Reference no: EM131017548

Let J ⊆ R be a compact interval and let A be a collection of continuous functions on J→R which satisfy the properties of the Stone-Weierstrass Theorem

[Stone-Weierstrass Theorem: Let K be a compact subset of R^p and let A be a collection of continuous functions on K to R with the properties:

a) The constant function e(x) = 1, x ∈ K belongs to A.

b) If f, g ∈ A, then αf + βg ∈ A, ∀ α, β ∈ R

c) if f, g ∈ A, then fg ∈ A

d) if x ≠ y, x, y ∈ K, ∃f ∈ A, st, f(x) ≠ f(y)

Then any continuous function K to R can be uniformly approximated on K by functions in A.]

Show that any continuous function on J x J ∈ R² to R can be uniformly approximated by functions of the form f₁(x)g₁(y) + ... + f_n(x)g_n(y) where f_i, g_i, ∈ A.