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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 mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its operator norm. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces .
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 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 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 bounded linear operator T : H → H is Hilbert–Schmidt if and only if the same is true of the operator | |:=, in which case the Hilbert–Schmidt norms of T and |T| are equal. [5] Hilbert–Schmidt operators are nuclear operators of order 2, and are therefore compact operators. [5]
Let X and Y be normed linear spaces, and denote by B(X,Y) the space of bounded operators of the form :.Let be any subset. We say that T is bounded below on whenever there is a constant (,) such that for all , the inequality ‖ ‖ ‖ ‖ holds.
Every bounded operator : can be written in the complex form = + where : and : are bounded self-adjoint operators. [ 6 ] Alternatively, every positive bounded linear operator A : H → H {\displaystyle A:H\to H} is self-adjoint if the Hilbert space H {\displaystyle H} is complex .