---- > [!definition] Definition. ([[sublinear functional]]) > Let $X$ be a [[vector space]] over $\mathbb{F}$, [[field|where]] $\mathbb{F}$ denotes $\mathbb{R}$ or $\mathbb{C}$. A function $s:X \to \mathbb{R}$ is called a **sublinear functional** if: >1. (positive homogeneity) $s(c x)=c s(x)$ for all $c > 0$; >2. (subadditivity/triangle inequality) $s(x_{1}+x_{2}) \leq s(x_{1})+s(x_{2})$. > [!basicproperties] > Like the [[norm|norms]] and [[seminorm|seminorms]] they generalize, sublinear functions are convex: for $x_{0},x_{1}$ in a [[convex set|convex subset]] of $X$ and $t \in [0,1]$: $s\big( (1-t)x_{0} + t x_ 1 \big)\leq s\big( (1-t)x_{0} \big)+s(t x_{1})=(1-t)s(x_{0})+ts(x_{1})$. ^properties ---- #### ---- #### 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 > ```