Today I plan to write a treatise on $\ell_p^n$ spaces. $\ell_p^n$ are normed spaces over $\Bbb{R}^n$ with the p-norm, or $\|\|_p$.

Say we have the $\ell_p^n$ space over $\Bbb{R}^n$. This just means that $\|x\|_p=\left( |x_1|^p + |x_2|^p+\dots +|x_n|^p\right)^{\frac{1}{p}}$, where $x\in \Bbb{R}^n$. That $\|\|_p$ is a norm is proved using standard arguments (including Minkowski’s argument, which is non-trivial).

Now we have a metric in $\ell_p^n$ spaces: $d(x,y)=\|x-y\|=\left( |x_1-y_1|^p + |x_2-y_2|^p+\dots +|x_n-y_n|^p\right)^{\frac{1}{p}}$.

Now we prove that every $\ell_p^n$ space is complete. Say we have a cauchy sequence $\{x_1,x_2,x_3,\dots\}$. This means that for every $\epsilon>0$, there exists an $N\in\Bbb{N}$ such that for $i,j>N$ $\|x_i-x_j\|<\epsilon$. This implies that for any $e\in\{1,2,\dots,n\}$, $\|x_e-y_e\|<\epsilon$. As $\Bbb{R}$ is complete, there exists a limit point for each coordinate. Using standard arguments from here, we can prove that $L_p$ spaces are complete. $L_p$ over $\Bbb{R}^n$ is called $l_p^n$

$\ell_p^\infty$ spaces are also complete.