Search results
Results from the WOW.Com Content Network
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 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]
Then a linear operator from U to V is called bounded if there exists c > 0 such that ‖ ‖ ‖ ‖ for every x in U. Bounded operators form a vector space. Bounded operators form a vector space.
By definition, the continuous linear functionals in the norm topology are the same as those in the weak Banach space topology. This dual is a rather large space with many pathological elements. On norm bounded sets of B( H ) , the weak (operator) and ultraweak topologies coincide.
A linear transformation between topological vector spaces, for example normed spaces, may be continuous. If its domain and codomain are the same, it will then be a continuous linear operator. A linear operator on a normed linear space is continuous if and only if it is bounded, for example, when the domain is finite-dimensional. [18]
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
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.