Lp Space

Definition

The $L^p$ space over a measure space $(X, \mu)$ is the set of measurable functions $f$ for which the $p$-norm is finite:

$$ L^p(X) = \left\{ f : \|f\|_p < \infty \right\} $$

where the $L^p$ norm is:

$$ \|f\|_p = \left( \int |f(x)|^p \, d\mu(x) \right)^{1/p} $$

Important Cases

Space	Norm	Intuition
$L^1$	$\int \|f\| \, dx$	Functions with finite “total variation”
$L^2$	$\sqrt{\int \|f\|^2 \, dx}$	Functions with finite “energy”
$L^\infty$	$\text{ess sup} \|f\|$	Bounded functions

$L^2$ — The Most Important Case

$L^2(\mathbb{R})$ is special because it is a Hilbert space — it has an inner product:

$$ \langle f, g \rangle = \int f(x) \overline{g(x)} \, dx $$

This makes $L^2$ the natural setting for:

• Fourier analysis (Plancherel theorem)
• Quantum mechanics (wavefunctions)
• Signal processing (finite-energy signals)

Intuition

• $L^1$: The function’s absolute values sum to something finite
• $L^2$: The function’s “energy” (squared magnitude) is finite
• $L^\infty$: The function stays bounded (doesn’t blow up)

A function can be in one $L^p$ space but not another. For example, $f(x) = 1/\sqrt{x}$ near $x=0$ is in $L^1$ locally but not $L^2$.

Related Concepts

• inner_product — exists on $L^2$
• norm — the $L^p$ norm
• Fourier_transform — is a unitary isomorphism on $L^2$
• Plancherel_theorem — $\|\mathcal{F}\{f\}\|_2 = \|f\|_2$