Metric Spaces

Surinder Pal Singh Kainth ()
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Surinder Pal Singh Kainth: Panjab University, Department of Mathematics

Chapter Chapter 2 in A Comprehensive Textbook on Metric Spaces, 2023, pp 39-61 from Springer

Abstract: Abstract Exploring the properties of real functions or sequences is just the beginning. A few answers lead to several questions. Can we extend our results from $$\mathbb {R}$$ R to more general spaces, such as the plane $$\mathbb {R}^2$$ R 2 or the three-dimensional space $$\mathbb {R}^3$$ R 3 or to $$\mathbb {R}^n?$$ R n ? Sometimes the proofs depend only upon a few properties of the underlying space. The ones which depend only upon the distance function can be extended to metric spaces. A metric space is defined to be a nonempty set along with a distance function having some particular properties. This chapter presents a vast collection of metric spaces, including the particular cases of normed spaces and sequence spaces. To provide a glimpse into generalizations from reals, we have included a section on convergence of sequences in metric spaces which also contains the case of finite-dimensional Euclidean spaces.

Date: 2023
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DOI: 10.1007/978-981-99-2738-8_2

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