Abstract
A new concept of orthogonality in real normed linear spaces is introduced. Typical properties of orthogonality (homogeneity, symmetry, additivity, ...) and relations between this orthogonality and other known orthogonalities (Birkhoff, Boussouis, Unitary-Boussouis and Diminnie) are studied. In particular, some characterizations of inner product spaces are obtained.
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References
J. Alonso, Ortogonalidad en espacios normados. Ph. D. Thesis, Univ. Extremadura, Badajoz, Spain, 1984.
J. Alonso and C. Benítez, Some characteristic and non-characteristic properties of inner product spaces. J. Approx. Theory 55, 318–325 (1988).
J. Alonso and C. Benítez, Orthogonality in normed linear spaces: A survey. Part I: Main properties. Extracta Math. 3 (1), 1–15 (1988).
J. Alonso and C. Benítez, OrthogonaHty in normed linear spaces: A survey. Part II: Relations between main orthogonalities. Extracta Math. 4 (3), 121–131 (1989).
J. Alonso and C. Benítez, Complements on Diminnie orthogonality. Math. Nachr. 165,99–106 (1994).
D. Amir, Characterizations of Inner Product Spaces. Basel, 1986.
C. Benítez, Una propiedad de la ortogonalidad de Birkhoff y una caracterización de espacios prehilbertianos. Collect. Math. 26, 211–218 (1975).
C. Benítez and M. Del Río, Characterization of inner product spaces through rectangle and square inequalities. Rev. Roumaine Math. Pures Appl. 29, 543–546 (1984).
C. Benítez, Orthogonality in normed linear spaces: a classification of the different concepts and some open problems. Rev. Mat. Univ. Complutense Madr. 2, 53–57 (1989).
G. Birkhoff, Orthogonality in linear metric spaces. Duke Math. J. 1, 169–172 (1935).
B. Boussouis, Orthogonalité et caractérisation des espaces préhilbertiens. Ph. D. Thesis, Université Sidi Mohamed Ben Abdellah, Fès, Morocco, 1995.
S. O. Carlsson, Orthogonality in normed linear spaces. Ark. Mat. 4, 297–318 (1962).
M. M. Day, Some characterizations of inner product spaces. Trans. Amer. Math. Soc. 62, 320–337 (1947).
C. R. Diminnie, A new orthogonality relation in normed linear spaces. Math. Nachr. 114, 197–203 (1983).
C.R. Diminnie, E.Z. Andalafte and R.W. Freese, Angles in normed linear spaces and characterization of real inner product spaces. Math. Nachr. 129, 197–204 (1986).
C. R. Diminnie, R. W. Freese and E. Z. Andalafte, An extension of Pythagorean and Isosceles orthogonality and a characterization of inner product spaces. J. Approx. Theory 39, 295–298 (1983).
R.C. James, Orthogonality in normed linear spaces. Duke Math. J. 12, 291–301 (1945).
R. C. James, Orthogonality and linear functionals in normed linear spaces. Trans. Amer. Math. Soc. 61, 265–292(1947).
R.C. James, Inner products in normed linear spaces. Bull. Amer. Math. Soc. 53, 559–566 (1947).
O. P. Kapoor and S. B. Mathur, Some geometric characterizations of inner product spaces. Bull. Austral. Math. Soc. 24, 239–246 (1981).
O. P. Kapoor and J. Prasad, Orthogonality and characterizations of inner product spaces. Bull. Austral. Math. Soc. 19, 403–416 (1978).
K. Ohira, On some characterizations of abstract euclidean spaces by properties of orthogonality. Kumamoto J. Sci. 1, 23–26 (1952).
J. Radon, Collected works, Vol. I. P. M. Gruber, E. Hlawka, W. Nöbauer, L. Schmetterer, eds. Basel-Boston 1987.
I. Singer, Unghiuri abstracte si functii trigonometrice in spatii Banach. Acad. Republ. Popul. Romine, Bul. Sti., Sect. Sti. Mat. Fiz. 9, 29–42 (1957).
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Alonso, J., Benítez, C. Area orthogonality in normed linear spaces. Arch. Math. 68, 70–76 (1997). https://doi.org/10.1007/PL00000397
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DOI: https://doi.org/10.1007/PL00000397