Titlebook: Lectures on Functional Analysis and the Lebesgue Integral; Vilmos Komornik Textbook 2016 Springer-Verlag London 2016 Functional analysis.H

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Obedient

MOAT

Cantankerous

cuticle

Locally Convex SpacesWe have seen in the preceding chapters the usefulness of weak convergence. From a theoretical point of view, it would be more satisfying to find a norm associated with weak convergence. In finite dimensions every norm is suitable because the weak and strong convergences are the same. In infinite dimensions the situation is quite different.

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Monotone Functions. (having more than one point).

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The Lebesgue Integral in ,In former times when one invented a new function it was for a practical purpose; today one invents them purposely to show up defects in the reasoning of our fathers and one will deduce from them only that.—H. Poincaré

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Generalized Newton–Leibniz FormulaOne of the (if not .) most important theorems of classical analysis is the Newton–Leibniz formula: . allowing us to compute many integrals. The purpose of this chapter is to extend its validity to Lebesgue integrable functions.

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Integrals on Measure SpacesIn Chap. 5 we defined the Lebesgue integral of functions defined on .. In this chapter we show that the theory remains valid in a much more general framework;moreover, almost all proofs can be repeated word for word. The results of this chapter include integrals of several variables and integrals on probability spaces