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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 Hahn–Banach theorem can be derived, was proved in ...
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.
Together with the Hahn–Banach 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.
In mathematics, specifically in functional analysis and Hilbert space theory, vector-valued Hahn–Banach theorems are generalizations of the Hahn–Banach theorems from linear functionals (which are always valued in the real numbers or the complex numbers) to linear operators valued in topological vector spaces (TVSs).
The above theorem can be used to extend a bounded linear transformation : to a bounded linear transformation from ¯ = to , if is dense in . If S {\displaystyle S} is not dense in X , {\displaystyle X,} then the Hahn–Banach theorem may sometimes be used to show that an extension exists .
The existence of a shift-invariant, finitely additive probability measure on the group Z also follows easily from the Hahn–Banach 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 ...
It shows that the analog of the Cauchy–Kovalevskaya theorem does not hold in the smooth category. The original example is not explicit, since it employs the Hahn–Banach theorem, but there since have been various explicit examples of the same nature found by Howard Jacobowitz. [1]
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.