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In mathematics, the isometry group of a metric space is the set of all bijective isometries (that is, bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. [1]
A global isometry, isometric isomorphism or congruence mapping is a bijective isometry. Like any other bijection, a global isometry has a function inverse. The inverse of a global isometry is also a global isometry. Two metric spaces X and Y are called isometric if there is a bijective isometry from X to Y.
In mathematical functional analysis a partial isometry is a linear map between Hilbert spaces such that it is an isometry on the orthogonal complement of its kernel. The orthogonal complement of its kernel is called the initial subspace and its range is called the final subspace. Partial isometries appear in the polar decomposition.
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem [1] (named after Stefan Banach and Juliusz Schauder), is a fundamental result that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map.
In the area of mathematics known as functional analysis, James' space is an important example in the theory of Banach spaces and commonly serves as useful counterexample to general statements concerning the structure of general Banach spaces.
In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) is either of two things: a way of representing commutative Banach algebras as algebras of continuous functions; the fact that for commutative C*-algebras, this representation is an isometric isomorphism.
The mapping j is linear, and it is isometric by the Hahn–Banach theorem. Another generalization was given by Kleiber and Pervin (1969): a metric space of density equal to an infinite cardinal α is isometric to a subspace of C 0 ([0,1] α, R), the space of real continuous functions on the product of α copies of the unit interval.
In mathematics, particularly in the area of functional analysis and topological vector spaces, the vague topology is an example of the weak-* topology which arises in the study of measures on locally compact Hausdorff spaces. Let be a locally compact Hausdorff space.