Page 365 - Advanced Linear Algebra

Hilbert Spaces 349

Notice also that

) ) ~ º ²%³Á ²%³» ~ ( ²%³ ( º%Á " » ~ i º%Á " »" i ) ~ % )
2 2
and so is an isometric isomorphism. We have proved the following theorem.

Theorem 13.29 If is a Hilbert space of Hilbert dimension and if is any
/
2

set of cardinality , then is isometrically isomorphic to ² M 2 . ³

/
The Riesz Representation Theorem
We conclude our discussion of Hilbert spaces by discussing the Riesz
representation theorem. As it happens, not all linear functionals on a Hilbert
space have the form “take the inner product withà ,” as in the finite-
dimensional case. To see this, observe that if & / , then the function
²%³ ~ º%Á &»
&
is certainly a linear functional on / . However, it has a special property. In
particular, the Cauchy–Schwarz inequality gives, for all %/ ,
( & ( ²%³ ~ ( (º%Á &»  % ) ) & )
)
or, for all %£ ,
( & ( ²%³ &
))
))
%
Noticing that equality holds if %~& , we have
( & ( ²%³
sup ~&
))
%£ ))%
This prompts us to make the following definition, which we do for linear
(
transformations between Hilbert spaces this covers the case of linear
functionals .
)
Definition Let be a linear transformation from / to / . Then is
¢/ ¦ /
said to be bounded if
) ) %
sup B
%£ ))%
If the supremum on the left is finite, we denote it by )) and call it the norm of

.