Titlebook: Measure, Integration & Real Analysis; Sheldon Axler Textbook‘‘‘‘‘‘‘‘ 2020 Sheldon Axler 2020 Measure theory textbook.Graduate real analysi

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Hilbert Spaces,In this chapter, we will see a clean description of the bounded linear functionals on a Hilbert space. We will also see that every Hilbert space has an orthonormal basis, which make Hilbert spaces look much like standard Euclidean spaces but with infinite sums replacing finite sums.

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Real and Complex Measures,A measure is a countably additive function from a .-algebra to [0,α]. In this chapter, we consider countably additive functions from a .-algebra to either . or .. The first section of this chapter shows that these functions, called real measures or complex measures, form an interesting Banach space with an appropriate norm.

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Riemann Integration,y presenting the definitions leading to the Riemann integral. The big result in the first section states that a continuous real-valued function on a closed bounded interval is Riemann integrable. The proof depends upon the theorem that continuous functions on closed bounded intervals are uniformly continuous.

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Differentiation,lewood maximal inequality. This tool is used to prove an almost everywhere version of the Fundamental Theorem of Calculus. These results lead us to an important theorem about the density of Lebesgue measurable sets.

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Banach Spaces,mplex-valued functions. After that, we rapidly review the framework of vector spaces, which allows us to consider natural collections of measurable functions that are closed under addition and scalar multiplication.

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