A Hahn-Banach theorem for semifields

Iseki and Kasahara (see [3]) have given a Hahn-Banach type theorem for semifield-valued linear functionals on real linear spaces. We shall generalize their result by considering linear spaces over semifields.

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Semiconvex geometry

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700017973 ◽

1981 ◽

Vol 30 (4) ◽

pp. 496-510 ◽

Cited By ~ 9

Author(s):

Joe Flood

Keyword(s):

Algebraic Variety ◽

Convex Sets ◽

Convex Geometry ◽

Algebraic Approach ◽

Algebraic Proof ◽

Linear Spaces ◽

Structural Decomposition ◽

Banach Theorem ◽

Real Linear ◽

Convex Subsets

AbstractSemiconvex sets are objects in the algebraic variety generated by convex subsets of real linear spaces. It is shown that the fundamental notions of convex geometry may be derived from an entirely algebraic approach, and that conceptual advantages result from applying notions derived from algebra, such as ideals, to convex sets. Some structural decomposition results for semiconvex sets are obtained. An algebraic proof of the algebraic Hahn-Banach theorem is presented.

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Linear Spaces, Linear Functionals, and the Hahn–Banach Theorem

Universitext - A Course in Functional Analysis and Measure Theory ◽

10.1007/978-3-319-92004-7_5 ◽

2018 ◽

pp. 137-158

Author(s):

Vladimir Kadets

Keyword(s):

Linear Functionals ◽

Linear Spaces ◽

Banach Theorem

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Algebras which represent their linear functionals

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100027973 ◽

1953 ◽

Vol 49 (1) ◽

pp. 1-14 ◽

Cited By ~ 2

Author(s):

F. F. Bonsall ◽

A. W. Goldie

Keyword(s):

Vector Space ◽

Linear Functional ◽

Real Field ◽

Linear Functionals ◽

The Real ◽

Banach Theorem ◽

Real Linear

This paper was originally intended to contain a generalization of a theorem of Banach on the extension of linear functionals. This generalized theorem now appears as a by-product of a study of a class of algebras which we believe to be of much greater interest than the theorem itself. Let X be a vector space over the real field and let π(x) be a sub-additive, positive-homogeneous functional on X. Banach ((2), pp. 27–9) proves that any real linear functional f on a subspace X0 of X which satisfies f(x) ≤ π(x) on X0 can be extended to a real linear functional F on X with F(x) ≤ π(x) on X. One of the essential differences between this theorem and the Hahn-Banach theorem is that π can take negative values.

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Dual Spaces and Hahn-Banach Theorem

Formalized Mathematics ◽

10.2478/forma-2014-0007 ◽

2014 ◽

Vol 22 (1) ◽

pp. 69-77 ◽

Cited By ~ 1

Author(s):

Keiko Narita ◽

Noboru Endou ◽

Yasunari Shidama

Keyword(s):

Extension Theorem ◽

Normed Spaces ◽

Linear Spaces ◽

Dual Spaces ◽

Basic Properties ◽

Banach Theorem ◽

Real Linear

Summary In this article, we deal with dual spaces and the Hahn-Banach Theorem. At the first, we defined dual spaces of real linear spaces and proved related basic properties. Next, we defined dual spaces of real normed spaces. We formed the definitions based on dual spaces of real linear spaces. In addition, we proved properties of the norm about elements of dual spaces. For the proof we referred to descriptions in the article [21]. Finally, applying theorems of the second section, we proved the Hahn-Banach extension theorem in real normed spaces. We have used extensively used [17].

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On the proof of the analytic form of the Hahn Banach theorem in real linear spaces

International Mathematical Forum ◽

10.12988/imf.2014.47135 ◽

2014 ◽

Vol 9 ◽

pp. 1423-1426

Author(s):

B. G. Akuchu

Keyword(s):

Analytic Form ◽

Linear Spaces ◽

Banach Theorem ◽

Real Linear

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Bidual Spaces and Reflexivity of Real Normed Spaces

Formalized Mathematics ◽

10.2478/forma-2014-0030 ◽

2014 ◽

Vol 22 (4) ◽

pp. 303-311

Author(s):

Keiko Narita ◽

Noboru Endou ◽

Yasunari Shidama

Keyword(s):

Real Number ◽

Normed Space ◽

Uniform Boundedness ◽

Linear Operators ◽

Linear Functionals ◽

Normed Spaces ◽

Boundedness Theorem ◽

Natural Mapping ◽

Real Normed Space ◽

Banach Theorem

Summary In this article, we considered bidual spaces and reflexivity of real normed spaces. At first we proved some corollaries applying Hahn-Banach theorem and showed related theorems. In the second section, we proved the norm of dual spaces and defined the natural mapping, from real normed spaces to bidual spaces. We also proved some properties of this mapping. Next, we defined real normed space of R, real number spaces as real normed spaces and proved related theorems. We can regard linear functionals as linear operators by this definition. Accordingly we proved Uniform Boundedness Theorem for linear functionals using the theorem (5) from [21]. Finally, we defined reflexivity of real normed spaces and proved some theorems about isomorphism of linear operators. Using them, we proved some properties about reflexivity. These formalizations are based on [19], [20], [8] and [1].

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