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  2. Hahn–Banach theorem - Wikipedia

    en.wikipedia.org/wiki/HahnBanach_theorem

    The theorem is named for the mathematicians Hans Hahn and Stefan Banach, who proved it independently in the late 1920s.The special case of the theorem for the space [,] of continuous functions on an interval was proved earlier (in 1912) by Eduard Helly, [1] and a more general extension theorem, the M. Riesz extension theorem, from which the HahnBanach theorem can be derived, was proved in ...

  3. Uniform boundedness principle - Wikipedia

    en.wikipedia.org/wiki/Uniform_boundedness_principle

    Together with the HahnBanach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. 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 .

  4. 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.

  5. Functional analysis - Wikipedia

    en.wikipedia.org/wiki/Functional_analysis

    Together with the HahnBanach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. 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.

  6. Hans Hahn (mathematician) - Wikipedia

    en.wikipedia.org/wiki/Hans_Hahn_(mathematician)

    Hahn's contributions to mathematics include the HahnBanach theorem and (independently of Banach and Steinhaus) the uniform boundedness principle. Other theorems include: the Hahn decomposition theorem; the Hahn embedding theorem; the Hahn–Kolmogorov theorem; the Hahn–Mazurkiewicz theorem; the Vitali–Hahn–Saks theorem.

  7. Continuous linear extension - Wikipedia

    en.wikipedia.org/wiki/Continuous_linear_extension

    Closed graph theorem (functional analysis) – Theorems connecting continuity to closure of graphs; Continuous linear operator; Densely defined operator – Function that is defined almost everywhere (mathematics) HahnBanach theoremTheorem on extension of bounded linear functionals

  8. Amenable group - Wikipedia

    en.wikipedia.org/wiki/Amenable_group

    The existence of a shift-invariant, finitely additive probability measure on the group Z also follows easily from the HahnBanach theorem this way. Let S be the shift operator on the sequence space ℓ ∞ ( Z ), which is defined by ( Sx ) i = x i +1 for all x ∈ ℓ ∞ ( Z ), and let u ∈ ℓ ∞ ( Z ) be the constant sequence u i = 1 for ...

  9. Banach fixed-point theorem - Wikipedia

    en.wikipedia.org/wiki/Banach_fixed-point_theorem

    In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach–Caccioppoli theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces and provides a constructive method to find those fixed points.