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.

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Graduate student

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