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Chapter 11

Hermitian Spaces

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

292

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