Spectral Theory in Hilbert Space

14 2. SPACES WITH SCALAR PRODUCT
Exercise 2.5. Show that 〈u, v〉 = 1
u(x)v(x) dx is a scalar prod-
0
uct on the space C[0, 1] of continuous, complex-valued functions defined
on [0, 1].
Exercise 2.6. Finish the proof of Lemma 2.2.
Exercise 2.7. Prove Lemma 2.3.
Exercise 2.8. Prove Bessel’s inequality!
Exercise 2.9. Prove Parseval’s formula!
Exercise 2.10. It is well known that the set of step functions which
are identically 0 outside a compact subinterval of an interval I are dense
in L 2 (I). Use this to show that L 2 (I) is separable.
Exercise 2.11. Prove Theorem 2.9.
Hint: Use Gram-Schmidt!
Exercise 2.12. Let L be the set of complex-valued functions u of
the form u(x) = k j=1 λjeiαjx where α1, . . . , αk are (a finite number of)
different real numbers and λ1, . . . , λk are complex numbers. Show that
L is a linear subspace of C(R) (the functions continuous on the real
uv serves as a scalar product.
line) on which 〈u, v〉 = limT →∞ 1
2T
T
−T
Then show that the norm of e iαx is 1 for any α ∈ R and that e iαx is
orthogonal to e iβx as soon as α = β. Conclude that L is not separable.
Exercise 2.13. Show that as metric spaces the set Q of rational
numbers is not complete but the set R of reals is.
Exercise 2.14. Suppose L is a space with scalar product which is
not complete, and that e1, e2, . . . is a complete orthonormal sequence
in L. Show that there exists a sequence λ1, λ2, . . . of complex numbers,
such that |λj| 2 < ∞ but λjej does not converge to any element
of L.