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Hilbert Spaces 327

vectors in an inner product space , then
=
²% ³ ¦ % if and only if ) % c % ¦ as ) ¦ B

The fact that the inner product is continuous as a function of either of its
coordinates is extremely useful.

Theorem 13.5 Let be an inner product space. Then
=
1) ²% ³¦%Á ²& ³¦& ¬ º% Á & »¦º%Á &»

)
2) ²% ³ ¦ % ¬ % ) ) ¦ % )
Complete inner product spaces play an especially important role in both theory
and practice.
Definition An inner product space that is complete under the metric induced by
the inner product is said to be a Hilbert space .

Example 13.1 One of the most important examples of a Hilbert space is the
space . Recall that the inner product is defined by

M
B
ºÁ » ~ % &
%&

~
(In the real case, the conjugate is unnecessary. The metric induced by this inner
³
product is
°2
B
%&
)
² Á ³ ~ % c & ) ~ ( % c & (
8 9
~
which agrees with the definition of the metric space given in Chapter 12. In

M
other words, the metric in Chapter 12 is induced by this inner product. As we
saw in Chapter 12, this inner product space is complete and so it is a Hilbert
(
space. In fact, it is the prototype of all Hilbert spaces, introduced by David
Hilbert in 1912, even before the axiomatic definition of Hilbert space was given
by John von Neumann in 1927.³
The previous example raises the question whether the other metric spaces M
² £ ), with distance given by
°
B
² Á ³ ~ % c & ) ~ 8 ( % c & ( 9 ( 13.1)
%&
)
~
are complete inner product spaces. The fact is that they are not even inner
product spaces! More specifically, there is no inner product whose induced
(
metric is given by 13.1 . To see this, observe that, according to Theorem 13.1,
)