---- > [!definition] Definition. ([[space of null sequences]]) > The **space of null sequences** is the [[linear subspace|linear]] [[subspace topology|subspace]] of $\ell^{\infty}$ given by $c_{0}:= \{ (a_{1},a_{2},\dots) \in \ell^{\infty} : \lim_{k \to \infty} a_{k}=0\}.$ > $c_{0}$ is a [[Banach space]], [[complete|being a]] [[closed set|closed subset]] [[Lp-norm|of]] $\ell^{\infty}$. ^definition > [!basicproperties] > - [[the dual of the space of null sequences naturally identifies with l1]] > - $\ell^{p} \subsetneq c_{0} \subsetneq \ell^{\infty}$ ^properties ---- #### ---- #### 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 > ```