Lebanese University
Faculty of Science (I) BSc Mathematics 2016-2017
Final exam June 9, 2017
Math 2206 : Introduction to functional analysis
120 minutes
Exercise 1 (Course question) : (14 points)
1. Let (H, h·, ·i) be a Hilbert space of infinite dimension. Show that :
(a) If Sn = {e1 , · · ·, en } is an orthonormal family of H and if Fn = span(Sn ) then, for all x ∈ H,
the orthogonal projection of x on Fn , PFn (x), verifies :
n
X n
X
PFn (x) = hx, ei iei and kPFn (x)k2 = |hx, ei i|2 .
i=1 i=1
(b) If H is a separable Hilbert space and if S = {e1 , · · ·, en , · · ·} is a Hilbert basis of H then, for all
x ∈ H, we have
∞
X
x= hx, ei iei in H.
i=1
2. Let (H, h·, ·i) be a real Hilbert space and H∗ = L(H, R) be its topological dual. Show that the mapping
∗
Φ :H 7−→ H
a 7−→ La , with La (f ) = ha, f i, ∀ f ∈ H,
is an isometry that means i.e. kLa k = kak .
Exercise 2 : (16 points)
Z 1
Let H = L2 (] − 1, 1[) with the usual inner product hf, gi = f (x)g(x)dx. Let
−1
Z 1
F = f ∈H : xf (x)dx = 0 .
−1
1. We define the linear form L : H → R by
Z 1
L(f ) = xf (x) dx.
−1
L ⊥
Show that L ∈ L(H, R). Deduce that H = F F .
2. Show that F = span{h}, where h(x) = x. (Hint : we may use the decomposition H = span{h} (span{h})⊥ )
⊥
L
3. For n ∈ N, we let gn (x) = xn . Calculate then PF (gn ), the orthogonal projection of gn on F .
Exercise 3 : (15 points)
Let f be the 2π-periodic function defined by
0 si −π ≤ x ≤ 0
f (x) =
x si 0 < x < π.
1. Sketch the curve of f .
2. Determine the Fourier series F S(f ) of the function f .
3. Deduce the values of the following numerical series :
+∞ +∞ +∞
X 1 X 1 X 1
U = , V = and W = .
k=0
(2k + 1)2 k=1
k2 k=0
(2k + 1)4
1/2
Exercise 4 : (25 points)
Let (H, h·, ·i) be a real Hilbert space and we designate by k · k the norm induced from h·, ·i. Let
B : H × H → R be a bilinear form, i.e., for all x, y, z ∈ H and λ, µ ∈ R :
B(λx + µy, z) = λB(x, z) + µB(y, z) and B(z, λx + µy) = λB(z, x) + µB(z, y).
Suppose there exist 0 < α ≤ β such that :
(
|B(x, y)| ≤ αkxkkyk, ∀ x, y ∈ H;
B(x, x) ≥ βkxk2 , ∀ x ∈ H.
Let f : H → R be a linear continueous form i.e f ∈ L(H, R) = H ∗ . The aim of this exercise is to
prove that there exist a unique element u0 ∈ H such that B(u0 , v) = f (v), for all v ∈ H.
1. For u ∈ H (fixed), we define the mapping Bu : H → R by
Bu (v) = B(u, v), ∀ v ∈ H.
Show that Bu ∈ L(H, R) = H ∗ . Deduce that there exist a unique w ∈ H such that
Bu (v) = hw, vi, ∀ v ∈ H. (∗)
2. As w is a function of u then we can define the mapping A : H → H by
w = A(u), where u and w verify (∗) i.e. hA(u), vi = B(u, v) for all v ∈ H.
(a) Show that A is linear.
(b) Show that , for all u ∈ H, we have kA(u)k ≤ αkuk.
(c) Show that , for all u, v ∈ H, hA(u) − A(v), u − vi ≥ βku − vk2 .
3. Say why, there exist a unique w0 ∈ H such that f (v) = hw0 , vi, for all v ∈ H.
4. Let λ > 0. We define the mapping Tλ : H → H by :
Tλ (u) = u − λA(u) + λw0 .
(a) Show that if u0 ∈ H then
A(u0 ) = w0 ⇐⇒ u0 is a fixed point for Tλ .
(b) Determine a value for λ so that Tλ is a contraction. (Hint : We may use part 2.)
5. Deduce that there exist a unique u0 ∈ H such that B(u0 , v) = f (v), for all v ∈ H.
2/2
