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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. 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.

  4. Local boundedness - Wikipedia

    en.wikipedia.org/wiki/Local_boundedness

    Let : a function between topological vector spaces is said to be a locally bounded function if every point of has a neighborhood whose image under is bounded. The following theorem relates local boundedness of functions with the local boundedness of topological vector spaces:

  5. Functional analysis - Wikipedia

    en.wikipedia.org/wiki/Functional_analysis

    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.

  6. Operator (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Operator_(mathematics)

    Operators on these spaces are known as sequence transformations. Bounded linear operators over a Banach space form a Banach algebra in respect to the standard operator norm. The theory of Banach algebras develops a very general concept of spectra that elegantly generalizes the theory of eigenspaces.

  7. Bounded function - Wikipedia

    en.wikipedia.org/wiki/Bounded_function

    A bounded operator: is not a bounded function in the sense of this page's definition (unless =), but has the weaker property of preserving boundedness; bounded sets are mapped to bounded sets (). This definition can be extended to any function f : X → Y {\displaystyle f:X\rightarrow Y} if X {\displaystyle X} and Y {\displaystyle Y} allow for ...

  8. Positive-definite function on a group - Wikipedia

    en.wikipedia.org/wiki/Positive-definite_function...

    In mathematics, and specifically in operator theory, a positive-definite function on a group relates the notions of positivity, in the context of Hilbert spaces, and algebraic groups. It can be viewed as a particular type of positive-definite kernel where the underlying set has the additional group structure.

  9. Boundedness - Wikipedia

    en.wikipedia.org/wiki/Boundedness

    Bounded operator, a linear transformation L between normed vector spaces for which the ratio of the norm of L(v) to that of v is bounded by the same number over all non-zero vectors v. Unbounded operator, a linear operator defined on a subspace; Bounded poset, a partially ordered set that has both a greatest and a least element