Spectral Theory in Hilbert Space

18 3. HILBERT SPACE
Two subspaces M and N are said to be orthogonal if every element
in M is orthogonal to every element in N. Then clearly M ∩ N = {0}
so the direct sum of M and N is defined. In the case at hand this is
called the orthogonal sum of M and N and denoted by M ⊕ N. Thus
M ⊕ N is the set of all sums u + v with u ∈ M and v ∈ N. If M and
N are closed, orthogonal subspaces of H, then their orthogonal sum is
also a closed subspace of H (Exercise 3.5). If A is an arbitrary subset
of H we define
A ⊥ = {u ∈ H | 〈u, v〉 = 0 for all v ∈ A}.
This is called the orthogonal complement of A. It is easy to see that
A ⊥ is a closed linear subspace of H, that A ⊂ B implies B ⊥ ⊂ A ⊥ and
that A ⊂ (A ⊥ ) ⊥ (Exercise 3.6).
When M is a linear subspace of H an alternative way of writing
M ⊥ is H ⊖ M. This makes sense because of the following theorem of
central importance.
Theorem 3.7. Suppose M is a closed linear subspace of H. Then
M ⊕ M ⊥ = H.
Proof. M ⊕ M ⊥ is a closed linear subspace of H so if it is not
all of H, then it has a non-trivial normal u by Lemma 3.6. But if u is
orthogonal to both M and M ⊥ , then u ∈ M ⊥ ∩ (M ⊥ ) ⊥ which shows
that u cannot be = 0. The theorem follows.
A nearly obvious consequence of Theorem 3.7 is that M ⊥⊥ = M
for any closed linear subspace M of H (Exercise 3.7).
A linear form ℓ on H is complex-valued linear function on H. Naturally
ℓ is said to be continuous if ℓ(uj) → ℓ(u) whenever uj → u. The
set of continuous linear forms on a Banach space B (or a more general
topological vector space) is made into a linear space in an obvious way.
This space is called the dual of B, and is denoted by B ′ . A continuous
linear form on a Banach space B has to be bounded in the sense that
there is a constant C such that |ℓ(u)| ≤ Cu for any u ∈ B. For
suppose not. Then there exists a sequence of elements u1, u2, . . . of
B for which |ℓ(uj)|/uj → ∞. Setting vj = uj/ℓ(uj) we then have
vj → 0 but |ℓ(vj)| = 1 → 0, so ℓ can not be continuous. Conversely, if
ℓ is bounded by C then |ℓ(uj) − ℓ(u)| = |ℓ(uj − u)| ≤ Cuj − u → 0 if
uj → u, so a bounded linear form is continuous. The smallest possible
bound of a linear form ℓ is called the norm of ℓ, denoted ℓ.
It is easy to see that provided with this norm B ′ is complete, so the
dual of a Banach space is a Banach space (Exercise 3.8). A familiar
example is given by the space L p (Ω, µ) for 1 ≤ p < ∞, where Ω is
a domain in R n and µ a Radon measure defined in Ω. The dual of
this space is Lq (Ω, µ), where q is the conjugate exponent to p, in the
sense that 1 1 + = 1. A simple example of a bounded linear form on
p q
a Hilbert space H is ℓ(u) = 〈u, v〉, where v is some fixed element of