----- > [!proposition] Proposition. ([[finite-dimensional normed vector spaces are complete]]) > Let $X$ be a [[dimension|finite-dimensional]] [[norm|normed]] [[vector space]] over $\mathbb{R}$ or $\mathbb{C}$. Then $X$ is [[complete]] (i.e. is a [[Banach space]]). ^proposition > [!proof]+ Proof. ([[finite-dimensional normed vector spaces are complete]]) > Let $(x_{m})_{m \in \mathbb{N}}$ be a [[Cauchy sequence]] in $X$. Write $n:=\text{dim }X$. Pick a [[basis]] $e_{1},\dots,e_{n}$ of $X$. Write $x_{m}=a_{1}^{(m)}e_{1}+\dots+a_{n}^{(m)}e_{n}$. If $x_{m}$ is [[Cauchy sequence|Cauchy]] in $X$, then so is each coordinate sequence $a_{i}^{(m)} \to a_{i}$. These coordinate sequences are in $\mathbb{R}$, which is [[complete]] by construction,[^1] hence they converge in $\mathbb{R}$. Therefore $(x_{m})$ converges. [^1]: To avoid being circular in these notes, here is a quick reproduction of the argument that [[Cauchy sequence|Cauchy sequences]] in $\mathbb{R}$ [[sequence|converge]]: combine the fact that [[Cauchy sequence|Cauchy sequence induced by norm is bounded]] with [[Bolzano-Weierstress]] and [[Cauchy sequence is convergent iff has 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 > ```