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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.
Example: A continuous and bounded linear map that is not bounded on any neighborhood: If : is the identity map on some locally convex topological vector space then this linear map is always continuous (indeed, even a TVS-isomorphism) and bounded, but is bounded on a neighborhood if and only if there exists a bounded neighborhood of the origin ...
In functional analysis, a branch of mathematics, a compact operator is a linear operator:, where , are normed vector spaces, with the property that maps bounded subsets of to relatively compact subsets of (subsets with compact closure in ).
For example, bijective operators preserving the structure of a vector space are precisely the invertible linear operators. They form the general linear group under composition. However, they do not form a vector space under operator addition; since, for example, both the identity and −identity are invertible (bijective), but their sum, 0, is not.
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
Corollary — If a sequence of bounded operators () converges pointwise, that is, the limit of (()) exists for all , then these pointwise limits define a bounded linear operator . The above corollary does not claim that T n {\displaystyle T_{n}} converges to T {\displaystyle T} in operator norm, that is, uniformly on bounded sets.
For example, if X is a Banach space and T is a strictly singular operator in B(X) then its spectrum satisfies the following properties: (i) the cardinality of () is at most countable; (ii) () (except possibly in the trivial case where X is finite-dimensional); (iii) zero is the only possible limit point of (); and (iv) every nonzero () is an ...
The commutator subspace of a two-sided ideal J of the bounded linear operators B(H) on a separable Hilbert space H is the linear span of operators in J of the form [A,B] = AB − BA for all operators A from J and B from B(H). The commutator subspace of J is a linear subspace of J denoted by Com(J) or [B(H),J].