Let β > 0, let G be a set of functions g : Rd → [−β, β], and let H be the set of all functions h : Rd ×R→R defined by
h(x, y) = |g(x) − Tβy| 2 ((x, y) ∈ Rd × R)
for some g ∈ G. Show that for any > 0 and any (x, y) n 1 = ((x1, y1),..., (xn, yn)) ∈ (Rd × [−β, β])n,
N1 (, H, (x, y) n 1 ) ≤ N1 4β , G, xn 1 .
Hint: Choose an L1 cover of G on xn 1 of minimal size. Show that you can assume w.l.o.g. that the functions in this cover are bounded in absolute value by β. Use this cover as in the proof of Theorem 10.3 to construct an L1 cover of H on (x, y) n 1 .