The Cauchy-Schwarz Inequality and the Induced Metrics on Real Vector Spaces Mainly on the Real Line

Autor/in	Ramasinghe, W.
Titel	The Cauchy-Schwarz Inequality and the Induced Metrics on Real Vector Spaces Mainly on the Real Line
Quelle	In: International Journal of Mathematical Education in Science and Technology, 36 (2005) 1, S.35-41 (7 Seiten)Infoseite zur Zeitschrift PDF als Volltext Verfügbarkeit
Sprache	englisch
Dokumenttyp	gedruckt; online; Zeitschriftenaufsatz
ISSN	0020-739X
Schlagwörter	Quantitative Daten; Trigonometry; Mathematical Concepts; Equations (Mathematics); Probability; Mathematical Formulas; Metric System + Suchen Sie Ihr Suchwort? Trigonometrie; Equations; Mathematics; Gleichungslehre; Wahrscheinlichkeitsrechnung; Wahrscheinlichkeitstheorie; Mathematische Formel; Metrischer Raum
Abstract	It is very well known that the Cauchy-Schwarz inequality is an important property shared by all inner product spaces and the inner product induces a norm on the space. A proof of the Cauchy-Schwarz inequality for real inner product spaces exists, which does not employ the homogeneous property of the inner product. However, it is shown that a real vector space with a product satisfying properties of an inner product except the homogeneous property induces a metric but not a norm. It is remarkable to see that the metric induced on the real line by such a product has highly contrasting properties relative to the absolute value metric. In particular, such a product on the real line is given so that the induced metric is not complete and the set of rational numbers is not dense in the real line. (Author).
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Erfasst von	ERIC (Education Resources Information Center), Washington, DC
Update	2017/4/10