1.6 Metric Spaces
9
Exercise 1.6.1 Show that the two functions defined above satisfies the four conditions of Definition 1.6.1 proving, therefore, that these functions are, in fact, two
metrics defined on R
n and on C
n , respectively.
We next define the concept of convergence in metric spaces.
Let (M, d) be a metric space. A sequence in M is a function x : N −→ M which
associates for every n ∈ N an element x(n) := x n ∈ M. The image of the function
x is denoted by (x n ) n or, simply, by (x n ). By abuse of notation we simply write (x n )
to denote a sequence.
Definition 1.6.2 Let (M, d) be a metric space and let (x n ) be a sequence in M. We
say that (x n ) converges if there exists an element x ∈ M such that
lim
n→∞
d(x n , x) = 0.
The element x is said to be the limit of (x n ) and it is written
lim
n→∞
x n = x.
One says that (x n ) converges to x or has limit x. On the other hand, if (x n ) does not
converge, then it is said to be divergent.
Definition 1.6.3 Let (M, d) be a metric space and (x n ) be a sequence in M. We say
that (x n ) is a Cauchy sequence if for every positive real number > 0, there exists a
n 0 = n 0 () ∈ N such that, for every m, n > n 0 one has d(x m , x n ) < . The space M
is called complete if every Cauchy sequence in M converges (to an element of M).
1.7 Inner Product Spaces
From now on, in order to fit our notation to quantum theory, we adapt the notation
of a vector as follows. Since the scenario of quantum mechanics is the vector space
C
n over the complex field C (or complex vector space C
n ), under usual addition
and scalar multiplication, we denote a vector v ∈ C
n by |v, which is called ket
(Dirac notation). This notation is the same utilized in [121] and is natural to quantum
mechanics. If |v ∈ C
n , then we represent |v by
|v ≡
⎡
⎢
⎢
⎢
⎣
z 1
z 2
. . .
z n
⎤
⎥
⎥
⎥
⎦
.
As usual, given a (complex) scalar z ∈ C, we define by z
∗ the complex conjugate of
z. Given a matrix A with complex entries, we denote A
† (the Hermitian conjugate
or adjoint of A) to be A
†
= (A
T
)
∗ , where A
T is the transpose of the matrix A.
9
Exercise 1.6.1 Show that the two functions defined above satisfies the four conditions of Definition 1.6.1 proving, therefore, that these functions are, in fact, two
metrics defined on R
n and on C
n , respectively.
We next define the concept of convergence in metric spaces.
Let (M, d) be a metric space. A sequence in M is a function x : N −→ M which
associates for every n ∈ N an element x(n) := x n ∈ M. The image of the function
x is denoted by (x n ) n or, simply, by (x n ). By abuse of notation we simply write (x n )
to denote a sequence.
Definition 1.6.2 Let (M, d) be a metric space and let (x n ) be a sequence in M. We
say that (x n ) converges if there exists an element x ∈ M such that
lim
n→∞
d(x n , x) = 0.
The element x is said to be the limit of (x n ) and it is written
lim
n→∞
x n = x.
One says that (x n ) converges to x or has limit x. On the other hand, if (x n ) does not
converge, then it is said to be divergent.
Definition 1.6.3 Let (M, d) be a metric space and (x n ) be a sequence in M. We say
that (x n ) is a Cauchy sequence if for every positive real number > 0, there exists a
n 0 = n 0 () ∈ N such that, for every m, n > n 0 one has d(x m , x n ) < . The space M
is called complete if every Cauchy sequence in M converges (to an element of M).
1.7 Inner Product Spaces
From now on, in order to fit our notation to quantum theory, we adapt the notation
of a vector as follows. Since the scenario of quantum mechanics is the vector space
C
n over the complex field C (or complex vector space C
n ), under usual addition
and scalar multiplication, we denote a vector v ∈ C
n by |v, which is called ket
(Dirac notation). This notation is the same utilized in [121] and is natural to quantum
mechanics. If |v ∈ C
n , then we represent |v by
|v ≡
⎡
⎢
⎢
⎢
⎣
z 1
z 2
. . .
z n
⎤
⎥
⎥
⎥
⎦
.
As usual, given a (complex) scalar z ∈ C, we define by z
∗ the complex conjugate of
z. Given a matrix A with complex entries, we denote A
† (the Hermitian conjugate
or adjoint of A) to be A
†
= (A
T
)
∗ , where A
T is the transpose of the matrix A.