----- > [!proposition] Proposition. ([[characterization of quotienting a Banach space]]) > Let $X$ and $Y$ be [[Banach space|Banach spaces]]. Let $W \leq X$ be a [[linear subspace]], so that $\overline{W}$ [[complete|is]] a sub-Banach space of $X$. > > Then $X / \overline{W}$ is a [[Banach space]] with respect to [[distance from point to set|the]] [[characterization of quotienting a (semi)normed vector space|quotient norm]] $\|x+\overline{W}\|=\text{dist}(x, \overline{W})$. > [!proposition] Corollary. (Universal property of quotient Banach spaces) > As an immediate consequence, we see the [[universal property|universal property]] [[characterization of quotienting a module|of]] [[quotient module|quotient vector spaces]] passes to both[^5] > > 1. The [[subcategory]] $\mathsf{Ban_{T}}$ of [[Banach space|Banach spaces]] [[continuous|and]] [[operator norm|bounded]] [[linear map|linear maps]]. > 2. The further [[subcategory]] $\mathsf{Ban}_{\mathsf{M}}$ of [[Banach space|Banach spaces]] and [[Lipschitz continuous|nonexpansive]] [[linear map|linear maps]]. > > Explicitly: let $X,Y$ be [[Banach space|Banach spaces]] and let $W$ be a [[linear subspace]]. By the [[universal property]] of [[quotient module|quotient vector spaces]], any [[operator norm|bounded]] [[linear map|linear map]] $T:X\to Y$ satisfying $W \subset \operatorname{ker }T$,[^1][^2] 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/\overline{W} \arrow[ru, "\exists ! \widetilde{T}"'] & & > \end{tikzcd} > \end{document} > ``` > > (We care about $\overline{W}$ here because the [[closed set|closed]] [[linear subspace|linear subspaces]] [[complete|are precisely]] the sub-Banach spaces.) > > **1. $(\mathsf{Ban_{T}})$** $\widetilde{T}$ is also [[operator norm|bounded]]. In fact, $\|T\|=\|\widetilde{T}\|$, thus: > **2. $(\mathsf{Ban_{M}})$** If $T$ is moreover [[Lipschitz continuous|nonexpansive]], then so is $\widetilde{T}$. > > Thus the [[diagram]] $(*)$, produced in the ambient [[category]], passes to $\mathsf{Ban_{T}}$ and $\mathsf{Ban_{M}}$. Everything discussed in this block more generally lives in [[characterization of quotienting a (semi)normed vector space|quotient seminormed space]]; the point is that the proposition statement lets us validly pass to Banach spaces. [^5]: Specifically: the [[universal property]] first passes to $\mathsf{Seminorm_{T}}$ and $\mathsf{Seminorm_{M}}$ (this is the content of [[characterization of quotienting a (semi)normed vector space]]); then since the [[category|categories]] $\mathsf{Ban_{T}}$ and $\mathsf{Ban_{M}}$ of [[Banach space|Banach spaces]] are [[subcategory|full subcategories]] [[norm|of]] $\mathsf{Norm_{T}}$ and $\mathsf{Norm_{M}}$, to check that the ambient [[diagram|diagrams]] all live in $\mathsf{Ban_{T}}$ resp. $\mathsf{Ban_{M}}$, we just have to check that their objects (namely, $X / \overline{W}$) do. > [!proposition] Corollary of the corollary. (First bijection theorem) > The [[universal property]] of [[quotient module|quotient vector spaces]] entails that if $T:X \to Y$ is any [[linear map|linear map]] between [[Banach space|Banach 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 for $\mathsf{Ban_{T}}$ resp. $\mathsf{Ban_{M}}$ leads to the following "[[first isomorphism theorem|first bijection theorem]]s" for $\mathsf{Ban_{T}}$ resp. $\mathsf{Ban_{M}}$. > > **1. (First bijection theorem for $\mathsf{Ban_{T}}$)** If $T$ is a [[operator norm|bounded linear map]], then so is $\widetilde{T}$ in the [[diagram]] $(* *)$. > > **2. (First bijection theorem for $\mathsf{Ban_{M}}$)** In fact, $\|T\|= \|\widetilde{T}\|$ in the diagram $(* *)$. So if $T$ is [[Lipschitz continuous|nonexpansive]] then so is $\widetilde{T}$. > > > Now, when is this a true [[first isomorphism theorem|first]] *[[isomorphism]]* [[first isomorphism theorem|theorem]] rather than merely a "first bijection theorem"? That is, when is $T$ a [[characterization of quotienting a (semi)normed vector space|quotient operator (onto its image)]], meaning $\widetilde{T}^{-1}: \operatorname{im }T \to X / \operatorname{ker }T$ is also a morphism in $\mathsf{Ban_{T}}$ resp. $\mathsf{Ban_{M}}$? > > The [[characterization of quotienting a (semi)normed vector space]], gives a host of equivalent conditions answering the question. For $\mathsf{Ban_{T}}$, the [[open mapping theorem]] immediately gives the best answer one could hope for: "if $\widetilde{T}^{-1}$ is a *map* between objects in $\mathsf{Ban_{T}}$ then it is automatically a *$\mathsf{Ban_{T}}$-morphism* between them". This boils down to: > > **1. ($\mathsf{Ban_{T}}$)** $\widetilde{T}^{-1}$ is a [[operator norm|bounded linear map]] iff $\operatorname{im }T$ is [[closed set|closed]] (i.e. iff $\operatorname{im }T$ is actually a [[Banach space]]). > > In summary, the [[bijection]] $\widetilde{T}$ may be upgraded to a $\mathsf{Ban_{T}}$-[[isomorphism]] if and only if $\operatorname{im }T$ is [[closed set|closed]]. > > For $\mathsf{Ban_{M}}$, this condition is not enough; I am not going to elaborate here. [^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]: Since $\operatorname{ker }T$ is [[closed set|closed]] and $A \subset B$ implies $\overline{A} \subset \overline{B}$ [[closure|it's]] equivalent to require $\overline{W} \subset \operatorname{ker } T$. > [!proof]- Proof. ([[characterization of quotienting a Banach space]]) > ~ ----- #### > [!note] Remark. > This feels a lot like the [[closed map lemma]] ([[closed map lemma|continuous bijection from compact to Hausdorff is a homeomorphism]]). Probably there is an explicit connection. ^note ---- #### 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 > ```