Countable-dimensional sphere

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Definition

As the sphere in countable-dimensional real vector space

Denote by ${\mathbb{R}}^{\omega }$ the space of sequences of real numbers (i.e., things of the form $(x_1,x_2,\dots ,x_n,\dots )$) with the property that at most finitely many of the numbers are nonzero. Denote by $S^{\mathrm{\infty }}$ the subset given by:

$S^{\omega }=\{(x_1,x_2,\dots ,x_n,\dots )\in {\mathbb{R}}^{\mathrm{\infty }}\mid \sum _{i=1}^{\mathrm{\infty }}{x}_i^2=1\}$

Note that the actual summation involves only finitely many nonzero terms, so it is not in fact an infinite sum.

This space $S^{\omega }$, also denoted $S^{\mathrm{\infty }}$, is termed the countable-dimensional sphere or infinite-dimensional sphere.

As an inductive limit of finite-dimensional spheres

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Properties

• Countable-dimensional sphere is contractible