Metric Space Examples

Why

We give examples of metric spaces.

Let $n$ be a natural number. Let $A$ be $\R ^n$ and define $d: \R ^n \times \R ^n \to \R $ by

\[ d(a, b) = \sqrt{(a_1 - b_1)^2 + \cdot s + (x_n - y_n)^2}. \]

$(A, d)$ is a metric space.

Let $A$ be the unit circle in $\R ^2$. So $A = \Set{x \in R^2}{x_1^2 + b^2 = 1}$. Let $d_1: A \times A \to \R $ defined by

\[ d(a, b) = \sqrt{(a_1 - b_1)^2 + (a_2 - b_2)^2}. \]

Let $d_2: A \times A \to \R $ defined as the arc length between the two points. Both $(A, d_1)$ and $(A, d_2)$ are metric spaces.

Let $A = C([0, 1], \R )$. Let $d_1: A \times A \to \R $ be such that

\[ d_1(a, b) = \underset{x \in [0, 1]}{\max} \abs{a(x) - b(x)}. \]

Let $\lambda $ be the outer cover measure. Let $d_2: A \times A \to R$ be such that

\[ d_2(a, b) = \int_{[0, 1]} \abs{f - g} d\lambda . \]

Both $(A, d_1)$ and $(A, d_2)$ metric spaces.