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Duality

The mapping x 0 ↦ f ( x 0 ) {\displaystyle x_{0}\mapsto f(x_{0})} is a function, where x 0 {\displaystyle x_{0}} is an argument of a function f . {\displaystyle f.} At the same time, the mapping of a function to the value of the function at a point f ↦ f ( x 0 ) {\displaystyle f\mapsto f(x_{0})} is a functional; here, x 0 {\displaystyle x_{0}} is a parameter.

Provided that f {\displaystyle f} is a linear function from a vector space to the underlying scalar field, the above linear maps are dual to each other, and in functional analysis both are called linear functionals.

Definite integral

Integrals such as f ↦ I [ f ] = ∫ Ω H ( f ( x ) , f ′ ( x ) , … ) μ ( d x ) {\displaystyle f\mapsto I[f]=\int _{\Omega }H(f(x),f'(x),\ldots )\;\mu (\mathrm {d} x)} form a special class of functionals. They map a function f {\displaystyle f} into a real number, provided that H {\displaystyle H} is real-valued. Examples include

• the area underneath the graph of a positive function f {\displaystyle f} f ↦ ∫ x 0 x 1 f ( x ) d x {\displaystyle f\mapsto \int _{x_{0}}^{x_{1}}f(x)\;\mathrm {d} x}
• L p {\displaystyle L^{p}} norm of a function on a set E {\displaystyle E} f ↦ ( ∫ E | f | p d x ) 1 / p {\displaystyle f\mapsto \left(\int _{E}|f|^{p}\;\mathrm {d} x\right)^{1/p}}
• the arclength of a curve in 2-dimensional Euclidean space f ↦ ∫ x 0 x 1 1 + | f ′ ( x ) | 2 d x {\displaystyle f\mapsto \int _{x_{0}}^{x_{1}}{\sqrt {1+|f'(x)|^{2}}}\;\mathrm {d} x}

Inner product spaces

Given an inner product space X , {\displaystyle X,} and a fixed vector x → ∈ X , {\displaystyle {\vec {x}}\in X,} the map defined by y → ↦ x → ⋅ y → {\displaystyle {\vec {y}}\mapsto {\vec {x}}\cdot {\vec {y}}} is a linear functional on X . {\displaystyle X.} The set of vectors y → {\displaystyle {\vec {y}}} such that x → ⋅ y → {\displaystyle {\vec {x}}\cdot {\vec {y}}} is zero is a vector subspace of X , {\displaystyle X,} called the null space or kernel of the functional, or the orthogonal complement of x → , {\displaystyle {\vec {x}},} denoted { x → } ⊥ . {\displaystyle \{{\vec {x}}\}^{\perp }.}

For example, taking the inner product with a fixed function g ∈ L 2 ( [ − π , π ] ) {\displaystyle g\in L^{2}([-\pi ,\pi ])} defines a (linear) functional on the Hilbert space L 2 ( [ − π , π ] ) {\displaystyle L^{2}([-\pi ,\pi ])} of square integrable functions on [ − π , π ] : {\displaystyle [-\pi ,\pi ]:} f ↦ ⟨ f , g ⟩ = ∫ [ − π , π ] f ¯ g {\displaystyle f\mapsto \langle f,g\rangle =\int _{[-\pi ,\pi ]}{\bar {f}}g}

Locality

If a functional's value can be computed for small segments of the input curve and then summed to find the total value, the functional is called local. Otherwise it is called non-local. For example: F ( y ) = ∫ x 0 x 1 y ( x ) d x {\displaystyle F(y)=\int _{x_{0}}^{x_{1}}y(x)\;\mathrm {d} x} is local while F ( y ) = ∫ x 0 x 1 y ( x ) d x ∫ x 0 x 1 ( 1 + [ y ( x ) ] 2 ) d x {\displaystyle F(y)={\frac {\int _{x_{0}}^{x_{1}}y(x)\;\mathrm {d} x}{\int _{x_{0}}^{x_{1}}(1+[y(x)]^{2})\;\mathrm {d} x}}} is non-local. This occurs commonly when integrals occur separately in the numerator and denominator of an equation such as in calculations of center of mass.

Functional equations

Main article: Functional equation

The traditional usage also applies when one talks about a functional equation, meaning an equation between functionals: an equation F = G {\displaystyle F=G} between functionals can be read as an 'equation to solve', with solutions being themselves functions. In such equations there may be several sets of variable unknowns, like when it is said that an additive map f {\displaystyle f} is one satisfying Cauchy's functional equation: f ( x + y ) = f ( x ) + f ( y )  for all  x , y . {\displaystyle f(x+y)=f(x)+f(y)\qquad {\text{ for all }}x,y.}

Derivative and integration

See also: Calculus of variations

Functional derivatives are used in Lagrangian mechanics. They are derivatives of functionals; that is, they carry information on how a functional changes when the input function changes by a small amount.

Richard Feynman used functional integrals as the central idea in his sum over the histories formulation of quantum mechanics. This usage implies an integral taken over some function space.

See also

• Linear form – Linear map from a vector space to its field of scalars
• Optimization (mathematics) – Study of mathematical algorithms for optimization problemsPages displaying short descriptions of redirect targets
• Tensor – Algebraic object with geometric applications

• Axler, Sheldon (December 18, 2014), Linear Algebra Done Right, Undergraduate Texts in Mathematics (3rd ed.), Springer (published 2015), ISBN 978-3-319-11079-0
• Kolmogorov, Andrey; Fomin, Sergei V. (1957). Elements of the Theory of Functions and Functional Analysis. Dover Books on Mathematics. New York: Dover Books. ISBN 978-1-61427-304-2. OCLC 912495626.
• Lang, Serge (2002), "III. Modules, §6. The dual space and dual module", Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, pp. 142–146, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001
• Wilansky, Albert (October 17, 2008) [1970]. Topology for Analysis. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-46903-4. OCLC 227923899.
• Sobolev, V.I. (2001) [1994], "Functional", Encyclopedia of Mathematics, EMS Press
• Linear functional at the nLab
• Nonlinear functional at the nLab
• Rowland, Todd. "Functional". MathWorld.
• Rowland, Todd. "Linear functional". MathWorld.

References

1. Lang 2002, p. 142 "Let E be a free module over a commutative ring A. We view A as a free module of rank 1 over itself. By the dual module E∨ of E we shall mean the module Hom(E, A). Its elements will be called functionals. Thus a functional on E is an A-linear map f : E → A." - Lang, Serge (2002), "III. Modules, §6. The dual space and dual module", Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, pp. 142–146, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001 https://mathscinet.ams.org/mathscinet-getitem?mr=1878556 ↩

2. Kolmogorov & Fomin 1957, p. 77 "A numerical function f(x) defined on a normed linear space R will be called a functional. A functional f(x) is said to be linear if f(αx + βy) = αf(x) + βf(y) where x, y ∈ R and α, β are arbitrary numbers." - Kolmogorov, Andrey; Fomin, Sergei V. (1957). Elements of the Theory of Functions and Functional Analysis. Dover Books on Mathematics. New York: Dover Books. ISBN 978-1-61427-304-2. OCLC 912495626. https://search.worldcat.org/oclc/912495626 ↩

3. Wilansky 2008, p. 7. - Wilansky, Albert (October 17, 2008) [1970]. Topology for Analysis. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-46903-4. OCLC 227923899. https://search.worldcat.org/oclc/227923899 ↩

4. Wilansky 2008, p. 7. - Wilansky, Albert (October 17, 2008) [1970]. Topology for Analysis. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-46903-4. OCLC 227923899. https://search.worldcat.org/oclc/227923899 ↩

5. Axler (2014) p. 101, §3.92 - Axler, Sheldon (December 18, 2014), Linear Algebra Done Right, Undergraduate Texts in Mathematics (3rd ed.), Springer (published 2015), ISBN 978-3-319-11079-0 ↩

6. Khelemskii, A.Ya. (2001) [1994], "Linear functional", Encyclopedia of Mathematics, EMS Press https://www.encyclopediaofmath.org/index.php?title=Linear_functional&oldid=51214 ↩

7. Kolmogorov & Fomin 1957, pp. 62-63 "A real function on a space R is a mapping of R into the space R1 (the real line). Thus, for example, a mapping of Rn into R1 is an ordinary real-valued function of n variables. In the case where the space R itself consists of functions, the functions of the elements of R are usually called functionals." - Kolmogorov, Andrey; Fomin, Sergei V. (1957). Elements of the Theory of Functions and Functional Analysis. Dover Books on Mathematics. New York: Dover Books. ISBN 978-1-61427-304-2. OCLC 912495626. https://search.worldcat.org/oclc/912495626 ↩