Page 367 - Advanced Linear Algebra

Hilbert Spaces 351

In particular,
) ) %
%~
)) ¬ 
%
))
and so
) ) ²%³ ) % )
)) ) %~ ) ¬  ¬ 
%~ ¬
) ) % ) ) %

Thus, is bounded.
Now we can state and prove the Riesz representation theorem.

(
Theorem 13.32 The Riesz representation theorem) Let / be a Hilbert

space. For any bounded linear functional on / , there is a unique '  /
such that
²%³ ~ º%Á ' »

)
for all %/ . Moreover, ')) ~ . )

Proof. If ~ , we may take ' ~ , so let us assume that £ . Hence,

2~ ker ² ³ £ / and since is continuous, 2 is closed. Thus

/~ 2 p 2 ž
Now, the first isomorphism theorem, applied to the linear functional ¢ / ¦ - ,
)
(
implies that /°2 š - as vector spaces . In addition, Theorem 3.5 implies that
ž
ž
ž
/°2 š 2 and so 2 š -. In particular, dim ²2 ³ ~ .
For any ' 2 ž , we have
%2 ¬ ²%³ ~ ~º%Á '»
ž
Since dim²2 ³ ~ , all we need do is find £ '  2 ž for which
²'³ ~ º'Á '»
for then ² '³ ~ ²'³ ~ º'Á '» ~ º 'Á '» for all  - , showing that
²%³ ~ º%Á '» for %  2 as well.

But if £'  2 ž , then

²'³
'~ '

º'Á '»

'
has this property, as can be easily checked. The fact that )) ~ ) ) has
already been established.