L _p -spaces

Abstract

In this chapter, we introduce the classicalL _p -spaces and study their basic properties. For a measure space (X, Σ, µ) and 0 <p < ∞, the space L _p (µ) is the collection of all equivalence classes of measurable functionsf for which thep-norm

$$ {\left\| f \right\|_p} = {(\int {{{\left| f \right|}^p}d\mu } )^{^{\frac{1} {p}}}}< \infty $$