Page 364 - Advanced Linear Algebra

348 Advanced Linear Algebra

M~ M ² ³ is a Hilbert space and the details are left to the reader. If we define
o
M ²2³ by

~ if
² ³ ~ Á ~ F
£ if
then the collection
E ~¸ “ 2¹

is a Hilbert basis for M²2³ , of cardinality 2(( . To see this, observe that
ºÁ » ~ ² ³ ² ³ ~ Á

2

E
and so is orthonormal. Moreover, if  M ²2³ , then ² ³ £ for only a
Z
countable number of  2 , say ¸ Á Á à ¹ . If we define by

B
Z
~ ² ³
~
Z
then  cspan ² ³ and ² ³ ~ ² ³ for all  2 , which implies that ~ Z .
Z
E

This shows that M²2³ ~ cspan E ² ³ and so is a total orthonormal set, that is, a
E

Hilbert basis for M²2³ .
Now let be a Hilbert space, with Hilbert basis ~ 8 ¸ " “  2 ¹ . We define
/
a map as follows. Since is a Hilbert basis, any %  / has the
8¢/ ¦ M ²2³
form
% ~ º%Á " »"
2
Since the series on the right converges, Theorem 13.22 implies that the series
( (º%Á " »
2
converges. Hence, another application of Theorem 13.22 implies that the
following series converges:
º%Á " » ²%³ ~
2
It follows from Theorem 13.19 that is linear and it is not hard to see that it is

also bijective. Notice that ²" ³ ~ and so takes the Hilbert basis for /
8

E
to the Hilbert basis for M²2³ .