MATHEMATICAL TRIPOS Part II PAPER 3 Before you begin read ...

21FLinear AnalysisState the Stone–Weierstrass Theorem for real-valued functions.State Riesz’s Lemma.11Let K be a compact, Hausdorff space and let A be a subalgebra of C(K) separatingthe points of K and containing the constant functions. Fix two disjoint, non-empty, closedsubsets E and F of K.(i) If x ∈ E show that there exists g ∈ A such that g(x) = 0, 0 g < 1 on K, and g > 0on F . Explain briefly why there is M ∈ N such that g 2 M on F .(ii) Show that there is an open cover U 1 , U 2 , . . . , U m of E, elements g 1 , g 2 , . . . , g m of A andpositive integers M 1 , M 2 , . . . , M m such that0 g r < 1 on K, g r 2M ron F, g r < 12M ron U rfor each r = 1, 2, . . . , m.(iii) Using the inequality1 − Nt (1 − t) N 1 Nt(0 < t < 1, N ∈ N) ,show that for sufficiently large positive integers n 1 , n 2 , . . . , n m , the elementof A satisfiesh r = 1 − (1 − g nrrnrr)M0 h r 1 on K, h r 1 4on U r , h r ( ) 13 m4on Ffor each r = 1, 2, . . . , m.(iv) Show that the element h = h 1 · h 2 · · · · · h m − 1 2of A satisfies− 1 2 h 1 2on K, h − 1 4 on E, h 1 4on F.Now let f ∈ C(K) with ‖f‖ 1. By considering the sets {x ∈ K : f(x) − 1 4 } and{x ∈ K : f(x) 1 4 }, show that there exists h ∈ A such that ‖f − h‖ 3 4. Deduce that Ais dense in C(K).Part II, Paper 3[TURN OVER