a) L1 norm = 1 + 2 + 3 + 4

= 10

b) L2 norm = sqrt(1^2 + 2^2 + 3^2 + 4^2)

= sqrt(30)

c) L_inf norm = max(1, 2, 3, 4)

= 4

a) If the L_inf norm of given points = 1, then the largest absolute value of any point is 1.

In Rd, x = {1, 1, … , 1} of length d. Thus the largest L1 norm is just 1 * d = d

For the largest L2 norm, x_2 = sqrt(1^2 + 1^2 + … + 1^2) = sqrt(d)

b) If the L2 norm of given points = 1, then the largest squared value of any point is 1

The largest L1 norm for all points where the L2 norm = 1 is sqrt(d)

The largest L_inf norm for all points where the L2 norm = 1 is 1

Checking the axioms of non-negativity, symmetry, triangle inequality, ****Identity of Indiscernibles:

1. Non-negativity:

   no entries are negative

2. symmetry:

   If we turn the table into a matrix, we see it is symmetric. More mathematically this means dist(x, y) = dist(y, x) for all x, y, which holds.

3. Indiscernible:

   Only time where distance = 0 between two points is when comparing a point to itself

4. Triangle Inequality:

   For all x, y, |x + y| ≤ |x| + |y| therefore all properties hold

Since all properties hold, it is considered a metric.