318 CHAPTER 12. BANACH SPACES

22. Given an inner product space H show that every orthonormal set of vectors is linearlyindependent. Now suppose V is a finite dimensional subspace of H. Show that V isspan(d1, ...,dn) where {dk}n

k=1 is a maximal orthonormal set of vectors contained inV and that n is the dimension of V . Show that Px ≡ ∑

nk=1 (x,dk)dk is the projection

of x to V . That is |x−∑nk=1 (x,dk)dk| ≤ |x−∑

nk=1 akdk| for every choice of ak. Also

show that this projection map is unique.

23. Show that if g ∈ C ([−π,π]) with g(−π) = g(π) then for x the point on the unitcircle S1 determined in the usual way by x, then g : S1 → C is continuous. Let-ting A be the algebra which consists of all products and linear combinations of thefunctions

{einx}

n∈Z, show that A is dense in D consisting of the set of functionsof C ([−π,π]) with g(−π) = g(π). Now if f ∈ L2 (−π,π) , show there is g ∈ Dsuch that ∥ f −g∥L2(−π,π) < ε . Show the functions x→ 1√

2πeinx for n ∈ Z are an or-

thonormal set of functions for the inner product ( f ,g)≡∫

π

−πf gdx and that therefore,

Sn f (x) gives the best approximation to f in L2 (−π,π) out of all linear combinationsof eikx for |k| ≤ n. Conclude from this that ∥ f −Sn f∥L2(−π,π)→ 0. Also explain why{

einx}

n∈Z is a maximal orthonormal set.