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[parent] every finite dimensional normed vector space is a Banach space (Theorem)
Theorem 1   Every finite dimensional normed vector space is a Banach space.

Proof. Suppose $ (V,\Vert\cdot\Vert)$ is the normed vector space, and $ (e_i)_{i=1}^N$ is a basis for $ V$. For $ x=\sum_{j=1}^N \lambda_j e_j$, we can then define

$\displaystyle \Vert x \Vert' = \sqrt{\sum_{j=1}^N \vert\lambda_j\vert^2} $
whence $ \Vert\cdot\Vert'\colon V\to \mathbb{R}$ is a norm for $ V$. Since all norms on a finite dimensional vector space are equivalent, there is a constant $ C>0$ such that
$\displaystyle \frac{1}{C} \Vert x \Vert' \le \Vert x \Vert \le C \Vert x \Vert', \quad x\in V. $
To prove that $ V$ is a Banach space, let $ x_1,x_2,\ldots$ be a Cauchy sequence in $ (V,\Vert\cdot \Vert)$. That is, for all $ \varepsilon>0$ there is an $ M\ge 1$ such that
$\displaystyle \Vert x_j-x_k \Vert <\varepsilon, \ $   for all$\displaystyle j,k\ge M. $
Let us write each $ x_k$ in this sequence in the basis $ (e_j)$ as $ x_k=\sum_{j=1}^N \lambda_{k,j} e_j$ for some constants $ \lambda_{k,j}\in \mathbbmss{C}$. For $ k,l\ge 1$ we then have
$\displaystyle \Vert x_k-x_l\Vert$ $\displaystyle \ge$ $\displaystyle \frac{1}{C} \Vert x_k-x_l \Vert'$  
  $\displaystyle \ge$ $\displaystyle \frac{1}{C} \sqrt{\sum_{j=1}^N \vert\lambda_{k,j}-\lambda_{l,j}\vert^2}$  
  $\displaystyle \ge$ $\displaystyle \frac{1}{C} \vert\lambda_{k,j}-\lambda_{l,j}\vert$  

for all $ j=1,\ldots, N$. It follows that $ (\lambda_{k,1})_{k=1}^\infty, \ldots, (\lambda_{k,N})_{k=1}^\infty$ are Cauchy sequences in $ \mathbbmss{C}$. As $ \mathbbmss{C}$ is complete, these converge to some complex numbers $ \lambda_1, \ldots, \lambda_N$. Let $ x=\sum_{j=1}^N \lambda_j e_j$.

For each $ k=1,2,\ldots$, we then have

$\displaystyle \Vert x-x_k\Vert$ $\displaystyle \le$ $\displaystyle C \Vert x-x_k\Vert'$  
  $\displaystyle \le$ $\displaystyle C \sqrt{\sum_{j=1}^N \vert\lambda_{j}-\lambda_{k,j}\vert^2}.$  

By taking $ k\to \infty$ it follows that $ (x_j)$ converges to $ x\in V$. $ \Box$



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Cross-references: complex numbers, converge, complete, sequence, Cauchy sequence, Banach space, norm, basis, normed vector space, proof
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This is version 7 of every finite dimensional normed vector space is a Banach space, born on 2005-01-09, modified 2005-02-18.
Object id is 6633, canonical name is EveryFiniteDimensionalNormedVectorSpaceIsABanachSpace.
Accessed 3018 times total.

Classification:
AMS MSC46B99 (Functional analysis :: Normed linear spaces and Banach spaces; Banach lattices :: Miscellaneous)

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