Question

Let C [0,1] be the set of all continuous functions from [0,1] to R. For any f,g ∈ C[0,1] define d_sup(f,g) = max_xE[0,1] |f(x)−g(x)| and d₁(f,g) = ∫¹₀ |f(x)−g(x)| dx. a) Prove that for any n≥1, one can find n points in C[0,1] such that, in d_sup metric, the distance between any two points is equal to 1. b) Can one find 100 points in C[0,1] such that, in d₁ metric, the distance between any two points is equal to 1? Prove your answers.

Answer 1

Solution: Let be the set of all continuous functions from [0,1] to R.

For any , define

and .

(a) we have to prove that for any , one can find n points in such that

in dsup metric, the distance between any two points is equal to 1.

In space , points are just a continuous functions, we can define operation on them as and multiplication as,, called point-wise addition and point-wise multiplication respectively.

Let .

Since are constant functions being continuous, so

.

Now .

Therefore

Again, let

Therefore,

Similarly, let

Therefore,

In this way let for .

Then

Therefore,

Therefore for any , one can find n points in such that

in dsup metric, the distance between any two points is equal to 1.(Proved)