---- > [!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 $\mathsf{Norm_{T}}$. > [!note] > The **degenerate space** of a [[norm|seminorm]] $\|\cdot\|$ is the [[linear subspace]] $D=\{ v:\|v\|=0 \}$ of $X$.[^1] [[quotient space|Quotienting]] by $D$ yields an earnest [[norm]] $\widetilde{\|\cdot\|}$. The new space is $X$ with the [[equivalence relation]] $a \sim b \iff \|a-b\|=0$.[^2] > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \usepackage{amsfonts} > \usepackage[mathscr]{euscript} > \usepackage{tikzit} > \usetikzlibrary{shapes.geometric} > > \begin{document} > % TiKZ style file generated by TikZiT. You may edit this file manually, > % but some things (e.g. comments) may be overwritten. To be readable in > % TikZiT, the only non-comment lines must be of the form: > % \tikzstyle{NAME}=[PROPERTY LIST] > > % Node styles > \tikzstyle{red dot}=[fill={rgb,255: red,191; green,0; blue,64}, draw=black, shape=circle] > \tikzstyle{green dot}=[fill={rgb,255: red,0; green,128; blue,128}, draw=black, shape=circle] > \tikzstyle{rectangle}=[fill=white, draw=black, shape=rectangle] > \tikzstyle{triangle}=[fill=white, draw=black, regular polygon sides=3, regular polygon, shape=regular polygon, tikzit shape=regular polygon, shape border rotate=180, minimum size=5mm, inner sep=0pt] > \tikzstyle{bullet}=[fill=black, draw=black, shape=circle, minimum size=2mm, inner sep=0pt] > \tikzstyle{noborder}=[fill=none, draw=none, shape=circle] > \tikzstyle{widerectangle}=[fill=white, draw=black, shape=rectangle, minimum width=16mm, minimum height=4mm, inner sep=1pt, tikzit shape=rectangle] > > % Edge styles > \tikzstyle{new edge style 0}=[-] > > > > > %% paste tikzpicture here > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAA0QBfU9TXfIRQBGclVqMWbADrSAtnRwALAEYrgAMS7deIDNjwEiZYePrNWiDgAIA9NeCyA1jABO12QB9ZAYygQcD2lPbS5xGCgAc3giUAAzVwg5JDIQHAgkACYeeMTkxFT0pFEJCxlgoL8AoM8dXKSs6iLEEoYsMEsQKDo4JQiQanMpK1kYAA8sOBw4awBCIIB3LFg8BlhHCo2qnC4vbWoGOhUYBgAFfkMhEFcsSKUcAbS6LAY2SA7uCi4gA > \begin{tikzcd} > X \arrow[d] \arrow[r, "\| \cdot \|"] & \mathbb{F} \\ > X / {D} \arrow[ru, "\exists ! \widetilde{\| {\cdot}\|}"', dashed] & > \end{tikzcd} > %% end paste > > \end{document} > ``` > > (See [[universal property]] of [[characterization of quotienting a module|quotient vector spaces]].) > A well-known example of this construction is the [[Lp-norm]]. ---- #### [^1]: Note that if $\|v\|=0$ and $\|w\|=0$ then $\|v+w\| \leq \|v\|+\|w\|=0$ and if $\|v\|=0$ then $\|cv\|=|c|\|v\|=0$, so this is indeed a [[linear subspace]] of $X$. [^2]: The direct linear algebra definition of quotienting by $D$ is [[quotient set|quotienting]] by the [[equivalence relation]] $a \sim b \iff a-b \in D \iff \|a-b\|=0$. 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 > ```