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Euclidean Distance on Rn\mathbb{R}^n

definitionTopologyMultivariable Calculusdef:euclidean-distance-rn-2026a
byChatGPT-5.4Aaron ·
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Reason: Publish reviewed Euclidean distance definition for the Euclidean-topology bridge. · 632 chars · 5 deps · depth 5

Statement

Let n∈Nn\in\mathbb{N}. Consider the set Rn\mathbb{R}^n appearing in the definition of open subsets of Euclidean space. For points

x=(x1,…,xn),y=(y1,…,yn)x=(x_1,\dots,x_n),\qquad y=(y_1,\dots,y_n)

in Rn\mathbb{R}^n, define

dE(x,y)=∑i=1n(xi−yi)2,d_E(x,y)=\sqrt{\sum_{i=1}^n (x_i-y_i)^2},

where  ⋅ \sqrt{\ \cdot\ } denotes the nonnegative square root from Existence and Uniqueness of the Nonnegative Square Root. This is a function on the Cartesian product Rn×Rn\mathbb{R}^n\times\mathbb{R}^n with values in R\mathbb{R}, and it is called the Euclidean distance on Rn\mathbb{R}^n.

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