----- > [!proposition] Proposition. ([[Hamel basis of infinite-dimensional Banach space is uncountable]]) > Let $X$ be an [[dimension|infinite-dimensional]] [[Banach space]]. Then any [[basis|Hamel basis]] for $X$ must be [[uncountably infinite|uncountable]]. ^proposition > [!proof]+ Proof. ([[Hamel basis of infinite-dimensional Banach space is uncountable]]) > BWOC suppose $B=\{ x_{n}: n \in \mathbb{N} \}$ is a [[countably infinite|countable]] [[basis|Hamel basis]] for $X$ and write $F_{n}=\span\{ x_{k}: 1 \leq k \leq n \}$, $n \in \mathbb{N}$. By [[finite-dimensional normed vector spaces are complete]] and [[complete|complete subspace of metric space is closed]], $F_{n}$ is [[closed set|closed]]. Moreover, $F_{n}$ has empty [[topological interior|interior]]: for any $r>0$, if $B_{r}(x )$ is an open ball around some $x$ in $F_{n}$, then e.g. $y:=x+ \varepsilon \frac{v}{\|v\|} \in B_{r}(x)$ for any $0<\varepsilon<r$ and $v \in X-F_{n}$ (e.g. $v=x_{n+1}$) with $y \not \in F_{n}$. [[Baire category theorem|Baire's Theorem]] then implies $\bigcup_{i=1}^{\infty}F_{n}$ has empty interior, but this union equals $X$ itself, 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 > ```