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The Toroidal Elton–Odell Theorem

The Elton–Odell theorem gives, in every infinite-dimensional normed space, a unit-sphere sequence with mutual distances at least $1+\varepsilon$. Over $\mathbb{C}$, identifying vectors differing by a unimodular scalar gives a toroidal distance. Does every infinite-dimensional complex normed space admit such a uniformly separated sequence for that distance? Yes.

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analysisJul 19, 2026Significance 15/100Registry: unreviewed

The Toroidal Elton–Odell Theorem

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The Elton–Odell theorem gives, in every infinite-dimensional normed space, a unit-sphere sequence with mutual distances at least $1+\varepsilon$. Over $\mathbb{C}$, identifying vectors differing by a unimodular scalar gives a toroidal distance. Does every infinite-dimensional complex normed space admit such a uniformly separated sequence for that distance? Yes.

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The Elton–Odell theorem gives, in every infinite-dimensional normed space, a unit-sphere sequence with mutual distances at least $1+\varepsilon$. Over $\mathbb{C}$, identifying vectors differing by a unimodular scalar gives a toroidal distance. Does every infinite-dimensional complex normed space admit such a uniformly separated sequence for that distance? Yes.

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The Toroidal Elton–Odell Theorem — Mathematical Frontier Network