---- > [!definition] Definition. ([[compact embedding of normed spaces]]) > Let $(X, \|\cdot\|_{X})$ and $(Y, \|\cdot\|_{Y})$ be two [[norm|normed spaces]] with $X \subset Y$. We say $X$ is **compactly embedded in $Y$**, and sometimes write $X \Subset Y$, if the set-inclusion $\iota:X \hookrightarrow Y$ is a [[continuous]] [[compact operator]], meaning > 1. ([[continuous]]) There exists $C>0$ such that $\|\cdot\|_{X} \leq C \|\cdot\|_{Y}$; > 2. ([[compact operator]]) any [[bounded set]] in $X$ is [[precompact]] in $Y$. > > (todo: swap roles of $X$ and $Y$ here for clarity) > ---- #### ---- #### 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 > ```