbb {R} ^{n}}$ ${\displaystyle V\cong V^{*}}$ ${\displaystyle I\colon \mathbb {R} ^{n}\rightarrow (\mathbb {R} ^{n})^{*}}$

${\displaystyle (I(x))(y):=\langle x,y\rangle =\sum _{i=1}^{n}x_{i}\cdot y_{i}.}$

The same applies to the vector space as in the first case, however the canonical mapping is semilinear in this case : ${\displaystyle V=\mathbb {C} ^{n}}$ ${\displaystyle I\colon \mathbb {C} ^{n}\rightarrow (\mathbb {C} ^{n})^{*}}$

${\displaystyle (I(x))(y):=\langle x,y\rangle =\sum _{i=1}^{n}{\overline {x_{i}}}\cdot y_{i}.}$

The dual space is the same size in this case, but has a different scalar multiplication with respect to the canonical mapping. In the sense of linear algebra one also says: The dual space is canonically isomorphic to the complex conjugated vector space.

For general finite-dimensional vector spaces, by choosing a basis and applying the first two cases, one can show that the dual space always has the same dimension as the original space. The mappings between the vector space and the dual space are then generally not canonical.

For infinitely dimensional vector spaces the case is much more complicated. In some important cases, e.g. B. for Hilbert spaces , the vector space is a canonical subspace, but in general this does not apply. The algebraic dual space of an infinitely dimensional vector space also has always larger dimensions (in the sense of the cardinality of an algebraic basis) than the original space.

Continuous linear functionals

As just seen, the algebraic dual space of an infinitely dimensional vector space is always greater than or equal to the original vector space. Last but not least, the aim of functional analysis is to extend the methods of multidimensional analysis to infinite-dimensional spaces and in particular to examine concepts such as convergence , continuity and differentiability . Therefore, a priori only vector spaces are considered which have at least one topological structure , i.e. the topological vector spaces . They include, among other things, all standardized vector spaces and in particular the Banach and Hilbert spaces.

In general, not all linear functionals are continuous in a topological vector space. The continuous linear functionals within the algebraic dual space, i.e. the given continuous linear forms, form a linear subspace of . This is the topological dual space of , which is one of the main objects in functional analysis. It is usually identified with the designation , but some authors also use the same designation as the algebraic dual space, i.e. also with . ${\displaystyle V}$ ${\displaystyle V^{*}}$ ${\displaystyle V}$ ${\displaystyle V'}$ ${\displaystyle V^{*}}$

Examples of topological dual spaces

For finite-dimensional vector spaces there is a natural topology ( norm topology ) that arises from the Euclidean norm (more precisely: from any Euclidean norm, if one chooses a basis). This is precisely the topology on which normal standard analysis is based, and in this every linear functional is continuous. That is, the algebraic dual space is equal to the topological dual space.

In the infinite-dimensional case, the topological dual space is (almost) always a real subspace of the algebraic dual space.

In normalized vector spaces a functional is continuous if and only if it is bounded, that is ${\displaystyle f}$

${\displaystyle \sup _{\|x\|\leq 1}|f(x)|<\infty .}$

The topological dual space is then automatically a Banach space with the supremum norm given above .

In Hilbert spaces, the topological dual space is canonically identifiable with the original space ( Fréchet-Riesz's theorem ). As in the finite-dimensional case, the identification takes place via the scalar product:

${\displaystyle (I(y)):=\langle \cdot ,y\rangle .}$

The topological dual space of the space of infinitely often continuously differentiable functions with a compact support on the real axis (the so-called test functions) with a certain topology (not explained in detail here) is called the space of distributions . The example of the Dirac delta functional mentioned above is also located in this space.

Nonlinear Functionals

Historically, nonlinear functionals appeared for the first time in the calculus of variations . Their studies are fundamentally different from that of the linear functionals described above. In the calculus of variations, for example, the goal is to determine the extreme points of such functional points. For this purpose one needs a generalization of the derivative concept of multidimensional analysis, i. H. a definition of the differential of the functional. In the calculus of variations and in its applications, this differential is known as the derivative of variation. B. by the Fréchet derivative and the Gateaux derivative .

Examples of nonlinear functionals

Nonlinear functionals on curve spaces are of great importance in the application, especially in classical mechanics , as in the example of the arc length functional above. One can easily generalize this example.

Again we consider a curve space and additionally a continuously differentiable function . With this we define: ${\displaystyle F\colon \mathbb {R} ^{3}\to \mathbb {R} }$

${\displaystyle L\colon {\mathcal {C}}([0,1],\mathbb {R} ^{3})\to \mathbb {R} \quad c\mapsto L[c]=\int _{0}^{1}F(c(t))\mathrm {d} t.}$

It is said that the functional has a stationary point on a curve if the differential ${\displaystyle L}$ ${\displaystyle c}$

${\displaystyle {\mathcal {D}}L_{c}(h):={\frac {\mathrm {d} }{\mathrm {d} s}}(F(c+sh))}$

for all variations , that is, curves with start and end point at zero, disappears. This is exactly the case here when the (ordinary) differential of vanishes on the whole curve : ${\displaystyle h}$ ${\displaystyle F}$ ${\displaystyle c}$

${\displaystyle DF(c(t))=0.}$

If one considers a curve space and doubly continuous functions with two arguments , one obtains analogously: ${\displaystyle F\colon \mathbb {R} ^{3}\times \mathbb {R} ^{3}\to \mathbb {R} }$

${\displaystyle L\colon {\mathcal {C}}^{1}([0,1],\mathbb {R} ^{3})\to \mathbb {R} \quad c\mapsto L[c]=\int _{0}^{1}F(c(t),{\dot {c}}(t))\mathrm {d} t,}$

stationary points on a curve when the differential ${\displaystyle c}$

${\displaystyle {\mathcal {D}}L_{c}(h):={\frac {\mathrm {d} }{\mathrm {d} s}}(F(c+sh))}$

for all variations , disappears. In this simple case this is the case if and only if the Euler-Lagrange equation holds, i.e. H. ${\displaystyle h}$ ${\displaystyle c}$

${\displaystyle D_{1}F(c(t))-{\frac {\mathrm {d} }{\mathrm {d} t}}D_{2}F(c(t))=0.}$

Occasionally, especially in application-oriented texts, a functional dependency (in contrast to the usual functional dependency) is written with square or curly brackets instead of round brackets, possibly naming a dummy argument of the argument function, i.e. or instead of . ${\displaystyle I[f]}$ ${\displaystyle I{f(x)}}$ ${\displaystyle I(f)}$

literature

• Philippe G. Ciarlet : Linear and Nonlinear Functional Analysis with Applications . Society for Industrial and Applied Mathematics , Philadelphia, PA 2013, ISBN 978-1-61197-258-0 ( MR3136903 ).
• Gerd Fischer : Linear Algebra . An introduction for first-year students. 16th, revised and expanded edition. Vieweg + Teubner , Wiesbaden 2008, ISBN 978-3-8348-0428-0 , p. 280-281 .
• Siegfried Großmann : functional analysis with regard to application in physics , AULA Verlag, 4th edition 1988, ISBN 978-3-89104-479-7
• Harro Heuser : Functional Analysis . Theory and application (=  mathematical guidelines ). 4th edition. BG Teubner , Wiesbaden 2006, ISBN 978-3-8351-0026-8 ( MR2380292 ).
• Friedrich Hirzebruch , Winfried Scharlau : Introduction to functional analysis . Unchanged reprint of the 1971 edition (=  BI university paperbacks . Volume 296 ). Bibliographisches Institut, Mannheim [u. a.] 1991, ISBN 3-411-00296-4 ( MR1183466 ).
• Peter Kosmol: Optimization and Approximation (=  De Gruyter Studies ). 2nd Edition. Walter de Gruyter & Co. , Berlin 2010, ISBN 978-3-11-021814-5 ( MR2599674 ).
• AP Robertson, WJ Robertson: Topological vector spaces . Translation from English by Horst S. Holdgrün (= BI university paperbacks  . 164 / 164a). Bibliographisches Institut, Mannheim 1967 ( MR0209926 ).
• Dirk Werner : Functional Analysis . 5th, exp. Edition. Springer-Verlag Berlin Heidelberg, Berlin 2005, ISBN 3-540-21381-3 .

Individual evidence

1. ↑ So here in particular the general linear forms on vector spaces over arbitrary bodies are not considered and neither are other functional types such as convex functionals discussed.