Answer to Question #239319 in Differential Geometry | Topology for Fozia Sayda

Question

The Euclidean space p² is metric space . Specially R^' (real line) R²( the complex plane) etc are metric .

Answer 1

Define d : $R^2 × R^2 → R \ by \ d(x, y) = |x_1 − y_1| + |x_2 − y_2| x = (x_1,x_2), y = (y_1,y_2)$ . Then d is a metric on $R^2$ , called the $ℓ_1$ metric. It is also referred to informally as the “taxicab” metric, since it's the distance one would travel by taxi on a rectangular grid of streets. , or maximum, metric.

$The \ Euclidean \ metric \ is \ the \ function \ d:R^n×R^n\to R \\\ that \ assigns \ to \ any \ two \ vectors \ in \ Euclidean \ n-space\\ \ x=(x_1,...,x_n) \ and \ y=(y_1,...,y_n) \ the \ number \ \\ d(x,y)=\sqrt{((x_1-y_1)^2+...+(x_n-y_n)^2)}, \\$

and so gives the "standard" distance between any two vectors in  $R^n$ .