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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.
A bounded wildcard is one with either an upper or a lower inheritance constraint. The bound of a wildcard can be either a class type, interface type, array type, or type variable. Upper bounds are expressed using the extends keyword and lower bounds using the super keyword. Wildcards can state either an upper bound or a lower bound, but not both.
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.
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 ...
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
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.
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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.