Page 362 - Advanced Linear Algebra

346 Advanced Linear Algebra

or equivalently,

( (º%Á " »  % )
)
2
with equality if and only if % cspan ²E . ³
A Characterization of Hilbert Bases

Recall from Theorem 13.15 that an orthonormal set E ~¸" “  2¹ in a

Hilbert space is a Hilbert basis if and only if
/
cspan²³ ~ /
E
Theorem 13.25, then leads to the following characterization of Hilbert bases.

Theorem 13.26 Let E ~¸" “  2¹ be an orthonormal family in a Hilbert

space . The following are equivalent:
/
)
(
1 E is a Hilbert basis a maximal orthonormal set)
ž
2) E ~ ¸ ¹
3 E ) is total that is, cspan E ( ²³ ~ / )
)
4 %~% V for all % /
)
5 Equality holds in Bessel's inequality for all %/ , that is,
))
))%~% V
for all %/
6) Parseval's identity
º%Á &» ~ º%Á &»
VV
holds for all %Á &  / , that is,
º%Á &» ~ º%Á " »º&Á " »

2
Proof. Parts 1 , 2 and 3 are equivalent by Theorem 13.15. Part 4 implies part
)
)
)
)
)
)
)
V
E
3 , since % cspan² ³ and 3 implies 4 since the unique best approximation of
)
)
any % cspan ² ³ is itself and so %~% V . Parts 3 and 5 are equivalent by
E
)
Theorem 13.25. Parseval's identity follows from part 4 using Theorem 13.19.
Finally, Parseval's identity for &~ % implies that equality holds in Bessel's
inequality.
Hilbert Dimension
We now wish to show that all Hilbert bases for a Hilbert space have the same
/
cardinality and so we can define the Hilbert dimension of / to be that
cardinality.