3. Let V be a normed linear space and W a subspace of V . Let f V . Prove that the set of best approximations to f by elements in W is a convex set.

4. Let f and g in C[0,1], and constants, and denote by Bnf the Bernstein polynomial of f of degree n. Prove that

(a) Bn(f + g) = Bnf + Bng, i.e. Bn is a linear operator in C[0,1]. (b) If f(x) g(x) for all x [0,1] then Bnf(x) Bng(x) for all x [0,1], i.e. Bn is a monotone operator.

. Let V be a normed linear space and W a subspace of V. Let f E V. Prove that the set of best approximations to f by elements in W is a convex set. . Let f and g in C[0, l], a and 6 constants, and denote by B" f the Bernstein polynomial of f of degree n. Prove that (a) Bn(ozf + [39) = aan + ,BBng, i.e. B" is a linear operator in C[0,1]. (b) If f(x) 2 9(115) for all 27 6 [0,1] then an(:c) 2 Bng(:r) for all as 6 [0,1], i.e. Bn is a monotone operator