Th: Let ${ \mathbb{F} }$ be ${ \mathbb{R} }$ or ${ \mathbb{C} }.$ Let ${ V }$ be a normed ${ \mathbb{F}- }$vector space with a basis ${ \mathcal{B} = (b _1, \ldots, b _k) }.$ Now ${ V }$ is complete.
Pf: We have an isomorphism ${ \mathbb{F} ^k \to V }$ sending ${ x = (x _1 , \ldots, x _k ) ^{t} \mapsto \mathcal{B}x = \sum _{1} ^{k} x _j b _j .}$
If ${ \mathscr{V} \overset{T}{\to} \mathscr{W} }$ is an isomorphism of ${ \mathbb{F}-}$vector spaces and ${ \mathscr{W} }$ is normed, then ${ \lVert v \rVert _{\mathscr{V}} := \lVert T(v) \rVert _{\mathscr{W}} }$ is a norm on ${ \mathscr{V} }.$ So via the isomorphism ${ \mathbb{F} ^k \to V }$ above, we get a norm ${ \lVert x \rVert ^{'} := \lVert \mathcal{B}x \rVert }$ on ${ \mathbb{F} ^k }.$
All norms on ${ \mathbb{F} ^k }$ are equivalent (same proof works over ${ \mathbb{C} }$) and ${ \mathbb{F} ^k }$ is complete wrt say euclidean norm. So ${ \mathbb{F} ^k }$ is complete wrt ${ \lVert \ldots \rVert ^{'} }.$ This implies ${ V }$ is complete :
Let ${ (v _n) }$ be a Cauchy seq in ${ V }.$ Using basis vectors, we can write ${ v _n = \mathcal{B} x _n }$ with ${ x _n \in \mathbb{F} ^k }.$ Now ${ \lVert v _m - v _n \rVert }$ ${ = \lVert \mathcal{B}(x _m - x _n) \rVert }$ ${ = \lVert x _m - x _n \rVert ^{'} }.$ So seq ${ (x _n) }$ is Cauchy in ${ (\mathbb{F} ^k , \lVert \ldots \rVert ^{'} ) },$ hence convergent. So there is a ${ p \in \mathbb{F} ^k }$ such that ${ \lVert x _n - p \rVert ^{'} \overset{n \to \infty}{\longrightarrow} 0 .}$ Now ${ \lVert \mathcal{B}(x _n - p) \rVert \to 0 },$ ie ${ \lVert v _n - (\mathcal{B}p) \rVert \to 0 },$ as needed.

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Eg: We have ${ \mathscr{B}([0,1], \mathbb{R}) }$ ${ = \mathscr{B}[0,1] },$ the space of bounded functions on ${ [0,1] }$ with sup norm. Consider subspace ${ \mathcal{P} _N [0,1], }$ the space of all polynomial functions on ${ [0,1] }$ of degree ${ \leq N }.$ Now ${ \mathcal{P} _N [0,1] }$ is complete, so closed in ${ \mathscr{B}[0,1] }.$
So if ${ f }$ is a bounded function on ${ [0,1] }$ not in ${ \mathcal{P} _N [0,1] },$ there is an ${ \epsilon }$ ${ (= \epsilon _N ) \gt 0 }$ such that ${ \lVert p - f \rVert _{\infty} \geq \epsilon }$ for all ${ p \in \mathcal{P} _N [0,1] }.$