---- > [!definition] Definition. ([[seminorm]]) > Let $X$ be a [[vector space]] over $\mathbb{R}$ or $\mathbb{C}$. A function $p:X \to \mathbb{R}$ is called a **seminorm** if it satisfies $p(x+y) \leq p(x)+p(y)$ for all $x,y \in X$ and $p(sx)=|s|p(x)$ for all $x \in X$ and scalars $s$. > > A [[norm]] is a positive definite seminorm. [[norms induce metrics|Norms induce metrics]], while *semi*norms induce *[[pseudometric|pseudometrics]].* > >The pair $(X, \|\cdot\|)$ is called a **seminormed (vector) space**. There are many (nonequivalent) ways to define a [[category]] $\mathsf{Seminorm}$ whose objects are normed spaces over $\mathbb{F}$. Here are two: > 1. As a [[subcategory|full subcategory]] $\mathsf{Seminorm_{T}}$ [[topological vector space|of]] $\mathsf{TVS}$[^4]; > 2. As a [[subcategory]] $\mathsf{Seminorm_{M}}$ [[metric space|of]] $\mathsf{Met}$ and $\mathsf{Vect}$, where morphisms are [[Lipschitz continuous|nonexpansive]] [[linear map|linear mappings]] (whence an [[isomorphism]] is a [[surjection|surjective]] ($\iff$ [[bijection|bijective]]) [[bijection|]] [[linear map|linear]] [[Lipschitz continuous|isometry]]). > > Note that $\mathsf{Norm_{M}}$ is a [[subcategory]] of $\mathsf{Norm_{T}}$. ---- #### ---- #### 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 > ```