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by mmmmmmmike
2903 days ago
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Sort of. A Euclidean structure has “addition” but not “multiplication”. Having an additional operation that’s required to be associative, have inverses, distribute over addition, and play nicely with the norm turns out to be such a constraint that, as he mentions in the previous paragraph, the real numbers, complex numbers, and quaternions are the only such structures. Literally speaking then, “... when R^d is an associative normed division algebra” just means “when d = 1, 2, or 4”, except of course that the idea is to use the multiplicative structure in the proof. |
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