Unit n-Sphere under Euclidean Metric is Metric Subspace of Euclidean Real Vector Space
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Theorem
Let $\Bbb S^n$ be the unit $n$-sphere.
Let $d_S: \Bbb S^n \times \Bbb S^n \to \R$ be the real-valued function defined as:
- $\ds \forall x, y \in \Bbb S^n: \map {d_S} {x, y} = \sqrt {\sum_{i \mathop = 1}^{n + 1} \paren {x_i - y_i}^2}$
where $x = \tuple {x_1, x_2, \ldots, x_{n + 1} }, y = \tuple {y_1, y_2, \ldots, y_{n + 1} }$.
Then $\struct {\Bbb S^n, d_S}$ is a metric subspace of $\struct {\R^{n + 1}, d}$, where $d$ is the Euclidean metric on the real vector space $\R^{n + 1}$.
Proof
The metric given is the Euclidean metric restricted to the subset $\Bbb S^n$ of the real vector space $\R^{n + 1}$.
The result follows from Subspace of Metric Space is Metric Space.
$\blacksquare$
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 7$: Subspaces and Equivalence of Metric Spaces: Example $3$