Duality

The mapping

$${\displaystyle x_{0}\mapsto f(x_{0})}$$

is a function, where ${\displaystyle x_{0}}$ is an argument of a function ${\displaystyle f.}$ At the same time, the mapping of a function to the value of the function at a point

$${\displaystyle f\mapsto f(x_{0})}$$

is a functional; here, ${\displaystyle x_{0}}$ is a parameter.

Provided that ${\displaystyle f}$ 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

$${\displaystyle f\mapsto I[f]=\int _{\Omega }H(f(x),f'(x),\ldots )\;\mu (\mathrm {d} x)}$$

form a special class of functionals. They map a function ${\displaystyle f}$ into a real number, provided that ${\displaystyle H}$ is real-valued. Examples include

• the area underneath the graph of a positive function ${\displaystyle f}$
  $${\displaystyle f\mapsto \int _{x_{0}}^{x_{1}}f(x)\;\mathrm {d} x}$$

  
• ${\displaystyle L^{p}}$ norm of a function on a set ${\displaystyle E}$
  $${\displaystyle f\mapsto \left(\int _{E}|f|^{p}\;\mathrm {d} x\right)^{1/p}}$$

  
• the arclength of a curve in 2-dimensional Euclidean space
  $${\displaystyle f\mapsto \int _{x_{0}}^{x_{1}}{\sqrt {1+|f'(x)|^{2}}}\;\mathrm {d} x}$$

  

Inner product spaces

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

For example, taking the inner product with a fixed function ${\displaystyle g\in L^{2}([-\pi ,\pi ])}$ defines a (linear) functional on the Hilbert space ${\displaystyle L^{2}([-\pi ,\pi ])}$ of square integrable functions on ${\displaystyle [-\pi ,\pi ]:}$

$${\displaystyle f\mapsto \langle f,g\rangle =\int _{[-\pi ,\pi ]}{\bar {f}}g}$$

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:

$${\displaystyle F(y)=\int _{x_{0}}^{x_{1}}y(x)\;\mathrm {d} x}$$

is local while

$${\displaystyle F(y)={\frac {\int _{x_{0}}^{x_{1}}y(x)\;\mathrm {d} x}{\int _{x_{0}}^{x_{1}}(1+[y(x)]^{2})\;\mathrm {d} x}}}$$

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.