Today I plan to write a treatise on spaces. are normed spaces over with the p-norm, or .

Say we have the space over . This just means that , where . That is a norm is proved using standard arguments (including Minkowski’s argument, which is non-trivial).

Now we have a metric in spaces: .

Now we prove that every space is complete. Say we have a cauchy sequence . This means that for every , there exists an such that for . This implies that for any , . As is complete, there exists a limit point for each coordinate. Using standard arguments from here, we can prove that spaces are complete. over is called

spaces are also complete.