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Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, inner product, norm, or topology) and the linear functions defined on these spaces and suitably respecting these structures.
The image of the function is the set of all output values it may produce, that is, the image of . The preimage of f {\displaystyle f} , that is, the preimage of Y {\displaystyle Y} under f {\displaystyle f} , always equals X {\displaystyle X} (the domain of f {\displaystyle f} ); therefore, the former notion is rarely used.
A specific element x of X is a value of the variable, and the corresponding element of Y is the value of the function at x, or the image of x under the function. The image of a function, sometimes called its range, is the set of the images of all elements in the domain. [6] [7] [8] [9]
Image analysis or imagery analysis is the extraction of meaningful information from images; mainly from digital images by means of digital image processing techniques. [1] Image analysis tasks can be as simple as reading bar coded tags or as sophisticated as identifying a person from their face .
The image of a function () is the set of all values of f when the variable x runs in the whole domain of f. For a continuous (see below for a definition) real-valued function with a connected domain, the image is either an interval or a single value. In the latter case, the function is a constant function.
The image of a function is always a subset of the codomain of the function. [ 5 ] As an example of the two different usages, consider the function f ( x ) = x 2 {\displaystyle f(x)=x^{2}} as it is used in real analysis (that is, as a function that inputs a real number and outputs its square).
That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis. [1] Holomorphic functions are also sometimes referred to as regular functions. [2] A holomorphic function whose domain is the whole complex plane is called an entire function.
However, the exponential function is a holomorphic function with a nonzero derivative, but is not one-to-one since it is periodic. [ 2 ] The Riemann mapping theorem , one of the profound results of complex analysis , states that any non-empty open simply connected proper subset of C {\displaystyle \mathbb {C} } admits a bijective conformal map ...