Takehome Exam, MATH 515, Spring 09
Problem 1) ( 75 pts) For a < b let W[a, b] be the vector space of continuously
differentiable functions on [a, b] with values in C. For f, g ∈W[a, b] let
hf, giW:= Zb
a
(f(t)g(t) + f0(t)g0(t))dt
and |f|W:= phf, f iW.
(a) Show that h,iWdefines a positive definite hermitian form on W[a, b], and
thus |.|Wis a norm on W[a, b].
(b) Consider W[0,1]. Let χn∈W[0,1] be defined by χn(t) := e2πint,n∈Z
and 0 ≤t≤1. Prove that hχn,χmiW= 0 when n6=m, and |χn−χm|W=
p2+4π2(n2+m2).
(c) Prove that in W[0,1]
hf, coshiW=f(1) sinh(1)
and deduce that
{f∈W[0,1] : f(1) = 0}
is a closed subspace of W[0,1].
(d) Prove that W[a, b] is not a Hilbert space. Hint: Consider indefinite integrals
of the sequence of functions fn(t) = 0 for t≤1
2−1
n,fn(t) = 1 for t≥1
2and
the graph of fn(t) is the line joining (1
2−1
n,0) and (1
2,1) for t∈[1
2−1
n,1
2].
Problem 3) (75 pts) page 108, Chapter V, §6, Exercise 8. The formula in (b)
should be:
hx, yi=1
2(|x+y|2− |x|2− |y|2) + i1
2(|x+iy|2− |x|2− |y|2)
Problem 3) (Bonus Problem 50 pts) page 108, Chapter V, §6, Exercise 7
1