# 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

and

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

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

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