(in-progress) ---- All [[vector space|vector spaces]] are implicitly assumed over $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$. > [!proposition] Proposition. ([[characterization of quotienting a (semi)normed vector space]]) > The [[universal property|universal property]] [[characterization of quotienting a module|of]] [[quotient module|quotient vector spaces]] passes to both > 1. The [[subcategory]] $\mathsf{Seminorm_{T}}$ of [[seminorm|seminormed spaces]] [[continuous|and]] [[operator norm|bounded]] [[linear map|linear maps]]. > 2. The further [[subcategory]] $\mathsf{Seminorm}_{\mathsf{M}}$ of [[norm|seminormed spaces]] and [[Lipschitz continuous|nonexpansive]] [[linear map|linear maps]]. > > Explicitly: let $X,Y$ be [[norm|seminormed spaces]] and let $W \leq X$ be a [[linear subspace]]. We know that the [[universal property]] of [[quotient module|quotient vector spaces]] states that for any [[linear map|linear map]] $T:X\to Y$ satisfying $W \subset \operatorname{ker }T$,[^1] there exists a unique [[linear map]] making the following [[diagram]] commute: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAA0QBfU9TXfIRQBGclVqMWbAJrdeIDNjwEiZYePrNWiDgHoAOvog0YAJwZYwMYAHUucvksFEATGOqapOgBQAqAJTc4jBQAObwRKAAZqYQALZIZCA4EEiiElpsACoOIDHxidQpSC480bEJiG7JqYjpntoghjAAHlhwOHAABACEXYYA7liweAywwFn21Ax0AEYwDAAK-MpCIKZYoQAWOEFcQA > \begin{tikzcd} > X \arrow[r, "T"] \arrow[d] & Y & (*) \\ > X/{W} \arrow[ru, "\exists ! \widetilde{T}"'] & & > \end{tikzcd} > \end{document} > ``` > Here, $X/W$ is endowed with the [[quotient topology]].[^2] > > Then: > > **1.** *(Objects)* The [[quotient topology|(quotient)]] [[topological space|topology]] on $X / W$ is [[metric topology|induced]] by the [[seminorm]] $\|x + W\|:= \text{dist}(x, W),$ > witnessing $X/W$ to be a [[seminorm|seminormed space]]. > > **2. ($\pi$ is in $\mathsf{T,M}$)** [[universal property of quotient topology|By (the quotient topology) construction]], we know $\pi:X \to X /W$ is [[continuous]] (thus $\pi$ is in $\mathsf{Seminorm_{T}}$). This is also [[characterizing continuity of linear maps|witnessed by]] $\|\pi\|=1$ if $\overline{W} \neq X$ and $\|\pi\|=0$ [[dense|if]] $\overline{W}=X$ (thus $\pi$ is in fact in $\mathsf{Seminorm_{M}}$). > > **3.** *$(\mathsf{Seminorm_{T}})$-morphisms)* In general, $\|\widetilde{T}\| =\|T\|$. Hence if $T$ is [[operator norm|bounded]], then so is $\widetilde{T}$. > > **4.** *$(\mathsf{Seminorm_{M}})$-morphisms).* Following from (3), if $T$ is [[Lipschitz continuous|nonexpansive]] ($\|T\| \leq 1$) then so is $\widetilde{T}$. > > Put together, $(1)-(4)$ show that the [[diagram]] $(*)$, produced by the [[universal property]] of [[quotient set|quotients]] in the ambient [[category]] $\mathsf{Vect}$, in fact constitutes a valid [[diagram]] in $\mathsf{Seminorm_{T}}$ and $\mathsf{Seminorm_{M}}$. > > Lastly: in $\mathsf{Norm_{T}}$ and $\mathsf{Norm_{M}}$, replace $W$ by $\overline{W}$; then $X / \overline{W}$ is [[norm|normed]] and the discussion above goes through. Suppose $\overline{W} \neq X$. Then if $x \in X$ with $\|x\|=1$, have $\|\pi(x)\|=\inf_{w \in \overline{W}} \|x-w\| \leq \|x\|$ by looking at $w=0$. OTOH, $\|\pi(x)\|$ > [!proposition] Corollary. (First bijection theorem for normed spaces) > The [[universal property]] of [[quotient module|quotient vector spaces]] entails that if $T:X \to Y$ is any [[linear map|linear map]] between [[seminorm|seminormed spaces]], then there is a unique [[linear map|linear map]] $\widetilde{T}$ making the diagram commute, and $\widetilde{T}$ is a [[linear isomorphism]] onto its image: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAA0QBfU9TXfIRQBGclVqMWbAJrdeIDNjwEiZYePrNWiDgAIA9LuAAdYwGsYAJ10AVLnL5LBRUeuqapO0xDRW6OCEswOgBbGGAsEK5bBwV+ZSFkACYxd0ltEAAKACpsgEpucRgoAHN4IlAAM0sIEKQyEACkUQktNhtY6tr66ibEFJAACxg6KDYcAHcIYdGEHiqauv7eiCQAZmoZsZ1J6ZGoBF66LAY2QYgIMxBqBiwwDKg6OGGxtLavYxgADyw4HDhdABCXSmCZYWB4BiwYB2a4gBh0ABGMAYAAV4s4dJYsCVBjhOot1itmkcTmcLlcuBQuEA > \begin{tikzcd} > X \arrow[r, "T"] \arrow[d, two heads] & Y & (**) \\ > X / {\ker T} \arrow[r, "\exists ! \widetilde{T}"', two heads, dashed, hook] & \operatorname{im} T \arrow[u, hook] & > \end{tikzcd} > \end{document} > ``` > > > Augmenting with the universal properties above for $\mathsf{Seminorm_{T}}$ resp. $\mathsf{Seminorm_{M}}$ automatically produces the following "[[first isomorphism theorem|first bijection theorem]]s" for $\mathsf{Seminorm_{T}}$ resp. $\mathsf{Seminorm_{M}}$. > > **1. (First bijection theorem for $\mathsf{Seminorm_{T}}$)** If $T$ is a [[operator norm|bounded linear map]], then so is $\widetilde{T}$ in the [[diagram]] $(* *)$. > > **2. (First bijection theorem for $\mathsf{Seminorm_{M}}$)** In fact, $\|T\|=\|\widetilde{T}\|$, so that if $T$ is [[Lipschitz continuous|nonexpansive]] then so is $\widetilde{T}$ in the [[diagram]] $(* *)$. > > > > > Now, when can we upgrade to a true [[first isomorphism theorem|first]] *[[isomorphism]]* [[first isomorphism theorem|theorem]] rather than merely a "first *bijection* theorem"? That is, when is $\widetilde{T}^{-1}: \operatorname{im }T \to X / \operatorname{ker }T$ also a morphism in $\mathsf{Seminorm_{T}}$ resp. $\mathsf{Seminorm_{M}}$? (We call $T$ a **quotient operator (onto its image)**[^9] in such a case.) This is characterized below. > > > > [!proposition] Characterization of quotient operators. (For what $T$ does FIT hold?) > > > > For $T:X \to (\operatorname{im }T \subset Y)$ a [[operator norm|bounded linear map]] between [[seminorm|seminormed spaces]], the following are equivalent: > > 1. $\widetilde{T}^{-1}$ is bounded (i.e. $T$ is a quotient operator onto its image) > > 2. $\widetilde{T}$ is an [[open map]] ($\iff$ $T$ is an [[open map]] onto its image) > > 3. There exists $M>0$ such that for all $y \in \operatorname{im }T$ there exists $x \in X$ with $Tx=y$ and $\|x\| \leq M \| y\|$ > > 4. $T(\mathbb{B})$ absorbs a neighborhood of $0$ in $\operatorname{im }T$ (that is, contains some ball around $0$ in $\operatorname{im } T$) > > 5. $T(\mathbb{B})$ has nonempty interior in $\operatorname{im }T$. > >> > > > Moreover, $T$ is an [[Lipschitz continuous|isometric]] quotient operator onto its image if and only if $T(\mathbb{B}_{X})=\mathbb{B}_{Y}$ . [^9]: "Onto its image" because quotient operators are required to be surjective. > [!proof] > > ^proof $(4)$ follows from $(3)$; so just have to show $(1)$-$(3)$. **1.** We have to show that the [[distance from point to set]] indeed defines a [[seminorm]] on $X / W$. Then we have to show this induces the [[quotient topology]]. - [ ] TODO **2.** If $\overline{W}=X$, then $\|\pi(x)\|=\text{dist}(x, W)=0$ for all $x,w$, hence $\|\pi\|=0$. Assume $\overline{W}\neq X$. In this case, $\frac{\|\pi(x)\|}{\|x\|}=$ **3.** - [ ] TODO **4.** (characterization of quotient operators) - [ ] todo ---- #### [^1]: We could equivalently [[quotient set|quotient]] by an [[equivalence relation]] $\sim$ (as is [[quotient space|common in]] [[topological space|topology]]) and place the condition $(x \sim y) \implies (Tx=Ty)$ on $T$ ([[universal property of quotient sets|cf.]]). The discussion in [[characterization of quotienting a group|quotient group]] carries over to show that the collection of equivalence relations on $X$ equals the collection of subspaces we could mod out by. [^2]: As induced by the [[equivalence relation]] $v_{1} \sim v_{2} \iff (v_{1}-v_{2}) \in \operatorname{ker }(\pi:V \to V / W)$, ---- #### 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 > ```