1 files changed, 22 insertions, 0 deletions
diff --git a/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md b/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md
index 8482e87..0424117 100644
--- a/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md
+++ b/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md
@@ -23,6 +23,28 @@ Endowed with pointwise addition and scalar multiplication
 $\mathscr{L}^p(X,\mathcal{A},\mu)$ becomes a vector space.
 {% endproposition %}
 
+{% proof %}
+We show that $\mathscr{L}^p := \mathscr{L}^p(X,\mathcal{A},\mu)$ is a linear subspace of
+the vector space of all $\KK$-valued functions on $X$.
+The set $\mathscr{L}^p$ is nonempty since
+it contains the zero function.
+Now, suppose $f$ and $g$ are in $\mathscr{L}^p$.
+Then the sum $f+g$ is measurable, because $f$ and $g$ are measurable.
+Moreover, the function $\abs{f+g}^p$ is integrable, because we have the estimate
+
+$$
+\abs{f+g}^p
+\le (\abs{f} + \abs{g})^p
+\le \big\lparen 2 \max(\abs{f},\abs{g}) \big\rparen^p
+\le 2^p (\abs{f}^p + \abs{g}^p),
+$$
+
+where $\abs{f}^p$ and $\abs{g}^p$ are integrable.
+This proves that $f+g$ lies in $\mathscr{L}^p$.
+Finally, it is easy to see that $\alpha f$ lies in $\mathscr{L}^p$
+for any scalar $\alpha \in \KK$.
+{% endproof %}
+
 {% proposition %}
 $\norm{\cdot}_p$ is a seminorm on $\mathscr{L}^p(X,\mathcal{A},\mu)$.
 {% endproposition %}