1  Show that ||f||=|f(0)|+sup|f′(t)| defines a norm on C¹[0,1], which is the space of (real) functions on [0,1] with continuous derivative.

 2  Show that the formula ⟨ (x_n),(y_n)⟩ =∑_n=1^∞x_ny_n/n² defines an inner product on l_∞, the space of bounded (complex) sequences. What norm does it produce?

 3  Use the Cauchy–Schwarz inequality for a suitable inner product to prove that for all f ∈ C[0,1] the inequality

⎪ ⎪ ⎪ ⎪

1 ∫ 0

f(x)x d x

⎪ ⎪ ⎪ ⎪

≤ C

⎛ ⎜ ⎜ ⎝

1 ∫ 0

|f(x)|2  d x

⎞ ⎟ ⎟ ⎠

1/2

holds for some constant C>0 (independent of f) and find the smallest possible C that holds for all functions f (hint: consider the cases of equality).

 4  We define the following norm on l_∞, the space of bounded complex sequences:

||(xn)||∞=

sup n ≥ 1

|xn|.

Show that this norm makes l_∞ into a Banach space (i.e., a complete normed space).

 5  Fix a vector (w₁,…,w_n) whose components are strictly positive real numbers, and define an inner product on ℂⁿ by

⟨ x,y ⟩ =

n ∑ k=1

wk xk

y

k.

Show that this makes ℂⁿ into a Hilbert space (i.e., a complete inner-product space).

 6  Show that the supremum norm on C[0,1] isn’t given by an inner product, by finding a counterexample to the parallelogram law.

 7  In l₂ let e₁=(1,0,0,…), e₂=(0,1,0,0,…), e₃=(0,0,1,0,0,…), and so on. Show that Lin (e₁,e₂,…)=c₀₀, and that CLin (e₁,e₂,…)=l₂. What is CLin (e₂,e₃,…)?

 8  Let C[−1,1] have the standard L₂ inner product, defined by

⟨ f, g⟩ =

1 ∫ −1

f(t)

g(t)

d t.

Show that the functions 1, t and t²−1/3 form an orthogonal (not orthonormal!) basis for the subspace P₂ of polynomials of degree at most 2 and hence calculate the best L₂-approximation of the function t⁴ by polynomials in P₂.

 9  Define an inner product on C[0,1] by

⟨ f,g⟩=

1 ∫ 0

√

t

f(t)

g(t)

d t.

Use the Gram–Schmidt process to find the first 2 terms of an orthonormal sequence formed by orthonormalising the sequence 1, t, t², ….

 10  Consider the plane P in ℂ⁴ (usual inner product) spanned by the vectors (1,1,0,0) and (1,0,0,−1). Find orthonormal bases for P and P^⊥, and verify directly that (P^⊥)^⊥=P.

 11  Let a and b be arbitrary real numbers with a < b. By using the fact that the functions 1/√2πe^inx, n ∈ ℤ, are orthonormal in L₂[0,2π], together with the change of variable x=2π(t−a)/(b−a), find an orthonormal basis in L₂[a,b] of the form e_n(t)=α e^{i n λ t}, n ∈ ℤ, for suitable real constants α and λ.

 12  For which real values of α is

∞ ∑ n=1

nαeint

the Fourier series of a function in L₂[−π,π]?

 13  Calculate the Fourier series of f(t)=e^t on [−π,π] and use Parseval’s identity to deduce that

∞ ∑ n=−∞

1  n2+1

=

π  tanhπ

.

 14  Using the fact that (e_n) is a complete orthonormal system in L₂[−π,π], where e_n(t)=exp(int)/√2π, show that e₀,s₁,c₁,s₂,c₂,… is a complete orthonormal system, where s_n(t)=sinnt/√π and c_n(t)= cosnt/√π. Show that every L₂[−π,π] function f has a Fourier series

a0+

∞ ∑ n=1

ancosnt + bn sinnt,

converging in the L₂ sense, and give a formula for the coefficients.

 15  Let C(T) be the space of continuous (complex) functions on the circle  T={ z ∈ ℂ: |z|=1 } with the supremum norm. Show that, for any polynomial f(z) in C(T)

∫

|z|=1

f(z)  d z=0.

Deduce that the function f(z)=z is not the uniform limit of polynomials on the circle (i.e., Weierstrass’s approximation theorem doesn’t hold in this form).

 16  Define a linear functional on C[0,1] (continuous functions on [0,1]) by α(f)=f(1/2). Show that α is bounded if we give C[0,1] the supremum norm. Show that α is not bounded if we use the L₂ norm, because we can find a sequence (f_n) of continuous functions on [0,1] such that ||f_n||₂ ≤ 1, but f_n(1/2) → ∞.

 17  The Hardy space H₂ is the Hilbert space of all power series f(z)=∑_n=0^∞a_n zⁿ, such that ∑_n=0^∞|a_n|² < ∞, where the inner product is given by

⟨ ⟨ ⟨ ⟨

∞ ∑ n=0

anzn,

∞ ∑ n=0

bnzn

⟩ ⟩ ⟩ ⟩

=

∞ ∑ n=0

an

bn

.

Show that the sequence 1, z, z², z³, … is an orthonormal basis for H₂.

Fix w with |w|<1 and define a linear functional on H₂ by α(f)=f(w). Write down a formula for the function g(z) ∈ H₂ such that α(f)=⟨ f, g ⟩. What is ||α||?

 18  The Volterra operator V: L₂[0,1] → L₂[0,1] is defined by

(Vf)(x)=

x ∫ 0

f(t)  d t.

Use the Cauchy–Schwarz inequality to show that |(Vf)(x)| ≤ √x||f||₂ (hint: write (Vf)(x)=⟨ f, J_x⟩ where J_x is a function that you can write down explicitly).

Deduce that ||Vf||₂² ≤ 1/ 2||f||₂², and hence ||V|| ≤ 1/√2.

 19  Find the adjoints of the following operators:

1. A:l₂ → l₂, defined by A(x₁,x₂,…)=(0,x₁ / 1, x₂/ 2, x₃/ 3, …);

   and, on a general Hilbert space H:

2. The rank-one operator R, defined by Rx=⟨ x,y ⟩ z, where y and z are fixed elements of H;
3. The projection operator P_M, defined by P_M(m+n)=m, where m ∈ M and n ∈ M^⊥, and H=M ⊕ M^⊥ as usual.

 20  Let U ∈ B(H) be a unitary operator. Show that (Ue_n) is an orthonormal basis of H whenever (e_n) is.

Let l₂(ℤ) denote the Hilbert space of two-sided sequences (a_n)_n=−∞^∞ with

||(an)||2=

∞ ∑ n=−∞

|an|2 < ∞.

Show that the bilateral right shift, V:l₂(ℤ) → l₂(ℤ) defined by V((a_n))=(b_n), where b_n=a_n−1 for all n∈ ℤ, is unitary, whereas the usual right shift S on l₂=l₂(ℕ) is not unitary.

 21  Let f∈ C[−π,π] and let M_f be the multiplication operator on L₂(−π,π), given by (M_fg)(t)=f(t) g(t), for g ∈ L₂(−π,π). Find a function f′ ∈ C[−π,π] such that M_f^*=M_f′.

Show that M_f is always a normal operator. When is it Hermitian? When is it unitary?

 22  Let T be any operator such that Tⁿ=0 for some integer n (such operators are called nilpotent). Show that I−T is invertible (hint: consider I+T+T²+…+Tⁿ⁻¹). Deduce that I−T/λ is invertible for any λ ≠ 0.

What is σ(T)? What is r(T)?

 23  Let (λ_n) be a fixed bounded sequence of complex numbers, and define an operator on l₂ by T((x_n))=((y_n)), where y_n=λ_nx_n for each n. Recall that T is a bounded operator and ||T||=||(λ_n)||_∞. Let Λ={λ₁,λ₂,…}. Prove the following:

1. Each λ_k is an eigenvalue of T, and hence is in σ(T).
2. If λ ∉Λ, then the inverse of T−λ I exists (and is bounded).

Deduce that σ(T)=Λ. Note, that then any non-empty compact set could be a spectrum of some bounden operator.

 24  Let S be an isomorphism between Hilbert spaces H and K, that is, S: H → K is a linear bijection such that S and S⁻¹ are bounded operators. Suppose that T ∈ B(H). Show that T and STS⁻¹ have the same spectrum and the same eigenvalues (if any).

 25  Define an operator U: l₂(ℤ) → L₂(−π,π) by U((a_n))=∑_n=−∞^∞a_n e^int/√2π. Show that U is a bijection and an isometry, i.e., that ||Ux||=||x|| for all x ∈ l₂(ℤ).

Let V be the bilateral right shift on l₂(ℤ), the unitary operator defined on Question 20. Let f ∈ L₂(−π,π). Show that (UVU⁻¹f)(t)=e^itf(t), and hence, using Question 24, show that σ(V)=T, the unit circle, but that V has no eigenvalues.

 26  Show that K(X) is a closed linear subspace of B(X), and that AT and TA are compact whenever T ∈ K(X) and A ∈ B(X). (This means that K(X) is a closed ideal of B(X).)

 27  Let A be a Hilbert–Schmidt operator, and let (e_n)_{n≥ 1} and (f_m)_{m≥ 1} be orthonormal bases of A. By writing each Ae_n as Ae_n=∑_m=1^∞⟨ Ae_n, f_m ⟩ f_m, show that

∞ ∑ n=1

||Aen||2=

∞ ∑ m=1

||A*fm||2.

Deduce that the quantity ||A||_HS²=∑_n=1^∞||Ae_n||² is independent of the choice of orthonormal basis, and that ||A||_HS=||A^*||_HS. (||A||_HS is called the Hilbert–Schmidt norm of A.)

 28 

1. Let T∈ K(H) be a compact operator. Using Question 26, show that T^*T and TT^* are compact Hermitian operators.
2. Let (e_n)_{n≥ 1} and (f_n)_{n ≥ 1} be orthonormal bases of a Hilbert space H, let (α_n)_{n ≥ 1} be any bounded complex sequence, and let T ∈ B(H) be an operator defined by

   Tx=

   ∞ ∑ n=1

   αn ⟨ x, en ⟩ fn.

   Prove that T is Hilbert–Schmidt precisely when (α_n) ∈ l₂. Show that T is a compact operator if and only if α_n → 0, and in this case write down spectral decompositions for the compact Hermitian operators T^*T and TT^*.

 29  Solve the Fredholm integral equation φ−λ Tφ=f, where f(x)=x and

(Tφ)(x)=

1 ∫ 0

xy2 φ(y)  d y    (φ ∈ L2(0,1)),

for small values of λ by means of the Neumann series.

For what values of λ does the series converge? Write down a solution which is valid for all λ apart from one exception. What is the exception?

 30  Suppose that h is a 2π-periodic L₂(−π,π) function with Fourier series ∑_n=−∞^∞a_n e^int. Show that each of the functions φ_k(y)=e^iky, k ∈ ℤ, is an eigenvector of the integral operator T on L₂(−π,π) defined by

(Tφ)(x)=

π ∫ −π

h(x−y) φ(y)  d y,

and calculate the corresponding eigenvalues.

Now let h(t)=−log(2(1−cost)). Assuming, without proof, that h(t) has the Fourier series ∑_{n ∈ ℤ, n ≠ 0} e^int/|n|, use the Hilbert–Schmidt method to solve the Fredholm equation φ−λ Tφ=f, where f(t) has Fourier series ∑_n=−∞^∞c_n e^int and 1/λ ∉σ(T).

 31  Use the Gram–Schmidt algorithm to find an orthonormal basis for the subspace X of L₂(−1,1) spanned by the functions t, t² and t⁴.

Hence find the best L₂(−1,1) approximation of the constant function f(t)=1 by functions from X.

 32  For n=1,2,… let φ_n denote the linear functional on l₂ defined by

φn(x)=x1+x2+…+xn,

where x=(x₁,x₂,…) ∈ l₂. Use the Riesz–Fréchet theorem to calculate ||φ_n||.

 33  Let T be a bounded linear operator on a Hilbert space, and suppose that T=A+iB, where A and B are self-adjoint operators. Express T^* in terms of A and B, and hence solve for A and B in terms of T and T^*.

Deduce that every operator T can be written T=A+iB, where A and B are self-adjoint, in a unique way.

Show that T is normal if and only if AB=BA.

 34  Let P_n be the subspace of L₂(−π,π) consisting of all polynomials of degree at most n, and let T_n be the subspace consisting of all trigonometric polynomials of the form f(t)=∑_k=−nⁿ a_k e^ikt. Calculate the spectrum of the differentiation operator D, defined by (Df)(t)=f′(t), when

1. D is regarded as an operator on P_n, and
2. D is regarded as an operator on T_n.

Note that both P_n and T_n are finite-dimensional Hilbert spaces.

Show that T_n has an orthonormal basis of eigenvectors of D, whereas P_n does not.

 35  Use the Neumann series to solve the Volterra integral equation φ−λ Tφ=f in L₂[0,1], where λ∈ ℂ, f(t)= 1 for all t, and (Tφ)(x)=∫₀^x t²φ(t)  d t. (You should be able to sum the infinite series.)