Spectral Theory in Hilbert Space

EXERCISES FOR CHAPTER 3 21
Exercises for Chapter 3
Exercise 3.1. Prove the completeness of ℓ 2 !
Hint: Given a Cauchy sequence show first that each coordinate converges.
Exercise 3.2. Prove that any Hilbert space is isometrically isomorphic
to ℓ 2 , i.e., there is a bijective (one-to-one and onto) linear
map H ∋ u ↦→ û ∈ ℓ 2 such that 〈u, v〉 = 〈û, ˆv〉 for any u and v in H.
Exercise 3.3. Suppose L is a linear space with norm · which
satisfies the parallelogram identity for all u, v ∈ L. Show that 〈u, v〉 =
1
4
3
k=0 ik u + i k v 2 is a scalar product on L.
Hint: Show first that 〈u, u〉 = u 2 , that 〈v, u〉 = 〈u, v〉 and that
〈iu, v〉 = i〈u, v〉. Then show that 〈u + v, w〉 − 〈u, w〉 − 〈v, w〉 = 0 and
from that 〈λu, v〉 = λ〈u, v〉 for any rational number λ. Finally use
continuity.
Exercise 3.4. Show that the semi-norm on the space Lc defined
in the text is well-defined, i.e., that the limit limuj exists for any
element (u1, u2, . . . ) ∈ Lc. Then verify that H = Lc/Nc can be given a
norm under which it is complete, that L may be viewed as isometrically
and densely embedded in H, and that H is a Euclidean space (a space
with scalar product) if L is.
Exercise 3.5. Show that if M and N are closed, orthogonal subspaces
of H, then also M ⊕ N is closed.
Exercise 3.6. Show that is A ⊂ H, then A ⊥ is a closed linear
subspace of H, that A ⊂ B implies B ⊥ ⊂ A ⊥ and that A ⊂ (A ⊥ ) ⊥ .
Exercise 3.7. Verify that M ⊥⊥ = M for any closed linear subspace
M of H, and also that for an arbitrary set A ⊂ H the smallest closed
linear subspace containing A is A ⊥⊥ .
Exercise 3.8. Show that a bounded linear form on a Banach space
B has a least bound, which is a norm on B ′ , and that B ′ is complete
under this norm.
Exercise 3.9. Show that an orthonormal sequence does not converge
strongly to anything but tends weakly to 0. Conclude that if in a
Euclidean space every weakly convergent sequence is convergent, then
the space is finite-dimensional.
Hint: Show that the distance between two arbitrary elements in the
sequence is √ 2 and use Bessel’s inequality to show weak convergence
to 0.