Abstract
The Lebesgue spaces L p(Ω) play a central role in many applications of functional analysis. This chapter focuses upon their basic properties, as well as certain attendant issues that will be important later. Notable among these are the semicontinuity of integral functionals, and the existence of measurable selections. We begin by identifying a geometric property of the norm which, when present, turns out to have a surprising consequence. A normed space X is said to be uniformly convex if it satisfies the following property:
In geometric terms, this is a way of saying that the unit ball is curved. The property depends upon the choice of the norm on X, even among equivalent norms, as one can see even in \({\mathbb{R}}^{\,2}\). Yet it implies an intrinsic property independent of the choice of norm: a uniformly convex Banach space is reflexive.
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Notes
- 1.
It turns out that the ball in \({\mathbb{R}}\) is curved in this sense, although it may seem rather straight to the reader.
- 2.
We follow Brézis [8, théorème IV.10].
- 3.
Let Ω be a bounded open subset of \({\mathbb{R}}^{ n}\), and let \(\varphi:\Omega\to\,{\mathbb{R}}\) be measurable, |φ(x)| ⩽ M a.e. For every ϵ>0 there exists \(g:\Omega\to\,{\mathbb{R}} \), continuous with compact support, having sup Ω| g | ⩽ M, such that meas { x∈ Ω:φ(x) ≠ g(x)}<ϵ. See Rudin [37, p. 53].
- 4.
We have used the following fact from integration: if two measurable functions f and g satisfy f ⩾ g, and if the integral of g is well defined, either finitely or as +∞, then the integral of f is well defined, either finitely or as +∞.
References
H. Brezis. Analyse fonctionnelle. Masson, Paris, 1983.
W. Rudin. Real and Complex Analysis. McGraw-Hill, New York, 1966.
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Clarke, F. (2013). Lebesgue spaces. In: Functional Analysis, Calculus of Variations and Optimal Control. Graduate Texts in Mathematics, vol 264. Springer, London. https://doi.org/10.1007/978-1-4471-4820-3_6
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DOI: https://doi.org/10.1007/978-1-4471-4820-3_6
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