----- Here, $\mathbb{B}_{X}$ resp. $\mathbb{B}_{Y}$ denotes the open unit balls in $X$ resp $Y$. (Just $\mathbb{B}$ when there's no risk for ambiguity.) > [!proposition] Proposition. ([[successive approximations lemma]]) > Let $X$ be a [[Banach space]], $Y$ a [[norm|normed]] [[vector space]], and $T \in B(X,Y)$ a [[operator norm|bounded linear map]]. Suppose there exist $0<\varepsilon<1$ and $M>0$ [[distance from point to set|such that]] $\text{dist}\big(y, T(M \mathbb{B}_{X})\big)<\varepsilon$ for all $y \in \mathbb{B}_{Y}$. Then $\mathbb{B}_{Y} \subset T(\frac{M\mathbb{B}_{X}}{1-\varepsilon}).$ Furthermore, if $T(M \mathbb{B}_{X})$ contains a [[dense]] subset of $\mathbb{B}_{Y}$, then $\mathbb{B}_{Y} \subset T(M \mathbb{B}_{X})$. Hence $T$ is a [[quotient operator]] onto $Y$ and $Y$ is [[complete]] ([[Banach space|Banach]]). ^proposition > [!proof]- Proof. ([[successive approximations lemma]]) > Part B. ----- #### ---- #### 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 > ```