A first example is the vector space ℓ ℓ p consisting of infinite vectors with real entries x = ( x 1 , x 2 , … … , x n , … … ) whose p -norm ( 1 ≤ ≤ p ≤ ≤ ∞ ∞ ) given by ‖ ‖ x ‖ ‖ ∞ ∞ := sup i | x i | for p = ∞ ∞ , and ‖ ‖ x ‖ ‖ p := ( ∑ ∑ i | x i | p ) 1 p for p < ∞ ∞ .