---
title: The L<sup>p</sup> Spaces 
parent: Lebesgue Integral
grand_parent: Measure and Integration
nav_order: 4
---

# {{ page.title }}

{% definition %}
Let $(X,\mathcal{A},\mu)$ be a measure space and let $p \in [1,\infty)$.
We write $\mathscr{L}^p(X,\mathcal{A},\mu)$ for the set of all
measurable functions $f : X \to \KK$ such that $\abs{f}^p$ is integrable.
For such $f$ we write

$$
\norm{f}_p = {\bigg\lparen\int_X \abs{f}^p \, d\mu\bigg\rparen}^{\!1/p}.
$$
{% enddefinition %}

{% proposition %}
Endowed with pointwise addition and scalar multiplication
$\mathscr{L}^p(X,\mathcal{A},\mu)$ becomes a vector space.
{% endproposition %}

{% proposition %}
$\norm{\cdot}_p$ is a seminorm on $\mathscr{L}^p(X,\mathcal{A},\mu)$.
{% endproposition %}

{% theorem * Young Inequality %}
Consider $p,q > 1$ such that $1/p + 1/q = 1$. Then

$$
a \cdot b \le \frac{a^p}{p} + \frac{b^q}{q} \qquad \forall a,b \ge 0.
$$
{% endtheorem %}