Page 239 - Advanced Linear Algebra

Real and Complex Inner Product Spaces 223

²$³
9~ $

$
))
For part 2), we have
º#Á 9 b » ~ ² b ³²#³
~ ²#³ b ²#³
~º#Á 9 » b º#Á 9 »

~º#Á 9 b 9 »

for all #= and so
9 ~ b 9 b 9
Note that if = ~ s , then 9 ~ ² ² ³Á à Á ² ³³ , where ² Á à Á ³ is the

standard basis for s .
Exercises
1. Prove that if a matrix 4 is unitary, upper triangular and has positive entries
on the main diagonal, must be the identity matrix.
2. Use the QR factorization to show that any triangularizable matrix is
unitarily (orthogonally) triangularizable.
3. Verify the statement concerning equality in the triangle inequality.
4. Prove the parallelogram law.
5. Prove the Apollonius identity

) )$c" b ) )$c# ) ~ ) " c# b h h $c ²" b#³

6. Let be an inner product space with basis . Show that the inner product
=
8
is uniquely defined by the values º"Á#» , for all "Á#  8 .
#
"
7. Prove that two vectors and in a real inner product space = are
orthogonal if and only if

) )"b# ) ) ~ " ) ) b #
8. Show that an isometry is injective.
9. Use Zorn's lemma to show that any nontrivial inner product space has a
Hilbert basis.
10. Prove Bessel's inequality.
11. Prove that an orthonormal set is a Hilbert basis for a finite-dimensional
E
=
vector space if and only if ~ # # , for all V # = .
12. Prove that an orthonormal set is a Hilbert basis for a finite-dimensional
E
vector space if and only if Bessel's identity holds for all  # = , that is, if
=
and only if