Mathematics Preliminaries
Terry Eleftherios Moschandreou · 2023
In this chapter, brief on real analysis material on sequences, limits and inequalities in the foundations of calculus is considered. Metric spaces, limit inferior and superior, Cauchy sequences are introduced and Lebesgue integration with the Monotone Convergence theorem and Fatou’s lemma is considered with examples. Uniform convergence is brought to fore, and convergence in the norm is discussed. Also, the completeness of metric spaces is considered with examples. Some inequalities such as the Gagliardo-Nirenberg and Prékopa-Leindler inequalities applied in Chapter 10 are introduced in this chapter for the first time.