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*what would be wrong in defining*##||X||##

*in an ##M^n## by*

$$||X||^2 = \sum_j (X_U^j)^2$$ The only problem I can see is that that definition is not independent of the chosen coordinate systems and thus not intrinsic to ##M^n##. Is there anything else I'm missing?

BTW, I'm not sure this is

*Differential Geometry*... Is this

*Topology*?