Minkowski's Inequality for L1(E), Lp(E), and L∞(E)

Minkowski's Inequality for L1(E), Lp(E), and L∞(E)

Recall from the Hölder's Inequality for L1(E) and Lp(E) page that if $1 \leq p < \infty$ and if $f \in L^p(E)$, $g \in L^q(E)$ then $fg \in L^1(E)$ and:

(1)
\begin{align} \quad \| fg \|_1 \leq \| f \|_p \| g \|_q \end{align}

We will now use this inequality to prove Minkowski's inequality, which will complete our proof in showing that $L^p(E)$ is a normed linear space.

 Theorem 1 (Minkowski's Inequality for $L^1(E)$, $L^p(E)$, and $L^{\infty}(E)$): Let $E$ be a measurable set and let $1 \leq p \leq \infty$. Then for all $f, g \in L^p(E)$ we have that $\| f + g \|_p \leq \| f \|_p + \| g \|_p$.
• So suppose that $1 < p < \infty$. Let $f, g \in L^p(E)$. Observe that since $q = \frac{p}{p-1}$ we have that $p - 1 = \frac{p}{q}$. Then:
(2)
\begin{align} \quad |f + g|^p = |f + g||f + g|^{p-1} \leq [|f| + |g|]|f + g|^{p-1} = |f||f + g|^{p-1} + |g||f + g|^{p-1} = |f||f + g|^{p/q} + |g||f + g|^{p/q} \end{align}
• Observe that $|f+g|^{p/q} \in L^q(E)$ since $\int_E (|f+g|^{p/q})^q = \int_E |f+g|^p < \infty$. Since $f \in L^p(E)$ and $g \in L^p(E)$, by Hölder's inequality we have that:
(3)
\begin{align} \quad \| f + g \|^p = \int_E |f + g|^p \leq \int_E |f||f + g|^{p/q} + \int_E |g||f + g|^{p/q} \leq \| f \|_p \| (f + g)^{p/q} \|_q + \| g \|_p \| (f + g)^{p/q} \|_q = [\| f \|_p + \| g \|_p] \| (f + g)^{p/q} \|_q \end{align}
• Observe that:
(4)
\begin{align} \quad \| (f + g)^{p/q} \|_q = \left ( \int_E (|f + g|^{p/q})^q \right )^{1/q} = \left ( \int_E |f+g|^p \right )^{1/q} = \| f + g \|_p^{p/q} \end{align}
• Hence:
(5)
\begin{align} \quad \| f + g \|^p \leq [\| f \|_p + \| g \|_p] \| f + g \|_p^{p/q} \end{align}
• Dividing both sides by $\| f + g \|^{p/q}$ and we get:
(6)
\begin{align} \quad \| f + g \|_p^{p - p/q} \leq \| f \|_p + \| g \|_p \end{align}
• But observe that $p - \frac{p}{q} = p - \frac{p(p-1)}{p} = p - (p - 1) = 1$. Hence:
(7)
\begin{align} \quad \| f + g \|_p \leq \| f \|_p + \| g \|_p \quad \blacksquare \end{align}