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  2. Bounded operator - Wikipedia

    en.wikipedia.org/wiki/Bounded_operator

    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.

  3. Unitary operator - Wikipedia

    en.wikipedia.org/wiki/Unitary_operator

    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

  4. Uniform boundedness principle - Wikipedia

    en.wikipedia.org/wiki/Uniform_boundedness_principle

    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.

  5. Open mapping theorem (functional analysis) - Wikipedia

    en.wikipedia.org/wiki/Open_mapping_theorem...

    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.

  6. Operator topologies - Wikipedia

    en.wikipedia.org/wiki/Operator_topologies

    On norm bounded sets of B(H), the weak (operator) and ultraweak topologies coincide. This can be seen via, for instance, the Banach–Alaoglu theorem . For essentially the same reason, the ultrastrong topology is the same as the strong topology on any (norm) bounded subset of B( H ) .

  7. Commutator subspace - Wikipedia

    en.wikipedia.org/wiki/Commutator_subspace

    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].

  8. Spectrum (functional analysis) - Wikipedia

    en.wikipedia.org/wiki/Spectrum_(functional_analysis)

    Since is a linear operator, the inverse is linear if it exists; and, by the bounded inverse theorem, it is bounded. Therefore, the spectrum consists precisely of those scalars λ {\displaystyle \lambda } for which T − λ I {\displaystyle T-\lambda I} is not bijective .

  9. Spectral theory of compact operators - Wikipedia

    en.wikipedia.org/wiki/Spectral_theory_of_compact...

    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 ...