---- > [!definition] Definition. ([[compact operator]]) > A **compact operator** between [[norm|normed spaces]] $X$ and $Y$ is a [[linear map|linear operator]] $T:X\to Y$ that sends [[bounded set|bounded subsets of]] $X$ to [[precompact]] subsets of $Y$. ^definition > [!equivalence] > > Let $T:X \to Y$ be a [[linear map|linear operator]]. TFAE: > 1. $T$ is a compact operator; > 2. $T(\mathbb{B})$ is [[precompact]] in $Y$; > 3. For any [[bounded set|bounded sequence]] $(x_{n}) \subset X$, $(Tx_{n})$ has a [[sequence|convergent]] [[subsequence]] > [!proof] > $(1) \iff (3)$ is immediate from the [[metrizable]]-space characterization of [[precompact|precompactness]]: suppose $T$ is compact operator. Any bounded sequence $(x_{n})$ lives in $r\mathbb{B}$ for some $r>0$, so $T(r\mathbb{B})$ is [[precompact]]; by the [[metrizable]]-space characterization of [[precompact|precompactness]] it has a [[sequence|convergent]] [[subsequence]]. Conversely suppose $(3)$. $B$ bounded, WTS $T(B)$ is [[precompact]], enough to show every [[sequence]] $(Tx_{n})$ in $T(B)$ has a convergent subsequence. Such a sequence is by definition the image of a sequence in $B$, which is bounded because $B$ is bounded. > > $(1) \implies (2)$ is obvious. For $(2) \implies (3)$, note that if $T(\mathbb{B})$ is precompact then so is $rT(\mathbb{B})$, hence so is $T(r\mathbb{B})$, for all $r>0$. Thus if $(x_{n})$ is a bounded sequence in $X$, we can find some $r\mathbb{B}$ containing $(x_{n})$, whose image is precompact in $Y$, and so $Tx_{n}$ has a convergent subsequence. > ---- #### ---- #### 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 > ```