Abstract
Geometry works, from ancient times, essentially with two instruments: a rule and a compass. The first one measures distances, the second, angles. Modern mathematics isolates these two activities in two subjects: metric spaces (for a theory of distances) and inner product spaces (that presents a theory of angles or, if the reader prefers, orthogonality based in the notion of an inner product, also called a dot product . Most interestingly, this notion—an inner product— is powerful enough to induce also a distance, and so the frame in which metric geometry can be done naturally is settled (see Sect. 11.1).
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Notes
- 1.
This statement implies the—apparently more precise—following one: A sequence \(\{x_n\}_{n=1}^{\infty}\) in M converges in the metric d 1 if and only if, it converges in the metric d 2, and both limits coincide. Indeed, if \(x_n\rightarrow x\) in the metric d 1, the sequence \(\{x_1,x,x_2,x,x_3,\ldots\}\) converges to x in the metric d 1, so \(\{x_1,x,x_2,x,x_3,\ldots\}\) converges in the metric d 2. Hence, \(x_n\rightarrow x\) in the metric d 2.
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Montesinos, V., Zizler, P., Zizler, V. (2015). Metric Spaces. In: An Introduction to Modern Analysis. Springer, Cham. https://doi.org/10.1007/978-3-319-12481-0_6
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DOI: https://doi.org/10.1007/978-3-319-12481-0_6
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