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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.
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In 1974, the big-budget film Hell River with Rod Taylor was screened with moderate success. This was also the case with Doctor Mladen (1975), The Peaks of Zelengora (1976), Maiden Bridge (1976), Dvoboj za južnu prugu (1978) and Arrive Before Daybreak (1978), the exception being Boško Buha, a film adaptation of eponymous novel by Dobrica ...
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.
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 the mathematical theory of Banach spaces, the closed range theorem gives necessary and sufficient conditions for a closed densely defined operator to have closed range. The theorem was proved by Stefan Banach in his 1932 Théorie des opérations linéaires .
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 ...
Let : a function between topological vector spaces is said to be a locally bounded function if every point of has a neighborhood whose image under is bounded. The following theorem relates local boundedness of functions with the local boundedness of topological vector spaces: