TECHNISCHE UNIVERSITEIT EINDHOVENFaculteit Wiskunde en Informatica

Exam Functional Analysis (2DE08)22 November 2006, 9.00-12.00 h

The answers of the problems have to be formulated in a clear way.Write readable.The use of a computer, graphical calculator, or a programmable calculatoris NOT ALLOWED.A calculator which has a key or a combination of keys for every letter of thealphabet is NOT ALLOWED.The use of a book NOT ALLOWED.

1. Let A : [a, b] → R be a continuous function on [a, b]. Define the operatorT : L2[a, b] → L2[a, b] by

(Tf)(t) = A(t)f(t).

a. Prove that T is a linear operator on L2[a, b].

b. Prove that T is a bounded linear operator on L2[a, b].

2. Show that there exist unique real numbers a0 and b0 such that for everya, b ∈ R holds∫ 1

0

|t3 − a0t− b0|2dt ≤∫ 1

0

|t3 − at− b|2dt.

Moreover, calculate the numbers a0 and b0.

3. Consider the inner product space C[0, 1] with the inner product

(f, g) =

∫ 1

0

f(t)g(t)dt.

The sequence of functions {fn}n∈N is defined by

fn(t) =

1 ; 0 ≤ t ≤ 1

2

1− n(t− 12) ; 1

2< t < 1

2+ 1

n

0 ; 12

+ 1n≤ t ≤ 1

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a. Sketch the graph of fn.

b. Prove that the sequence {fn} converges pointwise to a function,and that this limit function is not an element of C[0, 1].

c. Prove that the sequence {fn} is a Cauchy sequence.

d. Prove that the the sequence {fn} is not convergent.

4. Consider the normed linear space (c, ‖·‖∞) of all convergent sequences,i.e., the space of all sequences x = {λ1, λ2, λ3, . . .} for which there existsa scalar Lx such that λn → Lx as n →∞. Define the functional f on cby

f(x) = Lx.

a. Prove that |Lx| ≤ ‖x‖∞ for all x ∈ c.

b. Prove that f is a continous linear functional on (c, ‖ · ‖∞).

5. Consider the Hilbert space L2[0,∞) of square integrable real-valuedfunctions, with the standard inner product

〈f, g〉 =

∫ ∞

0

f(x)g(x)dx = limR→∞

∫ R

0

f(x)g(x)dx.

Define the linear operator T : L2[0,∞) → L2[0,∞) by

(Tf)(x) = f(x

5) where f ∈ L2[0,∞) and x ∈ [0,∞).

a. Calculate the Hilbert-adjoint operator T ∗.Recall that 〈Tf, g〉 = 〈f, T ∗g〉 for all f, g ∈ L2[0,∞).

b. Calculate the norm of ‖T ∗g‖ for all g ∈ L2[0,∞) with ‖g‖ = 1.

c. Calculate the norm of the operator T .

F or each part of the problems the following number of marks can beobtained:

1 a : 5 3b : 3 4b : 61 b : 5 3c : 3 5a : 52 : 10 3d : 2 5b : 33 a : 2 4a : 4 5c : 2

The final grade of this exam is the natural number which is closest to thescored number of marks divided by 5.

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