For x = ( x 1, x 2, …, x n ) and y = ( y 1, y 2, …, y n ) the following metrics are defined: L p metric: for p ∈ ℝ, p ≥ 1, ρ p ( x, y ) = ( | x 1 − y 1 | p + … + | x n − y n | p ) 1 ∕ p L ∞ metric: ρ ( x, y ) = max ( | x 1 − y 1 |, …, | x n − y n | ) Of course, we need to prove that both L p and L ∞ together with ℝ n satisfy axioms of a metric space.
Pick x, y ∈ ℝ n. We can see that max i | x i − y i | ≤ ( ∑ i | x i − y i | p ) 1 ∕ p ≤ n 1 ∕ p max i | x i − y i | Since: | x j − y j | ≤ max i | x i − y i | And: ∑ i | x i − y i | p ≤ n p max i | x i − y i |.