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+---
+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 %}