Spectral Theory in Hilbert Space

8 1. LINEAR SPACES
Exercises for Chapter 1
Exercise 1.1. Let L be a real linear space, and let V be the set
of ordered pairs (u, v) of elements of L with addition defined componentwise.
Show that V becomes a complex linear space if one defines
(x + iy)(u, v) = (xu − yv, xv + yu) for real x, y. Also show that L can
be ‘identified’ with the subset of elements of V of the form (u, 0), in
the sense that there is a one-to-one correspondence between the two
sets preserving the linear operations (for real scalars).
Exercise 1.2. Let M be an arbitrary set and let C M be the set
of complex-valued functions defined on M. Show that C M , provided
with the obvious definitions of the linear operations, is a complex linear
space.
Exercise 1.3. Prove Proposition 1.2.
Exercise 1.4. Let M be a non-empty subset of R n . Which of the
following choices of L make it into a linear subspace of C M ?
(1) L = {u ∈ C M | |u(x)| < 1 for all x ∈ M}.
(2) L = C(M) = {u ∈ C M | u is continuous in M}.
(3) L = {u ∈ C(M) | u is bounded on M}.
(4) L = L(M) = {u ∈ C M | u is Lebesgue integrable over M}.
Exercise 1.5. Let L be a linear space and uj ∈ L, j = 1, . . . , k.
Show that [u1, u2, . . . , uk] is a linear subspace of L.
Exercise 1.6. Show that if e1, . . . , en is a basis for L, then for
each u ∈ L there are uniquely determined complex numbers x1, . . . , xn,
called coordinates for u, such that u = x1e1 + · · · + xnen.
Exercise 1.7. Verify that L is infinite dimensional if and only if
every linearly independent subset of L can be extended to a linearly
independent subset of L with arbitrarily many elements. Then show
that u1, . . . , uk are linearly independent if and only if λ1u1+· · ·+λkuk =
0 only for λ1 = · · · = λk = 0. Also show that C M is finite-dimensional
if and only if the set M has finitely many elements.
Exercise 1.8. Let M be an open subset of R n . Verify that L is
infinite-dimensional for each of the choices of L in Exercise 1.4 which
make L into a linear space.
Exercise 1.9. Prove all statements in the penultimate paragraph
of the chapter.
Exercise 1.10. Prove that if L is a linear space and V a subspace,
then L/V is a well defined linear space.
Exercise 1.11. Prove Theorem 1.4.