According to me the formula satisfies all the properties of a norm.
(1st property) For both positive and negative numbers the formula returns positive output .
(2nd property) The only time square root of the square of a quantity /vector is zero is when the vector itself is zero.
(3rd property)|au| = sqrt((au1)^2+(au2)^2+...) = sqrt(a^2(u1^2+u2^2+...)) = sqrt(a^2)sqrt(u1^2+u2^2+...) = |a||u|
(4th property) |u|+|v| >= |u+v|
By taking the square on both the sides the equation reduces to
(sqrt(sum(ui^2))* sqrt(sum(vi^2))) ?? 2(sum(uivj)) by using Cauchy-Schwarz inequality we can show that the statement is true. Therefore the formula satisfies the fourth property also.