ANALYSIS QUALIFYING EXAM PROBLEMS BRIAN LEARY ...

Then for N sufficiently large, we have that
N−1
1
1
P (x + nα) − P (t)dt
N
Then
n=0
1
N
N−1
n=0
ANALYSIS QUALS 45
0
f(x + nα) −
≤ 1
N−1
|f(x + nα) − P (x + nα)| +
N
n=0
+
1
0
1
N
1
0
N−1
n=0
f(t)dt
|P (t) − f(t)| dt < ε.
< ε
3 .
P (x + nα) −
1
0
P (t)dt
Thus, the convergence is uniform, which proves part (2). Part (1) follows
by noting that continuous periodic functions are dense in L2 ([0, 1]).
n
Problem 5: Let An(f) = 1
n f(x)dx. Show that there exists a continuous
0
linear functional A : L∞ (R+) → R such that
A(f) = lim
n→∞ An(f)
whenever the limit exists. Here, R+ = (0, ∞).
Solution. We define A as above, and prove that it is a continuous linear
functional on L ∞ (R+). Clearly, A is a linear functional, as for c, d ∈ R,
A(cf + dg) = lim
n→ An(cf + dg) = lim
n→∞ cAn(f) + dAn(g) = cA(f) + dA(g).
Thus, it suffices to show that A is bounded. But we have that
|A(f)| =
lim
n
1
f(x)dx
n→∞ n
0
n
= lim
1
n→∞ f(x)dx
n
0
n
1
≤ lim |f(x)| dx
n→∞ n 0
n
1
≤ lim f
n→∞
L∞ dx
n
0
= lim
n→∞ f L ∞
= f L ∞ .
Thus, A is a bounded linear functional, with A ≤ 1.
Problem 6: Let X be a Banach space and let A : X → X be a linear map.
Define
ρ(A) = {λ ∈ C : (λ − A) maps X onto X}
Show that ρ(A) is an open subset of C.