Two mappings related to semi-inner products and their applications in geometry of normed linear spaces

Sever Silvestru Dragomir; Jaromír J. Koliha

Applications of Mathematics (2000)

  • Volume: 45, Issue: 5, page 337-355
  • ISSN: 0862-7940

Abstract

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In this paper we introduce two mappings associated with the lower and upper semi-inner product ( · , · ) i and ( · , · ) s and with semi-inner products [ · , · ] (in the sense of Lumer) which generate the norm of a real normed linear space, and study properties of monotonicity and boundedness of these mappings. We give a refinement of the Schwarz inequality, applications to the Birkhoff orthogonality, to smoothness of normed linear spaces as well as to the characterization of best approximants.

How to cite

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Dragomir, Sever Silvestru, and Koliha, Jaromír J.. "Two mappings related to semi-inner products and their applications in geometry of normed linear spaces." Applications of Mathematics 45.5 (2000): 337-355. <http://eudml.org/doc/33064>.

@article{Dragomir2000,
abstract = {In this paper we introduce two mappings associated with the lower and upper semi-inner product $(\cdot ,\cdot )_i$ and $(\cdot ,\cdot )_s$ and with semi-inner products $[\cdot ,\cdot ]$ (in the sense of Lumer) which generate the norm of a real normed linear space, and study properties of monotonicity and boundedness of these mappings. We give a refinement of the Schwarz inequality, applications to the Birkhoff orthogonality, to smoothness of normed linear spaces as well as to the characterization of best approximants.},
author = {Dragomir, Sever Silvestru, Koliha, Jaromír J.},
journal = {Applications of Mathematics},
keywords = {lower and upper semi-inner product; semi-inner products; Schwarz inequality; smooth normed spaces; Birkhoff orthogonality; best approximants; lower and upper semi-inner product; semi-inner product; Schwarz inequality; smooth normed spaces; Birkhoff orthogonality; best approximants},
language = {eng},
number = {5},
pages = {337-355},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Two mappings related to semi-inner products and their applications in geometry of normed linear spaces},
url = {http://eudml.org/doc/33064},
volume = {45},
year = {2000},
}

TY - JOUR
AU - Dragomir, Sever Silvestru
AU - Koliha, Jaromír J.
TI - Two mappings related to semi-inner products and their applications in geometry of normed linear spaces
JO - Applications of Mathematics
PY - 2000
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 45
IS - 5
SP - 337
EP - 355
AB - In this paper we introduce two mappings associated with the lower and upper semi-inner product $(\cdot ,\cdot )_i$ and $(\cdot ,\cdot )_s$ and with semi-inner products $[\cdot ,\cdot ]$ (in the sense of Lumer) which generate the norm of a real normed linear space, and study properties of monotonicity and boundedness of these mappings. We give a refinement of the Schwarz inequality, applications to the Birkhoff orthogonality, to smoothness of normed linear spaces as well as to the characterization of best approximants.
LA - eng
KW - lower and upper semi-inner product; semi-inner products; Schwarz inequality; smooth normed spaces; Birkhoff orthogonality; best approximants; lower and upper semi-inner product; semi-inner product; Schwarz inequality; smooth normed spaces; Birkhoff orthogonality; best approximants
UR - http://eudml.org/doc/33064
ER -

References

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  13. Inner Product Structures, D. Reidel, Dordrecht, 1987. (1987) MR0903846
  14. 10.1090/S0002-9947-1961-0133024-2, Trans. Amer. Math. Soc. 100 (1961), 29–43. (1961) Zbl0102.32701MR0133024DOI10.1090/S0002-9947-1961-0133024-2
  15. Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Die Grundlehren der math. Wissen. 171, Springer, Berlin, 1970. (1970) Zbl0197.38601MR0270044
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