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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 the context of computer science, the C Bounded Model Checker (CBMC) is a bounded model checker for C programs. [1] It was the first such tool. [2] CBMC has participated in the Competition on Software Verification (SV-COMP) in the years 2014–2022. [3] It came in first in at least one category in 2014, 2015, and 2017.
A Banach space is said to have bounded approximation property (BAP), if it has the -AP for some . A Banach space is said to have metric approximation property ( MAP ), if it is 1-AP. A Banach space is said to have compact approximation property ( CAP ), if in the definition of AP an operator of finite rank is replaced with a compact operator.
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.
The family of finite-rank operators () on a Hilbert space form a two-sided *-ideal in (), the algebra of bounded operators on . In fact it is the minimal element among such ideals, that is, any two-sided *-ideal I {\displaystyle I} in L ( H ) {\displaystyle L(H)} must contain the finite-rank operators.
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.
The space of bounded linear operators B(X) on a Banach space X is an example of a unital Banach algebra. Since the definition of the spectrum does not mention any properties of B ( X ) except those that any such algebra has, the notion of a spectrum may be generalised to this context by using the same definition verbatim.
Since the graph of T is closed, the proof reduces to the case when : is a bounded operator between Banach spaces. Now, factors as / .Dually, ′ is ′ () ′ ′ (/ ) ′ ′.