MATH 519 Homework
Fall 2013
31. In the Hilbert space L2 (−1, 1) what is M ⊥ if
a) M = {u : u(x) = u(−x) a.e.}
b) M = {u : u(x) = 0 a.e. for − 1 < x < 0}.
Give an explicit formula for the projection onto M in each case. (These are both closed
subspaces – you need not show this.)
32. Let M be a closed subspace of a Hilbert space H, and PM be the corresponding
projection. Show that
2
a) PM
= PM
b) hPM x, yi = hPM x, PM yi = hx, PM yi for any x, y ∈ H.
33. Show that `2 is a Hilbert space. (Discussion: The only property you need to check
is completeness, and you may freely use the fact that R is complete. A Cauchy sequence
in this case is a sequence of sequences, so use a notation like
(n)
(n)
x(n) = {x1 , x2 , . . . }
(n)
where xj denotes the j’th term of the n’th sequence x(n) . Given a Cauchy sequence
(n)
2
= xj for each fixed j.
{x(n) }∞
n=1 in ` you’ll first find a sequence x such that limn→∞ xj
2
You then must still show that x ∈ ` , and one good way to do this is by first showing
that x − x(n) ∈ `2 for some n.)
34. Let H be a Hilbert space.
a) If xn → x in H show that {xn }∞
n=1 is bounded in H.
b) If xn → x, yn → y in H show that hxn , yn i → hx, yi.
35. a) Compute orthogonal polynomials of degree 0,1,2,3 on [−1, 1] and on [0, 1] by
applying the Gram-Schmidt procedure to 1, x, x2 , x3 in L2 (−1, 1) and L2 (0, 1). (See
problem 6.12 in the text for more about these polynomials in the case of L2 (−1, 1), the
so-called Legendre polynomials.)
b) Using the result of part a) in the L2 (−1, 1) case, compute the best polynomial
approximations of degrees 0,1,2 and 3 to u(x) = ex in L2 (−1, 1). Feel free to use any
symbolic calculation tool you know to compute the necessary integrals, but give exact
coefficients, not calculator approximations. If possible, produce a graph displaying u
and the 4 approximations.