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Mathematics for Physics

226 Functional

A mapping from a vector space into the scalars that is used to assign a number to each vector in its domain.

Note: When the mapping is linear it is called a linear functional.

definition [d] (Functional) From Kreyszig: a functional is an operator whose range lies on the real line \(\mathbb{R}\) or in the complex plane \(\mathbb{C}\).

where

  • an operator is a mapping between vector spaces.
  • the range of the functional is a set of scalars in \(\mathbb{R}\) or \(\mathbb{C}\).

definition [d] (Functional) From Griffel: a functional on a real or complex normed space is a map from the space to the real or complex numbers.

where

  • the domain is a real or complex normed space.
  • the value of the functional at each point is a scalar.

definition [d] (Linear Functional) From Kreyszig: a linear functional \(f\) is a linear operator with domain in a vector space \(X\) and range in the scalar field \(K\) of \(X\); thus

  • \(f : \mathcal{D}(f) \rightarrow K\) ,

where

  • \(K = \mathbb{R}\) if \(X\) is real and \(K = \mathbb{C}\) if \(X\) is complex.
  • \(\mathcal{D}(f)\) is the domain of \(f\).

definition [d] (Linear Functional) From Griffel: a linear functional \(f\) is a functional such that

  • \(f(ax + by) = af(x) + bf(y)\)

for any scalars \(a,b\) and any \(x,y\) in the space.

where

  • \(f\) maps the space into the scalars.
  • \(a,b\) are real or complex numbers matching the scalar field of the space.

226.1 Elementary Example

226.1.1 Simple

A fixed force maps each displacement vector to a work value. On three displacements in \(\mathbb{R}^{3}\):

\[ F = (2,\ 0,\ 1) \]

\[ V = \{ v_{1},\ v_{2},\ v_{3} \} \]

\[ v_{1} = (1,0,0),\quad v_{2} = (0,1,0),\quad v_{3} = (0,0,3) \]

\[ W(v) = F \cdot v \]

\[ W(v_{1}) = 2,\quad W(v_{2}) = 0,\quad W(v_{3}) = 3 \]

where

  • \(W\) is the work functional.
  • \(F\) is the fixed force vector.
  • \(v\) is a displacement vector in \(V\).

226.1.2 General

The classical action maps a path \(x(t)\) to a real number by integrating kinetic energy minus potential energy.

\[ S[x] = \int_{a}^{b}\left(\dfrac{m}{2}\left|\dfrac{dx}{dt}\right|^{2} - V\bigl(x(t)\bigr)\right)\, dt \]

where

  • \(S\) is the action functional.
  • \(x(t)\) is a path over the time interval \([a,b]\).
  • \(m\) is the mass.
  • \(V\) is the potential energy.

226.2 References

  1. Kreyszig, E. Introductory Functional Analysis with Applications. Wiley, 1989. — §2.8: functional as operator with range in \(\mathbb{R}\) or \(\mathbb{C}\); linear functional \(f:\mathcal{D}(f)\rightarrow K\).
  2. Griffel, D. H. Applied Functional Analysis. Ellis Horwood, 1981. — Def. 7.52; fixed-\(u\) functional \(\ell_{u}(x)=u\cdot x\) on \(\mathbb{R}^{3}\).
  3. Hubbard, J. H., & Hubbard, B. B. Vector Calculus, Linear Algebra, and Differential Forms. — work \(W_{F}=F\cdot v\) as a scalar assigned to a displacement.
  4. Hall, B. C. Quantum Theory for Mathematicians. Springer, 2013. — action \(S[x]=\int_{a}^{b}\bigl(\dfrac{m}{2}\left|\dfrac{dx}{dt}\right|^{2}-V(x(t))\bigr)\,dt\).
  5. Hassani, S. Mathematical Physics. Springer. — integration as a linear functional on continuous functions.