O'Reilly logo

Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach by Hong Xie, Chaoqun Ma, Guojun Gan

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

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

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required