In mathematics, there are two theorems with the name "**open mapping theorem**". Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
Look up theorem in Wiktionary, the free dictionary. ...
## Functional analysis
In functional analysis, the **open mapping theorem**, also known as the **Banach-Schauder theorem**, is a fundamental result which states: if *A* : *X* → *Y* is a surjective continuous linear operator between Banach spaces *X* and *Y*, then *A* is an open map (i.e. if *U* is an open set in *X*, then *A*(*U*) is open in *Y*). Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...
In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ...
In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ...
In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...
In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...
In topology, an open map is a function between two topological spaces which maps open sets to open sets. ...
In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...
The proof uses the Baire category theorem. The Baire category theorem is an important tool in general topology and functional analysis. ...
The open mapping theorem has two important consequences: - If
*A* : *X* → *Y* is a bijective continuous linear operator between the Banach spaces *X* and *Y*, then the inverse operator *A*^{-1} : *Y* → *X* is continuous as well (this is called the *inverse mapping theorem*). - If
*A* : *X* → *Y* is a linear operator between the Banach spaces *X* and *Y*, and if for every sequence (*x*_{n}) in *X* with *x*_{n} → 0 and *Ax*_{n} → *y* it follows that *y* = 0, then *A* is continuous (Closed graph theorem). In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ...
In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ...
In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...
In mathematics, the closed graph theorem is a basic result in functional analysis which characterizes continuous linear operators between Banach spaces in terms of the operator graph. ...
## Complex analysis In complex analysis, the **open mapping theorem** states that if *U* is a connected open subset of the complex plane **C** and *f* : *U* → **C** is a non-constant holomorphic function, then *f* is an open map (i.e. it sends open subsets of *U* to open subsets of **C**). Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ...
Connected and disconnected subspaces of RÂ². The space A at top is connected; the shaded space B at bottom is not. ...
In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...
Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...
In topology, an open map is a function between two topological spaces which maps open sets to open sets. ...
The theorem for example implies that a non-constant holomorphic function cannot map an open disk *onto* a portion of a line. Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...
In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ...
### Proof
Blue dots represent zeros. Black spikes represent poles. The boundary of an open set is given by a dashed line. Note that all poles are exterior to the open set. First assume *f* is a non-constant holomorphic function and *U* is a connected open subset of the complex plane. If every point in *f*(*U*) is an interior point of *f*(*U*) then *f*(*U*) is open. Thus, if every point in *f*(*U*) is contained in a disk which is contained in *f*(*U*), then *f*(*U*) is open. Image File history File links No higher resolution available. ...
Wiktionary:Open - definition Open set (mathematics) Open (sport) - A type of competition in tennis and golf (among others) where entry is open to all qualifiers regardless of age. ...
A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ...
A spatial point is an entity with a location in space but no extent (volume, area or length). ...
In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ...
Around every point in *U*, there is a relevant ball in *U*. Consider an arbitrary *z*_{0} in *U*, and then consider its image point, *w*_{0} = *f*(*z*_{0}). Then *f*(*z*_{0}) − *w*_{0} = 0, making *z*_{0} a root of *f*(*z*) − *w*_{0}. The function *f*(*z*) − *w*_{0} may have another root at a distance *d*_{1} from *z*_{0}. Additionally, the distance from *z*_{0} to a point not in *U* shall be written *d*_{2}. Any ball *B* of radius less than the minimum of *d*_{1} and *d*_{2} will be contained in *U*, and at least one exists because *d*_{1},*d*_{2} > 0. In mathematics, a ball is the inside of a sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions, and for metric spaces in general. ...
In mathematics, the image of an element x in a set X under the function f : X â†’ Y, denoted by f(x), is the unique y in Y that is associated with x. ...
In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ...
Denote by *B*_{2} the ball around *w*_{0} with radius *e* whose elements are written *w*. By Rouché's theorem or the Argument principle, the function *f*(*z*) − *w*_{0} will have the same number of roots as *f*(*z*) − *w* for any *w* within a distance *e* of *f*(*z*_{0}). Let *z*_{1} be the root, or one of the roots of *f*(*z*) − *w* just shown to exist. Thus, for every *w* in *B*_{2}, there exists a *z*_{1} in *B* so that *f*(*z*_{1}) = *w*, The image of B_2 is a subset of the image of B, which is a subset of *f*(*U*). Remote Authentication Dial In User Service (RADIUS) is an AAA (authentication, authorization and accounting) protocol for applications such as network access or IP mobility. ...
In complex analysis, RouchÃ©s theorem tells us that if the complex-valued functions f and g are holomorphic inside and on some closed contour C, with |g(z)| < |f(z)| on C, then f and f + g have the same number of zeros inside C, where each zero is...
The contour C (black), the zeros of f (blue) and the poles of f (red). ...
Thus *w* is an interior point of *f*(*U*) for arbitrary *w*, and the theorem is proved. |