---- > [!definition] Definition. ([[nearly open map]]) > A map $f:X\to Y$ between [[topological space|topological]] [[topological vector space|vector]] [[vector space|spaces]] is said to be **nearly open** if, for each [[neighborhood]] $U$ of $0_{X}$, the [[closure]] $\overline{f(U)}$ contains a [[neighborhood]] $V$ of $0_{Y}$. ^definition > [!basicproperties] > A [[linear map|linear map]] $f: E\to F$ between [[norm|normed vector spaces]] is nearly open if $f(E)$ is [[Baire space|non-meager]] in $F$. > > > > [!proof]- Proof. > > It suffices to show the result assuming $U$ is an open ball centered at $0_{X}$. We have $f(E)=f(\bigcup_{n \in \mathbb{N}}nU)=\bigcup_{n \in \mathbb{N}}f(nU) \subset \bigcup_{n \in \mathbb{N}} \overline{f(nU)}$ > > Since $f(E)$ is [[Baire space|non-meager]] in $F$, [[Baire space|we know]] that some $\overline{f(nU)}$ has nonempty [[topological interior|interior]]; let $y$ be an interior point, and let $r>0$ be a radius witnessing this (i.e. $r$ is such that $B_{r}(y) \subset \overline{f(nU)}$). Then for any $v \in B_{r}(0_{Y})$, $v=\underbrace{ (v+y) }_{ \in \overline{f(nU)} }\underbrace{ -y }_{ \in \overline{f(-nU)} } \in \overline{f(2nU)}.$ > > It follows that $\frac{v}{2n} \in \overline{f(U)}$, witnessing that $B_{\frac{r}{2n}}(0) \subset \overline{f(U)}$ and thus the result. > ---- #### ---- #### 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 > ```