L^p Space - Lp Spaces

Lp Spaces

Let 1 ≤ p < ∞ and (S, Σ, μ) be a measure space. Consider the set of all measurable functions from S to C (or R) whose absolute value raised to the p-th power has finite integral, or equivalently, that

The set of such functions forms a vector space, with the following natural operations:

for every scalar λ.

That the sum of two pth power integrable functions is again pth power integrable follows from the inequality |f + g|p ≤ 2p-1 (|f|p + |g|p). In fact, more is true. Minkowski's inequality says the triangle inequality holds for || · ||_p. Thus the set of pth power integrable functions, together with the function || · ||_p, is a seminormed vector space, which is denoted by .

This can be made into a normed vector space in a standard way; one simply takes the quotient space with respect to the kernel of || · ||_p. Since for any measurable function f, we have that ||f||_p = 0 if and only if f = 0 almost everywhere, the kernel of || · ||_p does not depend upon p,

In the quotient space, two functions f and g are identified if f = g almost everywhere. The resulting normed vector space is, by definition,

For p = ∞, the space L∞(S, μ) is defined as follows. We start with the set of all measurable functions from S to C (or R) which are essentially bounded, i.e. bounded up to a set of measure zero. Again two such functions are identified if they are equal almost everywhere. Denote this set by L∞(S, μ). For f in L∞(S, μ), its essential supremum serves as an appropriate norm:

As before, we have

if f ∈ L∞(S, μ) ∩ Lq(S, μ) for some q < ∞.

For 1 ≤ p ≤ ∞, Lp(S, μ) is a Banach space. The fact that Lp is complete is often referred to as Riesz-Fischer theorem. Completeness can be checked using the convergence theorems for Lebesgue integrals.

When the underlying measure space S is understood, Lp(S, μ) is often abbreviated Lp(μ), or just Lp. The above definitions generalize to Bochner spaces.