Page 363 - Advanced Linear Algebra
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Hilbert Spaces 347
Theorem 13.27 All Hilbert bases for a Hilbert space / have the same
cardinality. This cardinality is called the Hilbert dimension of / , which we
denote by hdim²/³ .
Proof. If has a finite Hilbert basis, then that set is also a Hamel basis and so
/
all finite Hilbert bases have size dim²/³ . Suppose next that 8 ~ ¸ 2¹
and 9 ~¸ 1¹ are infinite Hilbert bases for . Then for each , we have
/
~ º Á »
1
where 1 is the countable set ¸ º Á » £ ¹ . Moreover, since no can be
orthogonal to every , we have 2 1 ~ 1 . Thus, since each 1 is countable,
we have
2
2
( (1~ e 1 e L (( ~ ((
2
¨
By symmetry, we also have ((2 ( ( and so the Schroder–Bernstein theorem
1
implies that ((1~ (2 . (
Theorem 13.28 Two Hilbert spaces are isometrically isomorphic if and only if
they have the same Hilbert dimension.
Proof. Suppose that hdim²/ ³ ~ hdim²/ ³ . Let E ~ ¸" 2¹ be a
Hilbert basis for / and E ~ ¸ # 2 ¹ a Hilbert basis for / . We may
as follows:
define a map ¢/ ¦ /
5
4 ~ " #
2 2
We leave it as an exercise to verify that is a bijective isometry. The converse
is also left as an exercise.
A Characterization of Hilbert Spaces
=
We have seen that any vector space is isomorphic to a vector space - ² ) ³ of
all functions from to that have finite support. There is a corresponding
)
-
result for Hilbert spaces. Let be any nonempty set and let
2
M²2³ ~ ¢ 2 ¦ d c ( ² ³ BE (
D
2
²
The functions in M²2³ are referred to as square summable functions . We can
also define a real version of this set by replacing by . We define an inner
³
d
s
product on M²2³ by
º Á » ~ ² ³ ² ³
2
The proof that M²2³ is a Hilbert space is quite similar to the proof that
