MATHEMATICAL TRIPOS Part II PAPER 4 Before you begin read ...

11
22G Linear Analysis
Let X be a Banach space and suppose that T : X → X is a bounded linear operator.
What is an eigenvalue of T What is the spectrum σ(T ) of T
Let X = C[0, 1] be the space of continuous real-valued functions f : [0, 1] → R
endowed with the sup norm. Define an operator T : X → X by
where
T f(x) =
G(x, y) =
Prove the following facts about T :
∫ 1
0
G(x, y)f(y) dy,
{
y(x − 1) if y x,
x(y − 1) if x y.
(i) T f(0) = T f(1) = 0 and the second derivative (T f) ′′ (x) is equal to f(x) for x ∈ (0, 1);
(ii) T is compact;
(iii) T has infinitely many eigenvalues;
(iv) 0 is not an eigenvalue of T ;
(v) 0 ∈ σ(T ).
[The Arzelà–Ascoli theorem may be assumed without proof.]
23I Algebraic Geometry
Let X be a smooth projective curve of genus 2, defined over the complex numbers.
Show that there is a morphism f : X → P 1 which is a double cover, ramified at six points.
Explain briefly why X cannot be embedded into P 2 .
For any positive integer n, show that there is a smooth affine plane curve which is
a double cover of A 1 ramified at n points.
[State clearly any theorems that you use.]
Part II, Paper 4
[TURN OVER