Compact operatorIn functional analysis, a branch of mathematics, a compact operator is a linear operator , where are normed vector spaces, with the property that maps bounded subsets of to relatively compact subsets of (subsets with compact closure in ). Such an operator is necessarily a bounded operator, and so continuous. Some authors require that are Banach, but the definition can be extended to more general spaces. Any bounded operator that has finite rank is a compact operator; indeed, the class of compact operators is a natural generalization of the class of finite-rank operators in an infinite-dimensional setting.
Basis functionIn mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors. In numerical analysis and approximation theory, basis functions are also called blending functions, because of their use in interpolation: In this application, a mixture of the basis functions provides an interpolating function (with the "blend" depending on the evaluation of the basis functions at the data points).
Operator (computer programming)In computer programming, operators are constructs defined within programming languages which behave generally like functions, but which differ syntactically or semantically. Common simple examples include arithmetic (e.g. addition with +), comparison (e.g. "greater than" with >), and logical operations (e.g. AND, also written && in some languages). More involved examples include assignment (usually = or :=), field access in a record or object (usually .), and the scope resolution operator (often :: or .).
Operators in C and C++This is a list of operators in the C and C++ programming languages. All the operators (except typeof) listed exist in C++; the column "Included in C", states whether an operator is also present in C. Note that C does not support operator overloading. When not overloaded, for the operators &&, ||, and , (the comma operator), there is a sequence point after the evaluation of the first operand. C++ also contains the type conversion operators const_cast, static_cast, dynamic_cast, and reinterpret_cast.
Particle number operatorIn quantum mechanics, for systems where the total number of particles may not be preserved, the number operator is the observable that counts the number of particles. The number operator acts on Fock space. Let be a Fock state, composed of single-particle states drawn from a basis of the underlying Hilbert space of the Fock space. Given the corresponding creation and annihilation operators and we define the number operator by and we have where is the number of particles in state . The above equality can be
Operator overloadingIn computer programming, operator overloading, sometimes termed operator ad hoc polymorphism, is a specific case of polymorphism, where different operators have different implementations depending on their arguments. Operator overloading is generally defined by a programming language, a programmer, or both. Operator overloading is syntactic sugar, and is used because it allows programming using notation nearer to the target domain and allows user-defined types a similar level of syntactic support as types built into a language.
Operator theoryIn mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators, and consideration may be given to nonlinear operators. The study, which depends heavily on the topology of function spaces, is a branch of functional analysis. If a collection of operators forms an algebra over a field, then it is an operator algebra.
Spaces of test functions and distributionsIn mathematical analysis, the spaces of test functions and distributions are topological vector spaces (TVSs) that are used in the definition and application of distributions. Test functions are usually infinitely differentiable complex-valued (or sometimes real-valued) functions on a non-empty open subset that have compact support. The space of all test functions, denoted by is endowed with a certain topology, called the , that makes into a complete Hausdorff locally convex TVS.
