7.2 Sets and Spaces
397
(1) x 0, and the necessary and sufficient conditions x = 0 are (x, x) = 0
namely x = θ ;
(2) αx =
√ (αx, αx) =
√ αα(x, x) = |α||x;
Now it needs only to prove that the triangle inequality is also valid. According to
the Schwarz inequality, there is
x + y
2
= (x + y, x + y)
= (x + y, x) + (x + y, y) x + y · x + x + y · y
Two sides are divided by x + y, there is
x + y x + y
Quod erat demonstrandum.
Let X be an inner product space, if any fundamental sequence in X is the convergent sequence, then X is called the complete inner product space, or it is called
the Hilbert space, it is usually denoted by H . The Hilbert space is a special Banach
space, namely the norm is the Banach space induced by the inner product, it is also
one of the most important space in functional analysis. The essence of Hilbert space
is a generalization of n-dimensional Euclidean space in infinite-dimensional space.
The Hilbert space is the important space that in functional analysis the geometric
structure is best, the research results is most, the application range is most widely.
From inequality (7.2.14) and Eq. (7.2.7), we get
|(x, y)| x · y
(7.2.16)
Inequality (7.2.16) is another kind of representation of the Schwarz inequality.
The norms also have the following properties
||x − y| x − y
(7.2.17)
Proof According to the triangle inequality of the norm, the following relation holds
(x − y) + y x − y + y
(1)
or
x x − y + y
(2)
From inequality (2) there is
x − y x − y
(3)