Page 227 - Advanced Linear Algebra

Real and Complex Inner Product Spaces 211

)
1 is an isometry if it preserves the inner product, that is, if

º"Á #» ~ º"Á #»

for all "Á #  = .
2 A bijective isometry is called an isometric isomorphism . When ¢= ¦ >
)
is an isometric isomorphism, we say that = and > are isometrically
isomorphic.

It is clear that an isometry is injective and so it is an isometric isomorphism
provided it is surjective. Moreover, if

dim²= ³ ~ dim²> ³  B
injectivity implies surjectivity and is an isometry if and only if is an

isometric isomorphism. On the other hand, the following simple example shows
that this is not the case for infinite-dimensional inner product spaces.

Example 9.3 The map ¢M ¦ M defined by
²% Á% Á% Áó ~ ² Á% Á% Áó

is an isometry, but it is clearly not surjective.
Since the norm determines the inner product, the following should not come as a
surprise.

B
Theorem 9.6 A linear transformation ²= Á > ³ is an isometry if and only if
it preserves the norm, that is, if and only if
) ) #~# )
)
for all #= .
Proof. Clearly, an isometry preserves the norm. The converse follows from the
polarization identities. In the real case, we have

º "Á #» ~ ² " b#) ) c " c# ³

)
)

~ ² ²" b #³) c ) ) ²" c #³ ³
)

)
)
~ ² "b# ) c "c# ³
)

~º"Á #»
and so is an isometry. The complex case is similar.

Orthogonality
The presence of an inner product allows us to define the concept of
orthogonality.