Examples of "functionals"
Advanced mesh refinement has been introduced via functionals. Functionals allow the ability to generate grids and provide mesh adaptation. Some advanced functionals include the Winslow and modified Liao functionals.
Functionals are allowed (and, usually, welcome) to use the services of other functionals.
due to linearity of scalar multiples of functionals and pointwise linearity of sums of functionals.  Then
These functionals {"b*"} are called biorthogonal functionals associated to the basis {"b"}. When the basis {"b"} is normalized, the coordinate functionals {"b*"} have norm ≤ 2"C" in the continuous dual of "V".
Statistics that can be represented as functionals formula_1 of the empirical distribution function formula_2 are called "statistical functionals". Differentiability of the functional "T" plays a key role in the von Mises approach; thus von Mises considers "differentiable statistical functionals".
Functionals of this type are, for example, TPSS and the Minnesota Functionals. These functionals include a further term in the expansion, depending on the density, the gradient of the density and the Laplacian (second derivative) of the density.
The continuous linear functionals on "B"("H") for the weak, strong, and strong (operator) topologies are the same, and are the finite linear combinations of the linear functionals
The ratio of customs officers to all functionals is surprisingly high. For example, Moscow has 17 customs officers for only 700 functionals.
The functionals they define have the following form:
formula_31 induces continuous linear functionals on formula_2, but not all.
Most functionals have regular human jobs (doctor, restaurant owner, etc.). Their services are available to other functionals and the select humans who are aware of their existence. These humans are usually the social elite, such as politicians, moguls, artists, and others. The relations between the social elite and the functionals differ from world to world.
Functionals began to appear on our Earth in the 1810s.
which extends verbatim to positive functionals on C*-algebras:
It follows from the Banach–Steinhaus theorem that the linear mappings are uniformly bounded by some constant . Let denote the coordinate functionals which assign to every in the coordinate of in the above expansion. They are called biorthogonal functionals. When the basis vectors have norm , the coordinate functionals have norm in the dual of .
Difficulties in expressing the exchange part of the energy can be relieved by including a component of the exact exchange energy calculated from Hartree–Fock theory. Functionals of this type are known as hybrid functionals.
The M06 suite of functionals, are a set of four meta-hybrid GGA and meta-GGA DFT functionals. They are constructed with empirical fitting of their parameters, but constraining to the uniform electron gas.
Minnesota Functionals (M"yz") are a group of approximate exchange-correlation energy functionals in density functional theory (DFT). They are developed by the group of Prof. Donald Truhlar at the University of Minnesota.
On every non-reflexive Banach space , there exist continuous linear functionals that are not "norm-attaining". However, the Bishop–Phelps theorem states that norm-attaining functionals are norm dense in the dual of .
In statistics, the plug-in principle is the method of estimation of functionals of a population distribution by evaluating the same functionals at the empirical distribution based on a sample.
The calculus of variations deals with functionals formula_1, where formula_2 is some function space and formula_3. The main interest of the subject is to find "minimizers" for such functionals, that is, functions formula_4 such that:formula_5