Math homework due in 3 hours

pedrocp

Let V be a linear space with norm ∥·∥, W a subspace of V, and f ∈ V. Prove that the set of best approximations to f by elements in W is convex.
Find the best uniform approximation to f(x) = sin2x on [0,2π] by polynomials of degree at most 2.
Let f ∈ C[a, b]. Find the best uniform approximation to f by a constant.
Let V = R3 with ∥·∥∞ and W = span{(0,1,0),(0,0,1)}. Let f = (3,6,4). Prove that
a best approximation to f is not unique.
Prove that every p ∈ Pn has a unique representation of the form
p(x)=a0 +a1T1(x)+...+anTn(x), (1) where Tj , for j = 1, . . . , n are the Chebyshev polynomials of degree j.
Plot T1, T2, T3 and T6.

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