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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.
Let B be a Banach space, V a normed vector space, and () [,] a norm continuous family of bounded linear operators from B into V. Assume that there exists a positive constant C such that for every t ∈ [ 0 , 1 ] {\displaystyle t\in [0,1]} and every x ∈ B {\displaystyle x\in B}
The resolvent set () of a bounded linear operator L is an open set. More generally, the resolvent set of a densely defined closed unbounded operator is an open set. Notes
A bounded linear operator T : X → Y is called completely continuous if, for every weakly convergent sequence from X, the sequence () is norm-convergent in Y (Conway 1985, §VI.3). Compact operators on a Banach space are always completely continuous.
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.
In functional analysis, compact operators are linear operators on Banach spaces that map bounded sets to relatively compact sets. In the case of a Hilbert space H, the compact operators are the closure of the finite rank operators in the uniform operator topology. In general, operators on infinite-dimensional spaces feature properties that do ...
A linear map is "bounded on a neighborhood" (of some point) if and only if it is locally bounded at every point of its domain, in which case it is necessarily continuous [2] (even if its domain is not a normed space) and thus also bounded (because a continuous linear operator is always a bounded linear operator). [6]
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.