11.1
Sesquilinear and Hermitian Forms, Pre-Hilbert
Spaces and Hermitian Spaces
In this chapter we generalize the basic results of Euclidean geometry presented in Chapter
9 to vector spaces over the complex numbers. Such a generalization is inevitable, and not
simply a luxury. For example, linear maps may not have real eigenvalues, but they always
have complex eigenvalues. Furthermore, some very important classes of linear maps can
be diagonalized if they are extended to the complexification of a real vector space. This
is the case for orthogonal matrices, and, more generally, normal matrices. Also, complex
vector spaces are often the natural framework in physics or engineering, and they are more
convenient for dealing with Fourier series. However, some complications arise due to complex
conjugation.
Recall that for any complex number z ∈ C, if z = x + iy where x, y ∈ R, we let z = x,
the real part of z, and
z = y, the imaginary part of z. We also denote the conjugate of
z = x + iy by z = x − iy, and the absolute value (or length, or modulus) of z by |z|. Recall
that |z|2 = zz = x2 + y2.
There are many natural situations where a map ϕ : E × E → C is linear in its first
argument and only semilinear in its second argument, which means that ϕ(u, µv) = µϕ(u, v),
as opposed to ϕ(u, µv) = µϕ(u, v). For example, the natural inner product to deal with
functions f : R → C, especially Fourier series, is
π
f, g =
f (x)g(x)dx,
−π
which is semilinear (but not linear) in g. Thus, when generalizing a result from the real case
of a Euclidean space to the complex case, we always have to check very carefully that our
proofs do not rely on linearity in the second argument. Otherwise, we need to revise our
proofs, and sometimes the result is simply wrong!
291
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CHAPTER 11. HERMITIAN SPACES
Before defining the natural generalization of an inner product, it is convenient to define
semilinear maps.
Definition 11.1. Given two vector spaces E and F over the complex field C, a function
f : E → F is semilinear if
f (u + v) = f (u) + f (v),
f (λu) = λf (u),
for all u, v ∈ E and all λ ∈ C.
Remark: Instead of defining semilinear maps, we could have defined the vector space E as
the vector space with the same carrier set E whose addition is the same as that of E, but
whose multiplication by a complex number is given by
(λ, u) → λu.
Then it is easy to check that a function f : E → C is semilinear iff f : E → C is linear.
We can now define sesquilinear forms and Hermitian forms.
Definition 11.2. Given a complex vector space E, a function ϕ : E ×E → C is a sesquilinear
form if it is linear in its first argument and semilinear in its second argument, which means
that
ϕ(u1 + u2, v) = ϕ(u1, v) + ϕ(u2, v),
ϕ(u, v1 + v2) = ϕ(u, v1) + ϕ(u, v2),
ϕ(λu, v) = λϕ(u, v),
ϕ(u, µv) = µϕ(u, v),
for all u, v, u1, u2, v1, v2 ∈ E, and all λ, µ ∈ C. A function ϕ: E × E → C is a Hermitian
form if it is sesquilinear and if
ϕ(v, u) = ϕ(u, v)
for all all u, v ∈ E.
Obviously, ϕ(0, v) = ϕ(u, 0) = 0. Also note that if ϕ : E × E → C is sesquilinear, we
have
ϕ(λu + µv, λu + µv) = |λ|2ϕ(u, u) + λµϕ(u, v) + λµϕ(v, u) + |µ|2ϕ(v, v),
and if ϕ : E × E → C is Hermitian, we have
ϕ(λu + µv, λu + µv) = |λ|2ϕ(u, u) + 2 (λµϕ(u, v)) + |µ|2ϕ(v, v).
11.1. HERMITIAN SPACES, PRE-HILBERT SPACES
293
Note that restricted to real coefficients, a sesquilinear form is bilinear (we sometimes say
R-bilinear). The function Φ : E → C defined such that Φ(u) = ϕ(u, u) for all u ∈ E is called
the quadratic form associated with ϕ.
The standard example of a Hermitian form on
n
C is the map ϕ defined such that
ϕ((x1, . . . , xn), (y1, . . . , yn)) = x1y1 + x2y2 + · · · + xnyn.
This map is also positive definite, but before dealing with these issues, we show the following
useful proposition.
Proposition 11.1. Given a complex vector space E, the following properties hold:
(1) A sesquilinear form ϕ : E × E → C is a Hermitian form iff ϕ(u, u) ∈ R for all u ∈ E.
(2) If ϕ : E × E → C is a sesquilinear form, then
4ϕ(u, v) = ϕ(u + v, u + v) − ϕ(u − v, u − v)
+ iϕ(u + iv, u + iv) − iϕ(u − iv, u − iv),
and
2ϕ(u, v) = (1 + i)(ϕ(u, u) + ϕ(v, v)) − ϕ(u − v, u − v) − iϕ(u − iv, u − iv).
These are called polarization identities.
Proof. (1) If ϕ is a Hermitian form, then
ϕ(v, u) = ϕ(u, v)
implies that
ϕ(u, u) = ϕ(u, u),
and thus ϕ(u, u) ∈ R. If ϕ is sesquilinear and ϕ(u, u) ∈ R for all u ∈ E, then
ϕ(u + v, u + v) = ϕ(u, u) + ϕ(u, v) + ϕ(v, u) + ϕ(v, v),
which proves that
ϕ(u, v) + ϕ(v, u) = α,
where α is real, and changing u to iu, we have
i(ϕ(u, v) − ϕ(v, u)) = β,
where β is real, and thus
α − iβ
α + iβ
ϕ(u, v) =
and ϕ(v, u) =
,
2
2
proving that ϕ is Hermitian.
(2) These identities are verified by expanding the right-hand side, and we leave them as
an exercise.
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CHAPTER 11. HERMITIAN SPACES
Proposition 11.1 shows that a sesquilinear form is completely determined by the quadratic
form Φ(u) = ϕ(u, u), even if ϕ is not Hermitian. This is false for a real bilinear form, unless
it is symmetric. For example, the bilinear form ϕ :
2
2
R × R → R defined such that
ϕ((x1, y1), (x2, y2)) = x1y2 − x2y1
is not identically zero, and yet it is null on the diagonal. However, a real symmetric bilinear
form is indeed determined by its values on the diagonal, as we saw in Chapter 9.
As in the Euclidean case, Hermitian forms for which ϕ(u, u) ≥ 0 play an important role.
Definition 11.3. Given a complex vector space E, a Hermitian form ϕ : E × E → C is
positive if ϕ(u, u) ≥ 0 for all u ∈ E, and positive definite if ϕ(u, u) > 0 for all u = 0. A
pair E, ϕ where E is a complex vector space and ϕ is a Hermitian form on E is called a
pre-Hilbert space if ϕ is positive, and a Hermitian (or unitary) space if ϕ is positive definite.
We warn our readers that some authors, such as Lang [67], define a pre-Hilbert space as
what we define as a Hermitian space. We prefer following the terminology used in Schwartz
[89] and Bourbaki [14]. The quantity ϕ(u, v) is usually called the Hermitian product of u
and v. We will occasionally call it the inner product of u and v.
Given a pre-Hilbert space E, ϕ , as in the case of a Euclidean space, we also denote
ϕ(u, v) by
u · v or
u, v
or (u|v),
and
Φ(u) by u .
Example 11.1. The complex vector space
n
C under the Hermitian form
ϕ((x1, . . . , xn), (y1, . . . , yn)) = x1y1 + x2y2 + · · · + xnyn
is a Hermitian space.
Example 11.2. Let l2 denote the set of all countably infinite sequences x = (xi)i∈ of
N
complex numbers such that
∞
i=0 |xi|2 is defined (i.e., the sequence
n
i=0 |xi|2 converges as
n → ∞). It can be shown that the map ϕ: l2 × l2 → C defined such that
∞
ϕ ((xi)i∈ , (y
) =
x
N
i)i∈N
iyi
i=0
is well defined, and l2 is a Hermitian space under ϕ. Actually, l2 is even a Hilbert space.
Example 11.3. Let Cpiece[a, b] be the set of piecewise bounded continuous functions
f : [a, b] → C under the Hermitian form
b
f, g =
f (x)g(x)dx.
a
It is easy to check that this Hermitian form is positive, but it is not definite. Thus, under
this Hermitian form, Cpiece[a, b] is only a pre-Hilbert space.
11.1. HERMITIAN SPACES, PRE-HILBERT SPACES
295
Example 11.4. Let C[a, b] be the set of complex-valued continuous functions f : [a, b] → C
under the Hermitian form
b
f, g =
f (x)g(x)dx.
a
It is easy to check that this Hermitian form is positive definite. Thus, C[a, b] is a Hermitian
space.
Example 11.5. Let E = Mn(C) be the vector space of complex n × n matrices. If we
view a matrix A ∈ Mn(C) as a “long” column vector obtained by concatenating together its
columns, we can define the Hermitian product of two matrices A, B ∈ Mn(C) as
n
A, B =
aijbij,
i,j=1
which can be conveniently written as
A, B = tr(A∗B) = tr(B∗A).
Since this can be viewed as the standard Hermitian product on n2
C
, it is a Hermitian product
on Mn(C). The corresponding norm
A
=
tr(A∗A)
F
is the Frobenius norm (see Section 7.2).
If E is finite-dimensional and if ϕ : E × E → R is a sequilinear form on E, given any
basis (e1, . . . , en) of E, we can write x =
n
x
y
i=1
iei and y =
n
j=1
j ej , and we have
n
n
n
ϕ(x, y) = ϕ
xiei,
yjej
=
xiyjϕ(ei, ej).
i=1
j=1
i,j=1
If we let G be the matrix G = (ϕ(ei, ej)), and if x and y are the column vectors associated
with (x1, . . . , xn) and (y1, . . . , yn), then we can write
ϕ(x, y) = x G y = y∗G x,
where y corresponds to (y1, . . . , yn).
Observe that in ϕ(x, y) = y∗G x, the matrix involved is the transpose of G = (ϕ(ei, ej)).
Furthermore, observe that ϕ is Hermitian iff G = G∗, and ϕ is positive definite iff the
matrix G is positive definite, that is,
x Gx > 0 for all x ∈ n
C , x = 0.
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CHAPTER 11. HERMITIAN SPACES
The matrix G associated with a Hermitian product is called the Gram matrix of the Hermi-
tian product with respect to the basis (e1, . . . , en).
Remark: To avoid the transposition in the expression for ϕ(x, y) = y∗G x, some authors
(such as Hoffman and Kunze [60]), define the Gram matrix G = (gij) associated with −, −
so that (gij) = (ϕ(ej, ei)); that is, G = G .
Conversely, if A is a Hermitian positive definite n × n matrix, it is easy to check that the
Hermitian form
x, y = y∗Ax
is positive definite. If we make a change of basis from the basis (e1, . . . , en) to the basis
(f1, . . . , fn), and if the change of basis matrix is P (where the jth column of P consists of
the coordinates of fj over the basis (e1, . . . , en)), then with respect to coordinates x and y
over the basis (f1, . . . , fn), we have
x Gy = x P GP y ,
so the matrix of our inner product over the basis (f1, . . . , fn) is P GP = (P )∗GP . We
summarize these facts in the following proposition.
Proposition 11.2. Let E be a finite-dimensional vector space, and let (e1, . . . , en) be a basis
of E.
1. For any Hermitian inner product −, − on E, if G = ( ei, ej ) is the Gram matrix of
the Hermitian product −, − w.r.t. the basis (e1, . . . , en), then G is Hermitian positive
definite.
2. For any change of basis matrix P , the Gram matrix of −, − with respect to the new
basis is (P )∗GP .
3. If A is any n × n Hermitian positive definite matrix, then
x, y = y∗Ax
is a Hermitian product on E.
We will see later that a Hermitian matrix is positive definite iff its eigenvalues are all
positive.
The following result reminiscent of the first polarization identity of Proposition 11.1 can
be used to prove that two linear maps are identical.
Proposition 11.3. Given any Hermitian space E with Hermitian product −, − , for any
linear map f : E → E, if f(x), x = 0 for all x ∈ E, then f = 0.
11.1. HERMITIAN SPACES, PRE-HILBERT SPACES
297
Proof. Compute f (x + y), x + y and f (x − y), x − y :
f (x + y), x + y = f (x), x + f (x), y + f (y), x + y, y
f (x − y), x − y = f(x), x − f(x), y − f(y), x + y, y ;
then, subtract the second equation from the first, to obtain
f (x + y), x + y − f(x − y), x − y = 2( f(x), y + f(y), x ).
If f (u), u = 0 for all u ∈ E, we get
f (x), y + f (y), x = 0 for all x, y ∈ E.
Then, the above equation also holds if we replace x by ix, and we obtain
i f (x), y − i f(y), x = 0, for all x, y ∈ E,
so we have
f (x), y + f (y), x = 0
f (x), y − f(y), x = 0,
which implies that f (x), y = 0 for all x, y ∈ E. Since −, − is positive definite, we have
f (x) = 0 for all x ∈ E; that is, f = 0.
One should be careful not to apply Proposition 11.3 to a linear map on a real Euclidean
space, because it is false! The reader should find a counterexample.
The Cauchy–Schwarz inequality and the Minkowski inequalities extend to pre-Hilbert
spaces and to Hermitian spaces.
Proposition 11.4. Let E, ϕ be a pre-Hilbert space with associated quadratic form Φ. For
all u, v ∈ E, we have the Cauchy–Schwarz inequality
|ϕ(u, v)| ≤
Φ(u) Φ(v).
Furthermore, if E, ϕ is a Hermitian space, the equality holds iff u and v are linearly de-
pendent.
We also have the Minkowski inequality
Φ(u + v) ≤
Φ(u) +
Φ(v).
Furthermore, if E, ϕ is a Hermitian space, the equality holds iff u and v are linearly de-
pendent, where in addition, if u = 0 and v = 0, then u = λv for some real λ such that
λ > 0.
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CHAPTER 11. HERMITIAN SPACES
Proof. For all u, v ∈ E and all µ ∈ C, we have observed that
ϕ(u + µv, u + µv) = ϕ(u, u) + 2 (µϕ(u, v)) + |µ|2ϕ(v, v).
Let ϕ(u, v) = ρeiθ, where |ϕ(u, v)| = ρ (ρ ≥ 0). Let F : R → R be the function defined such
that
F (t) = Φ(u + teiθv),
for all t ∈ R. The above shows that
F (t) = ϕ(u, u) + 2t|ϕ(u, v)| + t2ϕ(v, v) = Φ(u) + 2t|ϕ(u, v)| + t2Φ(v).
Since ϕ is assumed to be positive, we have F (t) ≥ 0 for all t ∈ R. If Φ(v) = 0, we must have
ϕ(u, v) = 0, since otherwise, F (t) could be made negative by choosing t negative and small
enough. If Φ(v) > 0, in order for F (t) to be nonnegative, the equation
Φ(u) + 2t|ϕ(u, v)| + t2Φ(v) = 0
must not have distinct real roots, which is equivalent to
|ϕ(u, v)|2 ≤ Φ(u)Φ(v).
Taking the square root on both sides yields the Cauchy–Schwarz inequality.
For the second part of the claim, if ϕ is positive definite, we argue as follows. If u and v
are linearly dependent, it is immediately verified that we get an equality. Conversely, if
|ϕ(u, v)|2 = Φ(u)Φ(v),
then the equation
Φ(u) + 2t|ϕ(u, v)| + t2Φ(v) = 0
has a double root t0, and thus
Φ(u + t0eiθv) = 0.
Since ϕ is positive definite, we must have
u + t0eiθv = 0,
which shows that u and v are linearly dependent.
If we square the Minkowski inequality, we get
Φ(u + v) ≤ Φ(u) + Φ(v) + 2 Φ(u) Φ(v).
However, we observed earlier that
Φ(u + v) = Φ(u) + Φ(v) + 2 (ϕ(u, v)).
11.1. HERMITIAN SPACES, PRE-HILBERT SPACES
299
Thus, it is enough to prove that
(ϕ(u, v)) ≤
Φ(u) Φ(v),
but this follows from the Cauchy–Schwarz inequality
|ϕ(u, v)| ≤
Φ(u) Φ(v)
and the fact that
z ≤ |z|.
If ϕ is positive definite and u and v are linearly dependent, it is immediately verified that
we get an equality. Conversely, if equality holds in the Minkowski inequality, we must have
(ϕ(u, v)) =
Φ(u) Φ(v),
which implies that
|ϕ(u, v)| =
Φ(u) Φ(v),
since otherwise, by the Cauchy–Schwarz inequality, we would have
(ϕ(u, v)) ≤ |ϕ(u, v)| <
Φ(u) Φ(v).
Thus, equality holds in the Cauchy–Schwarz inequality, and
(ϕ(u, v)) = |ϕ(u, v)|.
But then, we proved in the Cauchy–Schwarz case that u and v are linearly dependent. Since
we also just proved that ϕ(u, v) is real and nonnegative, the coefficient of proportionality
between u and v is indeed nonnegative.
As in the Euclidean case, if E, ϕ is a Hermitian space, the Minkowski inequality
Φ(u + v) ≤
Φ(u) +
Φ(v)
shows that the map u →
Φ(u) is a norm on E. The norm induced by ϕ is called the
Hermitian norm induced by ϕ. We usually denote
Φ(u) by u , and the Cauchy–Schwarz
inequality is written as
|u · v| ≤ u v .
Since a Hermitian space is a normed vector space, it is a topological space under the
topology induced by the norm (a basis for this topology is given by the open balls B0(u, ρ)
of center u and radius ρ > 0, where
B0(u, ρ) = {v ∈ E | v − u < ρ}.
If E has finite dimension, every linear map is continuous; see Chapter 7 (or Lang [67, 68],
Dixmier [27], or Schwartz [89, 90]). The Cauchy–Schwarz inequality
|u · v| ≤ u v
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CHAPTER 11. HERMITIAN SPACES
shows that ϕ : E × E → C is continuous, and thus, that
is continuous.
If E, ϕ is only pre-Hilbertian, u is called a seminorm. In this case, the condition
u = 0 implies u = 0
is not necessarily true. However, the Cauchy–Schwarz inequality shows that if u = 0, then
u · v = 0 for all v ∈ E.
Remark: As in the case of real vector spaces, a norm on a complex vector space is induced
by some psotive definite Hermitian product −, − iff it satisfies the parallelogram law:
u + v 2 + u − v 2 = 2( u 2 + v 2).
This time, the Hermitian product is recovered using the polarization identity from Proposi-
tion 11.1:
4 u, v = u + v 2 − u − v 2 + i u + iv 2 − i u − iv 2 .
It is easy to check that u, u = u 2, and
v, u = u, v
iu, v = i u, v ,
so it is enough to check linearity in the variable u, and only for real scalars. This is easily
done by applying the proof from Section 9.1 to the real and imaginary part of u, v ; the
details are left as an exercise.
We will now basically mirror the presentation of Euclidean geometry given in Chapter 9
rather quickly, leaving out most proofs, except when they need to be seriously amended.
11.2
Orthogonality, Duality, Adjoint of a Linear Map
In this section we assume that we are dealing with Hermitian spaces. We denote the Her-
mitian inner product by u · v or u, v . The concepts of orthogonality, orthogonal family of
vectors, orthonormal family of vectors, and orthogonal complement of a set of vectors are
unchanged from the Euclidean case (Definition 9.2).
For example, the set C[−π, π] of continuous functions f : [−π, π] → C is a Hermitian
space under the product
π
f, g =
f (x)g(x)dx,
−π
and the family (eikx)k∈ is orthogonal.
Z
Proposition 9.3 and 9.4 hold without any changes. It is easy to show that
n
2
n
u
2
i
=
ui
+
2 (ui · uj).
i=1
i=1
1≤i<j≤n
11.2. ORTHOGONALITY, DUALITY, ADJOINT OF A LINEAR MAP
301
Analogously to the case of Euclidean spaces of finite dimension, the Hermitian product
induces a canonical bijection (i.e., independent of the choice of bases) between the vector
space E and the space E∗. This is one of the places where conjugation shows up, but in this
case, troubles are minor.
Given a Hermitian space E, for any vector u ∈ E, let ϕlu : E → C be the map defined
such that
ϕlu(v) = u · v, for all v ∈ E.
Similarly, for any vector v ∈ E, let ϕrv : E → C be the map defined such that
ϕrv(u) = u · v, for all u ∈ E.
Since the Hermitian product is linear in its first argument u, the map ϕrv is a linear form
in E∗, and since it is semilinear in its second argument v, the map ϕlu is also a linear form
in E∗. Thus, we have two maps l : E → E∗ and r : E → E∗, defined such that
l(u) = ϕlu, and
r(v) = ϕrv.
Actually, ϕlu = ϕru and l = r. Indeed, for all u, v ∈ E, we have
l(u)(v) = ϕlu(v)
= u · v
= v · u
= ϕru(v)
= r(u)(v).
Therefore, we use the notation ϕu for both ϕlu and ϕru, and for both l and r.
Theorem 11.5. let E be a Hermitian space E. The map : E → E∗ defined such that
(u) = ϕlu = ϕru for all u ∈ E
is semilinear and injective. When E is also of finite dimension, the map : E → E∗ is a
canonical isomorphism.
Proof. That : E → E∗ is a semilinear map follows immediately from the fact that = r,
and that the Hermitian product is semilinear in its second argument. If ϕu = ϕv, then
ϕu(w) = ϕv(w) for all w ∈ E, which by definition of ϕu and ϕv means that
w · u = w · v
for all w ∈ E, which by semilinearity on the right is equivalent to
w · (v − u) = 0 for all w ∈ E,
which implies that u = v, since the Hermitian product is positive definite. Thus, : E → E∗
is injective. Finally, when E is of finite dimension n, E∗ is also of dimension n, and then
: E → E∗ is bijective. Since is semilinar, the map : E → E∗ is an isomorphism.
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CHAPTER 11. HERMITIAN SPACES
The inverse of the isomorphism : E → E∗ is denoted by : E∗ → E.
As a corollary of the isomorphism : E → E∗, if E is a Hermitian space of finite dimen-
sion, then every linear form f ∈ E∗ corresponds to a unique v ∈ E, such that
f (u) = u · v, for every u ∈ E.
In particular, if f is not the null form, the kernel of f , which is a hyperplane H, is precisely
the set of vectors that are orthogonal to v.
Remark: The “musical map” : E → E∗ is not surjective when E has infinite dimension.
This result can be salvaged by restricting our attention to continuous linear maps, and by
assuming that the vector space E is a Hilbert space.
The existence of the isomorphism : E → E∗ is crucial to the existence of adjoint maps.
Indeed, Theorem 11.5 allows us to define the adjoint of a linear map on a Hermitian space.
Let E be a Hermitian space of finite dimension n, and let f : E → E be a linear map. For
every u ∈ E, the map
v → u · f(v)
is clearly a linear form in E∗, and by Theorem 11.5, there is a unique vector in E denoted
by f ∗(u), such that
f ∗(u) · v = u · f(v),
that is,
f ∗(u) · v = u · f(v), for every v ∈ E.
The following proposition shows that the map f ∗ is linear.
Proposition 11.6. Given a Hermitian space E of finite dimension, for every linear map
f : E → E there is a unique linear map f∗ : E → E such that
f ∗(u) · v = u · f(v),
for all u, v ∈ E. The map f∗ is called the adjoint of f (w.r.t. to the Hermitian product).
Proof. Careful inspection of the proof of Proposition 9.6 reveals that it applies unchanged.
The only potential problem is in proving that f ∗(λu) = λf ∗(u), but everything takes place
in the first argument of the Hermitian product, and there, we have linearity.
The fact that
v · u