Abstract
In this chapter, we gather definitions and results that will be used in the sequel. Most of these are covered in courses on linear algebra and real analysis. We shall prove some of the nontrivial statements to give a flavour of the kind of arguments that are involved. The second section of this chapter on linear spaces and the third section on metric spaces constitute the main prerequisites for this book. The last section on Lebesgue measure and integration is not required for developing the main results given in this book, but it is very much useful for illustrating them. Readers familiar with the contents of this chapter can directly go to the next chapter and look up the relevant material in this chapter as and when a reference is made. There are no exercises at the end of this chapter.
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Notes
- 1.
Here we have adopted the convention \(\infty +n=n+\infty =\infty +\infty =\infty \) for \(n\in {\mathbb N}\).
- 2.
If a set E represents a town, and a point x represents a watchman, then we may say that a town is ‘totally bounded’ if it can be guarded by a finite number of watchmen having an arbitrarily short sight.
- 3.
This can be seen as follows. By the definition of the Lebesgue measure of \(E\setminus G\), there are open intervals \(I_1, I_2,\ldots \) such that \(E\setminus G\subset \bigcup _{n=1}^\infty I_n\) and \(\sum _{n=1}^\infty \ell (I_n)<m(E\setminus G)+\epsilon \). Let \(J:=\bigcup _{n=1}^\infty I_n\) and \(F:=E\cap ({\mathbb R}\setminus J)\).
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© 2016 Springer Science+Business Media Singapore
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Limaye, B.V. (2016). Prerequisites. In: Linear Functional Analysis for Scientists and Engineers. Springer, Singapore. https://doi.org/10.1007/978-981-10-0972-3_1
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DOI: https://doi.org/10.1007/978-981-10-0972-3_1
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Online ISBN: 978-981-10-0972-3
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