Lebesgue space

Definition of $L^{p} (Ω)$

Let $Ω \subset R^{n}$ be a measurable set and let $1 \leq p \leq \infty$ . The Lebesgue space $L^{p} (Ω)$ is the set of all measurable functions $f : Ω \to R$ (or $C$ ) such that the $p$ -th power of the absolute value of $f$ is integrable (or essentially bounded, if $p = \infty$ ).

For $1 \leq p < \infty$ :

L^{p} (Ω) = {f : Ω \to R | \int_{Ω} | f (x) |^{p} d x < \infty}

The associated norm is:

∥ f ∥_{L^{p} (Ω)} = {(\int_{Ω} | f (x) |^{p} d x)}^{1 / p}

For $p = \infty$ :

L^{\infty} (Ω) = {f : Ω \to R | \underset{x \in Ω}{ess \sup} | f (x) | < \infty}

The essential supremum norm is:

∥ f ∥_{L^{\infty} (Ω)} = \underset{x \in Ω}{ess \sup} | f (x) |

Key Properties

$L^{p} (Ω)$ is a Banach space for all $1 \leq p \leq \infty$ .
When $p = 2$ , $L^{2} (Ω)$ is a Hilbert space with the inner product:

⟨ f, g ⟩ = \int_{Ω} f (x) g (x) d x

These spaces form the foundational building blocks for Sobolev spaces, where not only the function but also its derivatives (in a weak sense) lie in $L^{p} (Ω)$ .

Lebesgue space

Definition of Lp(Ω)

Key Properties

Definition of $L^{p} (Ω)$