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Orthogonality Types in Normed Linear Spaces

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Surveys in Geometry I

Abstract

This chapter gives a comparing exposition of the large family of orthogonality concepts in normed linear spaces. With the help of fundamental properties that such concepts can have, or cannot have, we show their differences, similarities and direct connections. Based on this framework, we try to structurize this little subfield of the theory of real Banach spaces. For example, many characterizations of inner product spaces or of other special norm classes are presented. We also try to emphasize the geometric side of the given theoretical setting. Various open problems are posed, and a final survey can be taken as an update of the large amount of existing related literature. Hence our exposition should be useful for beginners in this research direction.

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Notes

  1. 1.

    If \(\dim X=+\infty \), it is necessary to assume that X is complete.

  2. 2.

    If \(\dim X=+\infty \), in property (b) it is necessary to assume that X is complete.

  3. 3.

    If \(\dim X=+\infty \), in property (b) it is necessary to assume that X is complete.

  4. 4.

    In fact, this property characterizes inner product spaces (see the footnote on page 46).

  5. 5.

    Similarly if P − is replaced by P +.

  6. 6.

    Similarly if P − is replaced by P +.

  7. 7.

    Joly [91] proved his result for gauges.

  8. 8.

    In [5] it is also proved that the corresponding curve for area orthogonality has the same property. But nevertheless, this is not a general property of all orthogonalities. If we consider the Carlsson orthogonality, x ⊥C y  :⇔ 5∥x − y∥2 = ∥x − 2y∥2 + ∥y − 2x∥2, then \(A(\mathcal {S}_C)\approx 2.3381 A(S_X)\).

  9. 9.

    Joly [91] assumed that \(\mathcal {S}_B=\mathcal {S}_B^{\prime }=\sqrt {2}\,S_X\).

  10. 10.

    Later, in Theorem 4.8.21, we will see that any of these properties characterizes two-dimensional inner product spaces .

  11. 11.

    A Banach space X is said to be uniformly non-square if there exists δ ∈ (0,  1) such that for any x, y ∈ S X, either ∥x + y∥≤ 2(1 − δ) or ∥x − y∥≤ 2(1 − δ).

  12. 12.

    Some years after the paper from Del Río–Benítez [63] had been published, Baronti [34] showed that this property characterizes inner product spaces .

  13. 13.

    Similarly if P + is replaced by P −.

  14. 14.

    Also coincide with area orthogonality.

  15. 15.

    See the comments before Theorem 4.5.17.

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Authors and Affiliations

  1. Instituto de Matemáticas de la Universidad de Extremadura (IMUEX), Badajoz, Spain

    Javier Alonso

  2. Technische Universität Chemnitz, Fakultät für Mathematik, Chemnitz, Germany

    Horst Martini

  3. North University of China, Taiyuan, China

    Senlin Wu

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  1. Javier Alonso
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Correspondence to Horst Martini .

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    Athanase Papadopoulos

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Alonso, J., Martini, H., Wu, S. (2022). Orthogonality Types in Normed Linear Spaces. In: Papadopoulos, A. (eds) Surveys in Geometry I. Springer, Cham. https://doi.org/10.1007/978-3-030-86695-2_4

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