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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.
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.
Since the graph of T is closed, the proof reduces to the case when : is a bounded operator between Banach spaces. Now, T {\displaystyle T} factors as X → p X / ker T → T 0 im T ↪ i Y {\displaystyle X{\overset {p}{\to }}X/\operatorname {ker} T{\overset {T_{0}}{\to }}\operatorname {im} T{\overset {i}{\hookrightarrow }}Y} .
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 ...
The commutators [δ h, ∆ 1] are second order differential operators from H k+1 (Ω) to H k−1 (Ω). Their operators norms are uniformly bounded for h near 0; for the computation can be carried out on the annulus where the commutator just replaces the coefficients of ∆ 1 by their difference quotients composed with S h. On the other hand, v ...
A T ∈ L(H) is a Fredholm operator if and only if T is invertible modulo compact perturbation, i.e. TS = I + C 1 and ST = I + C 2 for some bounded operator S and compact operators C 1 and C 2. In other words, an operator T ∈ L(H) is Fredholm, in the classical sense, if and only if its projection in the Calkin algebra is invertible.
In operator theory, a bounded operator T: X → Y between normed vector spaces X and Y is said to be a contraction if its operator norm ||T || ≤ 1. This notion is a special case of the concept of a contraction mapping, but every bounded operator becomes a contraction after suitable scaling. The analysis of contractions provides insight into ...
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.