Orthogonality in normed linear spaces

James, Robert C.

doi:10.1215/s0012-7094-45-01223-3

Cited by 249 publications

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“…Isosceles (1945): ¡¡x-yIL= ¡Ix+yl¡ Pythagorean (1945): ¡¡x-y¡¡'= ¡¡xII'-s-I¡yIk both introduced by James (14].…”

Section: -'unclassified

Orthogonality in normed linear spaces: A classification of the different concepts and some open problems

Rodríguez¹

1989

Rev. Mat. Complut.

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Orthogonality in normed linear spaces: a class¿fication of the dtfJ'erent concepis and sorne open problerns CARLOS BENíTEZABSTRAer. Orthogonality in inner product tpaces is a binary relation that can be expressed in many ways without explicit mention to the inner product of the space.Oreat pan of such definitions have alto tense in normed linear spaces. This simple observation it at the bate of ntany concepts of orthogonality in thete more general structuret.Various authors introduced such concepts ayer the last fifty years, although the origins of sorne of the mott interesting retults that can be obtained for these generalized concepts are, as usual in Mathematics, in previous or parallel works about convex sets, elliptis Gr elliptoids, duality, etc. (see, e.g., Gruber's paper [13]about the prior contributions of Caratheodory, Blaschke Gr Radon). DIFFERENT CONCEPTS OF ORTHOGONALITYAccording to its greater frecuency in tite literature this exposition is lirnited to tite case in wicit E is a real normed linear space.When tite norm of E is induced by an inner product, the ortitogonality of two points ir and y ob E is equivalent to each one of next (main or secondary) propositions.In tite more general context of normed linear spaces any one of such propositions is a definition of ot-thogonality between ir andy. ROBERTS (1934): ¡¡x-Xy¡I

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“…Isosceles (1945): ¡¡x-yIL= ¡Ix+yl¡ Pythagorean (1945): ¡¡x-y¡¡'= ¡¡xII'-s-I¡yIk both introduced by James (14].…”

Section: -'unclassified

Orthogonality in normed linear spaces: A classification of the different concepts and some open problems

Rodríguez¹

1989

Rev. Mat. Complut.

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“…In this section we use the idea of norming functionals in combination with James' nonlinear orthogonality [9] to give a geometric characterization of the countably additive members of sa(E, X), where sa(E, X) denotes the strongly additive members of the space ba(E, X) of all bounded finitely additive X-valued measures defined over E. The reader may consult Chapter I of Diestel and Uhl [6] for a discussion of many of the properties of ba(E, X) and sa(E, X). We continue to let \\p\\ denote the semivariation norm of a member of ba(E,X).…”

Section: R G Bllyeu and P W Lewismentioning

confidence: 99%

Norm Attaining Operators and Norming Functionals

Bilyeu¹,

Lewis²

1982

Proceedings of the American Mathematical Society

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ABSTRACT.The question of whether a countably additive measure with values in a Banach space attains the diameter of its range was unresolved. In this paper an example is given of a countably additive vector measure, taking values in a C(K) space, for which the diameter of the range is not attained. A property stronger than the attainment of the diameter, but which is possessed by many measures taking values in L-spaces, is shown to fail for infinite-dimensional measures into a space having smooth dual. As an application of the concept of norming functional (the existence of which is equivalent to the attainment of diameter), a characterization is given of the countably additive measures into space having smooth dual.

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“…It is also clear that an inner product space is an orthogonality vector space under the standard orthogonality relation. Furthermore, any normed space is an orthogonality vector space under the James [8] Since a linear functional is automatically hemicontinuous, hemicontinuity is no restriction on linear functional. Thus ƒ and f 2 in the above corollary need not be norm continuous.…”

Section: ±_ L Y) If {T:x(t)y(t)j£0} Is 0 In Case (I) or Of Measure mentioning

confidence: 99%

Orthogonality and nonlinear functionals

Gudder¹,

Strawther²

1974

Bull. Amer. Math. Soc.

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Let X be a vector space of functions. In practice, Zis taken to be (i) the set of continuous functions on some type of topological space or (ii) a set of measurable functions on a measure space. We say that x, y e X are orthogonal in the lattice theoretic sense (x ±_ L y) if {t: [13], in this note we consider orthogonalities which are standard in inner product and normed spaces. To some extent our results are less general than those in the above papers since the standard orthogonality is weaker than lattice theoretic orthogonality. On the other hand, some of our theorems apply to more general vector spaces than the above and furthermore we have obtained results for a more general class of functionals which we call orthogonally monotone functionals. Finally, we use our results to solve a nonlinear functional equation and give an application for the solution.An orthogonality vector space is a real vector space X with dim X^.2 on which there is defined a relation x_]_y such that (01) 0_L*, xj_0 for all xeX y (02) if x_[_y and x, y^O then x, y are linearly independent, (03) if x±y then ouc±py for all a 9 peR 9 (04) if B is a two-dimensional subspace of X, then for every Ojéx e B, there exists Ojéy e B such that x±_y and x+y J_x-y.It is easily seen that any real vector space of dimension ^2 is an orthogonality vector space if we define 0J_x, *_i_0 for all x and if x, y5^0 then xA_y iff x, y are linearly independent. It is also clear that an inner product space is an orthogonality vector space under the standard orthogonality AMS (MOS) subject classifications (1970). Primary 46B99, 46C05, 47H15; Secondary 46F05, 81A12.

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Orthogonality in normed linear spaces

Cited by 249 publications

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Orthogonality in normed linear spaces: A classification of the different concepts and some open problems

Orthogonality in normed linear spaces: A classification of the different concepts and some open problems

Norm Attaining Operators and Norming Functionals

Orthogonality and nonlinear functionals

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