# CHAPTER 8

*L*^{p} SPACES

*L*^{p} spaces are spaces of functions whose *p*th power is integrable. For functions in a *L*^{p} space, we can define norms and metrics and study the convergence of sequences of functions. In this chapter, we introduce the concepts of *L*^{p} spaces and some important inequalities for functions in the *L*^{p} spaces.

# 8.1 Basic Concepts and Facts

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

and

**Definition 8.2** (*L*^{p} Norm). Let *p* (0, ∞] and *f* *L*^{p}(*S*, ∑, μ). The norm of *f* on *L*^{P}(*S*, ∑, μ) is defined as

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*,

*d*(

*x, y*) =

*d*

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