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.

Citation

Ramasinghe, W. (2005). The Cauchy-Schwarz Inequality and the Induced Metrics on Real Vector Spaces Mainly on the Real Line. International Journal of Mathematical Education in Science and Technology, 36(1), 35-41. Retrieved August 7, 2024 from https://www.learntechlib.org/p/165755/.

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Keywords

• Equations (Mathematics)
• Mathematical Concepts
• Mathematical Formulas
• Metric System
• Probability
• Trigonometry