Abstract
This chapter is devoted to a review of standard topics from functional analysis such as quasinormed and normed linear spaces and closed, compact and Fredholm linear operators on Banach spaces. These topics form a necessary background for what follows. In Sects. 3.1–3.3 we study linear operators and functionals, quasinormed and normed linear spaces. In a normed linear space we consider continuous linear functionals as generalized coordinates of the space. In Sect. 3.4 we prove the Riesz–Markov representation theorem which describes an intimate relationship between Radon measures and non-negative linear functionals on the spaces of continuous functions. This fact constitutes an essential link between measure theory and functional analysis, providing a powerful tool for constructing Markov transition functions in Chap. 9.
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Taira, K. (2014). Elements of Functional Analysis. In: Semigroups, Boundary Value Problems and Markov Processes. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43696-7_3
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DOI: https://doi.org/10.1007/978-3-662-43696-7_3
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