1. Let Ut , t ∈ R, be the operator on L2 (R) defined by
(Ut f )(x) = f (x + t).
Prove that kUt − Us k = 2 for any t 6= s.
2. Let T ∈ L(X, Y ) be such that kT xk ≥ ckxk for some c > 0. Assume X is a Banach
space. Show that the image of T is closed.
3. Let X be a Banach space, Y ⊂ X a closed subspace. Prove that X/Y is a Banach
space. (Hint: use P
that if {zn }n is a Cauchy sequence in X/Y then there exists a subsequence
{znk }k such that k kznk − znk+1 k < ∞.)
4. Give an example of a Banach space X and a closed subspace Y ⊂ X ∗ such that
(Y⊥ )⊥ 6= Y .
5. Let X be a Banach space, T ∈ L(X), λ ∈ C. Show that λ ∈ σ(T ) if and only if one
of the following properties holds:
(i) there exists a sequence {xn }n ⊂ X such that kxn k = 1 and T xn − λxn → 0;
(ii) there exists f ∈ X ∗ such that kf k = 1 and T ∗ f = λf .