---- > [!definition] Definition. ([[separable space]]) > A [[topological space]] $X$ is said to be **separable** if it contains a [[countably infinite|countable]] [[dense]] subset, [[closure|that is]], there exists a [[sequence]] $(x_{n})_{n=1}^{\infty}$ of elements of the space such that every nonempty open subset contains at least one element in the sequence. > > Further assuming $X$ is endowed with a [[metrizable|metric]] $d$, an **r-separated subset** of $X$ is a subset $S \subset X$ for which any two distinct points in $S$ are at least $r$ apart: $d(x,y) \geq r$ for all $x \neq y$ in $S$. > [!equivalence] > For [[metrizable]] spaces the notions of separable and [[second-countable space|second-countable]] coincide. ^equivalence > [!equivalence] Equivalence. > ^equivalence > [!basicexample] > - Euclidean space $\mathbb{R}^{n}$ is separable > - Euclidean [[Lp-norm|Lp spaces]] $L^{p}(E \subset \mathbb{R}^{n})$ [[Euclidean Lp spaces are separable|are separable]] for $1 \leq p < \infty$ ($L^{\infty}$ is not separable) > - The [[sequence]] spaces $\ell^{p}$ are separable for $1 \leq p < \infty$ ($\ell^{\infty}$ is not separable, [[space of null sequences|but]] $c_{0}$ is) ^basic-example - show the last bit > [!basicproperties] > - In a separable [[metric space]], every $r$-separated set is [[countably infinite|countable]]. > > > [!proof]- Proof. > > Let $(M,d)$ be separable and $(x_{n})_{n=1}^{\infty}$ be a countable dense subset. [[cover|Cover]] $M$ by the countable family of open balls $\left\{ B_{\frac{r}{2}}(x_{n}) \right\}_{n=1}^{\infty}$. If $S \subset M$ is $r$-separated, then each ball contains at most one point of $S$. Hence $S$ injects into $\mathbb{N}$. >- (Lemma 6.4) A [[norm|normed space]] $X$ over $\mathbb{F}$, is separable [[dual vector space|if]] $X^{*}$ is. Converse fails in general (take $\ell^{1}$), though clearly holds e.g. if $X$ is [[reflexive space|reflexive]]. > > > [!proof]- > > Use [[linear functional characterization of linear subspace closure|linear subspace is dense iff annihilator is trivial]]. > > Let $(f_{n})_{n=1}^{\infty } \subset \mathbb{S}_{X^{*}}$ be dense in $\mathbb{S}_{X^{*}}$. Obtain $(x_{n}) \subset \mathbb{S}_{X}$ with $|f_{n}(x_{n})|> \frac{1}{59}$; this we can do because $\|f_{n}\|=1$. Put $Y:=\span(x_{n})$; we claim $\overline{Y}=X$ and to will shows this by showing $Y^{\circ}=(0)$. Indeed, suppose $f \in X^{*}$ with $f(x_{n})=0$ for all $n$. Then, normalizing $f$ to the sphere if necessary, there exists $n$ such that $\|f-f_{n}\|< \frac{1}{59}$. In particular, $ | \, |\cancel{ f(x_{n}) }^{=0}|- \overbrace{ |f_{n}(x_{n})| }^{ >1/59 } \, | \leq |f(x_{n})-f_{n}(x_{n})|< \frac{1}{59},$ > > a contradiction. > > > ---- #### ---- #### 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 > ```