CHAPTER 8

Lp SPACES

Lp spaces are spaces of functions whose pth power is integrable. For functions in a Lp space, we can define norms and metrics and study the convergence of sequences of functions. In this chapter, we introduce the concepts of Lp spaces and some important inequalities for functions in the Lp spaces.

8.1 Basic Concepts and Facts

Definition 8.1 (Lp Space). Let (S, ∑, μ) be a measure space and p (0, ∞]. Then Lp(S, ∑, μ) is defined as

equation

and

equation

Definition 8.2 (Lp Norm). Let p (0, ∞] and f Lp(S, ∑, μ). The norm of f on LP(S, ∑, μ) is defined as

equation

When p = ∞, the norm is called the infinite norm or the maximum norm.

Definition 8.3 (Metric Space). Let S be a set and d : S × S → [0, ∞] be a function. (S, d) is called a metric space if for all x, y, z S,

(a) (Symmetry) d(x, y) = d

Get Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach now with the O’Reilly learning platform.

O’Reilly members experience live online training, plus books, videos, and digital content from nearly 200 publishers.