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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.
The Parsons problem format is used in the learning and teaching of computer programming. Dale Parsons and Patricia Haden of Otago Polytechnic developed Parsons's Programming Puzzles to aid the mastery of basic syntactic and logical constructs of computer programming languages, in particular Turbo Pascal, [1] although any programming language ...
Word problem for linear bounded automata [25] Word problem for quasi-realtime automata [26] Emptiness problem for a nondeterministic two-way finite state automaton [27] [28] Equivalence problem for nondeterministic finite automata [29] [30] Word problem and emptiness problem for non-erasing stack automata [31]
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.
Thus a unitary operator is a bounded linear operator that is both an isometry and a coisometry, [1] or, equivalently, a surjective isometry. [2] An equivalent definition is the following: Definition 2. A unitary operator is a bounded linear operator U : H → H on a Hilbert space H for which the following hold: U is surjective, and
Therefore, a counterexample to the invariant subspace problem would be a Banach space and a bounded operator : for which every non-zero vector is a cyclic vector for . (Where a "cyclic vector" x {\displaystyle x} for an operator T {\displaystyle T} on a Banach space H {\displaystyle H} means one for which the orbit [ x ] {\displaystyle [x]} of ...
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 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.