Lp Spaces

L^{p}

-Norm

Suppose that

(X, 𝒮, μ)

is a measure space,

p \in [1, \infty)

, and

f : X \to 𝔽

is measurable. Define

{‖ f ‖}_{p} : = {(\int_{X} {| f |}^{p} d μ)}^{1 / p} .

L^{p}

-Space

Suppose that

p \in [1, \infty)

. The $L^{p}$ -space of

μ

is the vector space of equivalence classes of measurable functions

f : X \to 𝔽

such that

{‖ f ‖}_{p} < \infty

, where functions equal almost everywhere are identified.

Essential Supremum Norm

Suppose that

f : X \to 𝔽

is measurable. Define

{‖ f ‖}_{\infty} : = \inf {c \in [0, \infty] : | f (x) | \leq c for almost every x \in X} .

L^{\infty}

-Space

The $L^{\infty}$ -space of

μ

is the vector space of equivalence classes of measurable functions

f : X \to 𝔽

such that

{‖ f ‖}_{\infty} < \infty

, where functions equal almost everywhere are identified.

Holder's Inequality

Suppose that

p, q \in [1, \infty]

and

1 / p + 1 / q = 1

. If

f \in L^{p} (μ)

and

g \in L^{q} (μ)

, then

f g \in L^{1} (μ)

and

{‖ f g ‖}_{1} \leq {‖ f ‖}_{p} {‖ g ‖}_{q} .

If ${‖ f ‖}_{p} = 0$ or ${‖ g ‖}_{q} = 0$ , the result is immediate after modifying on a null set. Otherwise normalize so ${‖ f ‖}_{p} = {‖ g ‖}_{q} = 1$ . Young's inequality gives $a b \leq a^{p} / p + b^{q} / q$ for $a, b \geq 0$ , so $| f g | \leq \frac{{| f |}^{p}}{p} + \frac{{| g |}^{q}}{q} .$ Integrating and using the definition of the integral gives ${‖ f g ‖}_{1} \leq 1$ . Rescaling proves the stated inequality.

Minkowski's Inequality

Suppose that

p \in [1, \infty]

. If

f, g \in L^{p} (μ)

, then

{‖ f + g ‖}_{p} \leq {‖ f ‖}_{p} + {‖ g ‖}_{p} .

For $1 < p < \infty$ , apply Holder's inequality with conjugate exponent $q$ to ${| f + g |}^{p} \leq | f | {| f + g |}^{p - 1} + | g | {| f + g |}^{p - 1} .$ This gives ${‖ f + g ‖}_{p}^{p} \leq ({‖ f ‖}_{p} + {‖ g ‖}_{p}) {‖ f + g ‖}_{p}^{p - 1}$ , and division yields the result. The cases $p = 1$ and $p = \infty$ follow directly from the triangle inequality for absolute value and the essential supremum norm.

L^{p}

is Complete

Suppose that

p \in [1, \infty]

. Then

L^{p} (μ)

, with its

L^{p}

-norm, is a Banach space.

Let $(f_{n})$ be Cauchy in $L^{p}$ . Choose a subsequence with ${‖ f_{n_{k + 1}} - f_{n_{k}} ‖}_{p} < 2^{- k}$ . The series $\sum_{k} | f_{n_{k + 1}} - f_{n_{k}} |$ has finite $L^{p}$ -norm by Minkowski's inequality and the monotone convergence theorem, so it is finite almost everywhere. Thus the subsequence converges pointwise almost everywhere to a measurable $f$ . Another application of Fatou's lemma gives ${‖ f_{n_{k}} - f ‖}_{p} \to 0$ , and the original Cauchy sequence then converges to $f$ . Hence $L^{p}$ is complete.

Density of Simple Functions in

L^{p}

Suppose that

p \in [1, \infty)

. The simple functions in

L^{p} (μ)

are dense in

L^{p} (μ)

For $f \in L^{p} (μ)$ , the function ${| f |}^{p}$ is in $L^{1} (μ)$ . By approximation by simple functions, choose simple $s_{n} \to f$ pointwise with $| s_{n} | \leq | f |$ . Then ${| f - s_{n} |}^{p} \leq 2^{p} {| f |}^{p}$ , and the dominated convergence theorem gives ${‖ f - s_{n} ‖}_{p} \to 0$ .

Density of Continuous Compactly Supported Functions in

L^{p}

Suppose that

p \in [1, \infty)

. Continuous compactly supported functions are dense in

L^{p} (ℝ)

By density of simple functions, approximate $f$ in $L^{p}$ by a finite linear combination of characteristic functions of finite-measure Lebesgue measurable sets. The regularity characterization lets us approximate those sets by finite unions of intervals. Replacing interval characteristic functions by continuous tent functions with compact support changes the $L^{p}$ -norm by an arbitrarily small amount. Combining the finitely many errors proves density.

Riesz-Fischer Theorem

Suppose that

p \in [1, \infty]

. Every Cauchy sequence in

L^{p} (μ)

converges in

L^{p} (μ)

This is the sequential form of $L^{p}$ completeness. In a normed space, completeness means precisely that every Cauchy sequence converges in the norm, so the theorem follows immediately.