Spectral Theory in Hilbert Space

10 2. SPACES WITH SCALAR PRODUCT
one speaks about a semi-scalar product. Note that (2) implies that
〈u, u〉 is real so that (3) makes sense. Also note that by combining (1)
and (2) we have 〈w, λu + µv〉 = λ〈w, u〉 + µ〈w, v〉. One says that the
scalar product is anti-linear in its second argument (Warning: In the
so called Dirac formalism in quantum mechanics the scalar product is
instead anti-linear in the first argument, linear in the second). Together
with (1) this makes the scalar product into a sesqui-linear (=1 1
2 -linear)
form. In words: A scalar product is a Hermitian, sesqui-linear and
positive definite form. We now assume that we have a scalar product
on L and define u = 〈u, u〉 for any u ∈ L. To show that this
definition makes · into a norm we need the following basic theorem.
Theorem 2.1. (Cauchy-Schwarz) If 〈·, ·〉 is a semi-scalar product
on L, then for all u, v ∈ L holds |〈u, v〉| 2 ≤ 〈u, u〉〈v, v〉.
Proof. For arbitrary complex λ we have 0 ≤ 〈λu + v, λu + v〉 =
|λ| 2 〈u, u〉 + λ〈u, v〉 + λ〈v, u〉 + 〈v, v〉. For λ = −r〈v, u〉 with real r
we obtain 0 ≤ r 2 |〈u, v〉| 2 〈u, u〉 − 2r|〈u, v〉| 2 + 〈v, v〉. If 〈u, u〉 = 0 but
〈u, v〉 = 0 this expression becomes negative for r > 1〈v,
v〉|〈u, v〉|−2
2
which is a contradiction. Hence 〈u, u〉 = 0 implies that 〈u, v〉 = 0 so
that the theorem is true in the case when 〈u, u〉 = 0. If 〈u, u〉 = 0
we set r = 〈u, u〉 −1 and obtain, after multiplication by 〈u, u〉, that
0 ≤ −|〈u, v〉| 2 + 〈u, u〉〈v, v〉 which proves the theorem.
In the case of a scalar product, defining u = 〈u, u〉, we may
write the Cauchy-Schwarz inequality as |〈u, v〉| ≤ uv. In this
case it is also easy to see when there is equality in Cauchy-Schwarz’
inequality. To see that · is a norm on L the only non-trivial point
is to verify that the triangle inequality holds; but this follows from
Cauchy-Schwarz’ inequality (Exercise 2.4).
Recall that in a finite dimensional space with scalar product it is
particularly convenient to use an orthonormal basis since this makes
it very easy to calculate the coordinates of any vector. In fact, if
x1, . . . , xn are the coordinates of u in the orthonormal basis e1, . . . , en,
then xj = 〈u, ej〉 (recall that e1, . . . , en is called orthonormal if all basis
elements have norm 1 and 〈ej, ek〉 = 0 for j = k). Given an arbitrary
basis it is easy to construct an orthonormal basis by use of the Gram-
Schmidt method (see the proof of Lemma 2.2).
In an infinite-dimensional space one can not find a (finite) basis.
The best one can hope for are infinitely many vectors e1, e2, . . . such
that each finite subset is linearly independent, and any vector is the
limit in norm of a sequence of finite linear combinations of e1, e2, . . . .
Again, it will turn out to be very convenient if e1, e2, . . . is an orthonormal
sequence, i.e., ej = 1 for j = 1, 2, . . . and 〈ej, ek〉 = 0 for j = k.
The following lemma is easily proved by use of the Gram-Schmidt procedure.