Lp Spaces - Core Definitions

The $L^p$ spaces are fundamental function spaces in analysis, providing natural settings for studying integration, convergence, and functional analysis. They generalize the notion of square-integrable functions to $p$ -th power integrable functions.

DefinitionLp Spaces

Let $(X, \mathcal{F}, \mu)$ be a measure space and $1 \leq p < \infty$ . The space $L^p(X, \mu)$ (or simply $L^p(\mu)$ or $L^p$ ) consists of all measurable functions $f: X \to \mathbb{R}$ (or $\mathbb{C}$ ) such that: $\int_X |f|^p \, d\mu < \infty$

More precisely, $L^p$ consists of equivalence classes of functions that are equal almost everywhere. The $L^p$ norm is defined by: $\|f\|_p = \left(\int_X |f|^p \, d\mu\right)^{1/p}$

For $p = \infty$ , $L^\infty(X, \mu)$ consists of essentially bounded functions: $L^\infty = \{f : \text{there exists } M \text{ such that } |f(x)| \leq M \text{ a.e.}\}$

with norm $\|f\|_\infty = \inf\{M : |f| \leq M \text{ a.e.}\}$ , called the essential supremum.

ExampleCommon Lp Spaces

$L^1([0,1])$ : Integrable functions on $[0,1]$ . Includes $f(x) = 1/\sqrt{x}$ near $0$ .
$L^2([0,1])$ : Square-integrable functions. This is a Hilbert space with inner product $\langle f, g \rangle = \int_0^1 fg \, d\lambda$ . The function $1/\sqrt{x}$ is NOT in $L^2$ .
$L^\infty([0,1])$ : Essentially bounded functions. Includes all continuous functions on $[0,1]$ .
$\ell^p$ : When $X = \mathbb{N}$ with counting measure, $L^p$ consists of sequences with: $\|x\|_p = \left(\sum_{n=1}^{\infty} |x_n|^p\right)^{1/p} < \infty$

Remark

Key observations about $L^p$ spaces:

Inclusion relations: If $\mu(X) < \infty$ and $p < q$ , then $L^q \subseteq L^p$ . The reverse inclusion holds for counting measure on $\mathbb{N}$ .
$L^2$ is special: $L^2$ is the only $L^p$ space (for $1 \leq p < \infty$ ) that is a Hilbert space. It has an inner product structure that makes it particularly amenable to geometric intuition.
Completeness: All $L^p$ spaces are complete, meaning every Cauchy sequence converges to an element in $L^p$ . This makes them Banach spaces.
Duality: The dual space of $L^p$ (for $1 < p < \infty$ ) is $L^q$ where $1/p + 1/q = 1$ . The functional $F(f) = \int fg \, d\mu$ for $g \in L^q$ represents all bounded linear functionals on $L^p$ .