Relevant elements of functional analysis 49Lem1na 4 If A is a seH-adjoint positive bounded operator, then the esti-
mate is valid:(13) II Ay 112 < II A II (Ay, y)'
Since A* =A> 0, there always exists the operator A^112. By merely
setting v = A^112 Y we obtain(Ay, Ay) = (Av, v) <II A II · II v 112 = II A II (Ay, Y) ·
- Linear operators in finite-dimensional spaces. It is supposed that an
n-dimensional vector space Rn is equipped with an inner product ( , ) and
associated norm 11 x 11 = ~- By the definition of finite-dimensional
space, any vector x E Rn can uniquely be represented as a linear combina-
tion x = cl el + ''' en en of linearly independent vectors el '' '' 'en' which
constitute a basis for the space Rn. The numbers ck are called the coordi-
nates of the vector x. One can always choose as a basis an orthogonal and
normed system of vectors el'. '. ' en:
if:-k,
i = k'by means of which it is possible to write C1,; = (x, ek)·
Let A be a linear operator in the space Rn, Any operator A in the basis
el'. ' ' 'en can be put in conespondence with an n x n matrix 2[ = ( aik)'
whose element aik is the ith component of the vector A~k· Conversely, any
matrix 2l = (aik), i, k = 1, ... , n, specifies a linear operator.
The matrix of a self-adjoint operator in any orthonormal basis is a
symmetric matrix.
Let us dwell on the properties of eigenvalues and eigenvectors of a
linear self-adjoint operator A. A number ,\ such that there exists a vector
e f:- 0 with Ae = >-e is called an eigenvalue of the operator A. This vector
e is called an eigenvector conesponding to the given eigenvalue ,\,- A self-adjoint operator A in the space Rn possesses n mutually
orthogonal eigenvectors e 1 , ... , en. We assume that all the ek 's are nor-
malized, that is, II ek II = 1 for k = 1,. '' 'n. Then (ei, ek) = bik• The
corresponding eigenvalues are ordered with respect to absolute values: - If a linear operator A given on Rn possesses n mutually orthogonal
eigenvalues, then A is a self-adjoint operator: A = A*,