Page 225 - Advanced Linear Algebra

Real and Complex Inner Product Spaces 209

and so

º#Á"»º"Á#» ( º"Á#» (
º"Á "» c ~ ")) c
º#Á #» # ))
which is equivalent to the Cauchy–Schwarz inequality. Furthermore, equality
holds if and only if ) )"c # ~ , that is, if and only if "c # ~ , which is
equivalent to and being scalar multiples of one another.
#
"
To prove the triangle inequality, the Cauchy–Schwarz inequality gives
) )"b# ~ º" b#Á " b#»
~ º"Á "» b º"Á #» b º#Á "» b º#Á #»
#
)) b
"
#
" ))) ) b ) )
"
#
~² )) b ) )³
from which the triangle inequality follows. The proof of the statement
concerning equality is left to the reader.
Any vector space = , together with a function )) ¢ = ¦ s that satisfies
h
properties 1), 2) and 4) of Theorem 9.3, is called a normed linear space and the
h
function )) is called a norm . Thus, any inner product space is a normed linear
space, under the norm given by 9.1 .
(
)
It is interesting to observe that the inner product on can be recovered from the
=
norm. Thus, knowing the length of all vectors in is equivalent to knowing all
=
inner products of vectors in .
=
Theorem 9.4 (The polarization identities )
)
1 If is a real inner product space, then
=

)
)
º"Á #» ~ ² " b#) c " c# ³
)

2 If is a complex inner product space, then
)
=

)
)
)
º"Á #» ~ ² " b#) c " c# ³b ² "b #) c " c # ³
)
)
)

The norm can be used to define the distance between any two vectors in an
inner product space.
Definition Let be an inner product space. The distance ² " Á # ³ between any
=
two vectors and in is
#
=
"
²"Á #³ ~ " c # ) (9.2 )
)
Here are the basic properties of distance.