A densely defined operator is a mathematical object that maps a certain set of inputs, known as the domain, to a set of outputs, known as the range. It is called "densely defined" because the domain is a dense subset of the underlying space, meaning that for any point in the space, there is a sequence of points in the domain that converge to it.
A densely defined operator can be thought of as a generalization of a continuous operator. While a continuous operator must map every point in the domain to a point in the range, a densely defined operator only needs to map a dense subset of the domain to the range. In other words, a densely defined operator can have "holes" in its domain, while a continuous operator cannot.
Some common examples of densely defined operators include differentiation and integration operators, as well as operators that map functions to their Fourier or Laplace transforms. In functional analysis, densely defined operators are often used to study the properties of certain function spaces, such as the space of continuous functions or the space of square-integrable functions.
Densely defined operators play a crucial role in many areas of mathematics, including functional analysis, differential equations, and calculus of variations. They provide a powerful framework for studying the properties of functions and function spaces, and have applications in physics, engineering, and other scientific fields.
The main properties of a densely defined operator include linearity, continuity, and closedness. Linearity means that the operator satisfies the superposition principle, while continuity means that small changes in the input result in small changes in the output. Closedness is a topological property that ensures the operator is well-behaved with respect to the underlying space.