----- > [!proposition] Proposition. ([[Minkowski's inequality]]) > Suppose $(X, \Sigma, \mu)$ is a [[measure|measure space]], $1 \leq p <\infty$, and $f,g \in \mathcal{L}^{p}(\mu)$. Then $\|f+g\|_{p}\leq \|f\|_{p}+\|g\|_{p}.$ ^proposition > [!proof]- Proof. ([[Minkowski's inequality]]) > First a lemma. > > [!proposition] Lemma. > > Suppose $(X, \Sigma, \mu)$ is a [[measure|measure space]] and $0<p<\infty$. Then $\|f+g\|_{p}^{p} \leq 2^{p}(\|f\|_{p}^{p}+ \|g\|_{p}^{p}).$ > > > Indeed, let $f,g \in \mathcal{L}^{p}(\mu)$. If $x \in X$, then $\begin{align} > |f(x)+g(x)|^{p} &\leq (|f(x)|+|g(x)|)^{p} \\ > & \leq (2 \max \{ |f(x)|, |g(x)| \})^{p} \\ > & \leq 2^{p} (|f(x)|^{p}+|g(x)|^{p}). > \end{align}$ > [[integral|Integrating]] both sides above wrt $\mu$ gives the desired inequality. > ^proposition > > > Assume $1 \leq p < \infty$, leaving the case $p=\infty$ for later me. We'll work with the [[Lp-norm#^equivalence|equivalent definition]] of $\|\cdot\|_{p}$. The lemma implies that $f+g \in \mathcal{L}^{p}(\mu)$. Suppose $h \in \mathcal{L}^{p'}(\mu)$ with $\|h\|_{p'} \leq 1$. Then $\begin{align}\left|\int (f+g)h \, d\mu \right| & \leq \int |fh| \, d\mu + \int |gh| \, d\mu \\ > & \leq (\|f\|_{p} + \|g\|_{p}) \|h\|_{p'} \\ > & \leq \|f\|_{p}+ \|g\|_{p}, > \end{align}$ > where the first inequality comes from the [[triangle inequality for integrals]], the second inequality comes from [[Hölder's inequality]], and the final inequality uses that $\|h\|_{p'} \leq 1$. Passing to the [[supremum]] over $h \in \mathcal{L}^{p}(\mu)$ s.t. $\|h\|_{p'} \leq 1$ on the left side yields the result. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```