----- > [!proposition] Proposition. ([[norms are convex functions]]) > Let $(V, \|\cdot\|)$ be a [[norm|normed]] [[vector space]]. The function $\|x\|:V \to \mathbb{R}$ is [[convex function|convex]]. > [!proof]- Proof. ([[norms are convex functions]]) > Let $t \in [0,1)$ and $\b u,\b v \in (V, \|\cdot\|)$. We have $\begin{align} \|t \b u+(1-t)\b v\| \leq & \|t\b u \| + \|(1-t)\b v\| \\ = & t\|\b u\| + (1-t)\|\b v\|, \end{align}$ as required. ----- #### ---- #### 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 > ```