3.8 Rademacher identities. Fix m  1. Prove the following identities for any 2 R and any two hypothesis sets H and H0 of functions mapping from X to R:

(a) Rm(H) = jjRm(H):

(b) Rm(H + H0) = Rm(H) + Rm(H0):

(c) Rm(fmax(h; h0) : h 2 H; h0 2 H0g)  Rm(H) + Rm(H0);

where max(h; h0) denotes the function x 7! maxx2X(h(x); h0(x)) (Hint: you could use the identity max(a;

b) = 1 2 [a+b+ja????bj] valid for all a; b 2 R and Talagrand's contraction lemma (see lemma 5.7)).