Lp Spaces - Key Properties

The $L^p$ spaces satisfy fundamental inequalities that establish their norm properties and enable their use in analysis. These inequalities are essential tools throughout mathematics.

TheoremHolder's Inequality

Let $1 \leq p, q \leq \infty$ with $\frac{1}{p} + \frac{1}{q} = 1$ (we say $p$ and $q$ are conjugate exponents). If $f \in L^p$ and $g \in L^q$ , then $fg \in L^1$ and: $\int_X |fg| \, d\mu \leq \|f\|_p \|g\|_q$

For $p = q = 2$ , this reduces to the Cauchy-Schwarz inequality: $\int_X |fg| \, d\mu \leq \|f\|_2 \|g\|_2$

TheoremMinkowski's Inequality

For $1 \leq p \leq \infty$ , if $f, g \in L^p$ , then $f + g \in L^p$ and: $\|f + g\|_p \leq \|f\|_p + \|g\|_p$

This is the triangle inequality for the $L^p$ norm, establishing that $\|\cdot\|_p$ is indeed a norm.

These two inequalities are cornerstones of $L^p$ theory. Holder's inequality provides a way to bound products, while Minkowski's inequality establishes the norm structure.

ExampleApplications

Bounding integrals: To estimate $\int_0^1 x^{-1/3} \sin(x) \, dx$ , note that $x^{-1/3} \in L^{3/2}([0,1])$ and $\sin(x) \in L^3([0,1])$ . By Holder's inequality with $p = 3/2$ , $q = 3$ : $\left|\int_0^1 x^{-1/3} \sin(x) \, dx\right| \leq \|x^{-1/3}\|_{3/2} \|\sin\|_3 < \infty$
$L^2$ orthogonality: In $L^2$ , if $\langle f, g \rangle = \int fg = 0$ , then by Pythagorean theorem (derived from Minkowski): $\|f + g\|_2^2 = \|f\|_2^2 + \|g\|_2^2$

Remark

Additional important properties:

Interpolation: For $1 \leq p_0 < p < p_1 \leq \infty$ and $\theta \in (0, 1)$ with $\frac{1}{p} = \frac{1-\theta}{p_0} + \frac{\theta}{p_1}$ , if $f \in L^{p_0} \cap L^{p_1}$ , then $f \in L^p$ and: $\|f\|_p \leq \|f\|_{p_0}^{1-\theta} \|f\|_{p_1}^{\theta}$
Completeness (Riesz-Fischer): $L^p$ is complete for all $1 \leq p \leq \infty$ . If $\{f_n\}$ is Cauchy in $L^p$ , then there exists $f \in L^p$ with $\|f_n - f\|_p \to 0$ .
Dense subsets: On $\mathbb{R}^n$ , continuous functions with compact support are dense in $L^p$ for $1 \leq p < \infty$ .
Separability: $L^p(\mathbb{R}^n)$ is separable for $1 \leq p < \infty$ , but $L^\infty$ is not separable.