2.4 Problems

Problem 1. Let V be a vector space and define the function d : V ×V → [0,∞) by

(a) Show that d is a metric (see Definition 8).

(b) Show that d cannot come from a norm.

Problem 2. Let S_n be the set of all ASCII strings of n character length and define the Hamming distance h(x,y) for any two x,y ∈S_n by the number of corresponding positions where x and y are different.

For instance,

Show that h satisfies the three defining properties of a metric. (Note that S_n is not a vector space so, technically, the Hamming distance is not a metric.)

Problem 3. Let ∥⋅∥ be a norm on the vector space ℝⁿ, and define the mapping f : ℝⁿ →ℝⁿ,

Show that

is a norm on ℝⁿ.

Problem 4. Let a₁,…,a_n/span>0 be arbitrary positive numbers. Show that

is an inner product, where x = (x₁,…,x_n) and y = (y₁,…,y_n).

Problem 5. Let V be a finite-dimensional inner product space, let v₁,…...