Page 234 - Advanced Linear Algebra

218 Advanced Linear Algebra

(entrywise) to an upper triangular matrix , which therefore has the eigenvalues
<
of on its main diagonal. Results can be obtained in the real case as well. For
(
more details, we refer the reader to [48], page 115.
Hilbert and Hamel Bases
Definition A maximal orthonormal set in an inner product space is called a
=
=
Hilbert basis for .
Zorn's lemma can be used to show that any nontrivial inner product space has a
Hilbert basis. We leave the details to the reader.

Some care must be taken not to confuse the concepts of a basis for a vector
space and a Hilbert basis for an inner product space. To avoid confusion, a
vector space basis, that is, a maximal linearly independent set of vectors, is
referred to as a Hamel basis . We will refer to an orthonormal Hamel basis as an
orthonormal basis.

To be perfectly clear, there are maximal linearly independent sets called
(Hamel) bases and maximal orthonormal sets (called Hilbert bases). If a
maximal linearly independent set (basis) is orthonormal, it is called an
orthonormal basis.

Moreover, since every orthonormal set is linearly independent, it follows that an
orthonormal basis is a Hilbert basis, since it cannot be properly contained in an
orthonormal set. For finite-dimensional inner product spaces, the two types of
bases are the same.

Theorem 9.13 Let = be an inner product space. A finite subset
E ~¸" Á Ã Á " ¹ of is an orthonormal Hamel basis for if and only if it is( ) =
=

a Hilbert basis for .
=
Proof. We have seen that any orthonormal basis is a Hilbert basis. Conversely,
if E is a finite maximal orthonormal set and E ‰ F , where F is linearly
independent, then we may apply part 1) to extend E to a strictly larger
orthonormal set, in contradiction to the maximality of . Hence, is maximal
E
E
linearly independent.
The following example shows that the previous theorem fails for infinite-
dimensional inner product spaces.
Example 9.5 Let =~ M and let 4 be the set of all vectors of the form

~ ² ÁÃÁ Á Á Áó

where has a in the th coordinate and 's elsewhere. Clearly, 4 is an

orthonormal set. Moreover, it is maximal. For if #~²% ³M has the property

that #ž4 , then