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A linear operator : between two topological vector spaces (TVSs) is called a bounded linear operator or just bounded if whenever is bounded in then () is bounded in . A subset of a TVS is called bounded (or more precisely, von Neumann bounded ) if every neighborhood of the origin absorbs it.
Operators on these spaces are known as sequence transformations. Bounded linear operators over a Banach space form a Banach algebra in respect to the standard operator norm. The theory of Banach algebras develops a very general concept of spectra that elegantly generalizes the theory of eigenspaces.
In its basic form, it asserts that for a family of continuous linear operators (and thus bounded operators) whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm. The theorem was first published in 1927 by Stefan Banach and Hugo Steinhaus, but it was also proven independently by Hans Hahn.
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A bounded operator: is not a bounded function in the sense of this page's definition (unless =), but has the weaker property of preserving boundedness; bounded sets are mapped to bounded sets (). This definition can be extended to any function f : X → Y {\displaystyle f:X\rightarrow Y} if X {\displaystyle X} and Y {\displaystyle Y} allow for ...
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.
An operator (,) is called strictly singular whenever it fails to be bounded below on any infinite-dimensional subspace of X. Denote by S S ( X , Y ) {\displaystyle {\mathcal {SS}}(X,Y)} the set of all strictly singular operators in B ( X , Y ) {\displaystyle B(X,Y)} .
Bounded convergence theorem (measure theory) Bounded inverse theorem (operator theory) Bourbaki–Witt theorem (order theory) Brahmagupta theorem (Euclidean geometry) Branching theorem (complex manifold) Brauer–Nesbitt theorem (representation theory of finite groups) Brauer–Siegel theorem (number theory) Brauer–Suzuki theorem (finite groups)