Metamath Proof Explorer


Theorem nmfnxr

Description: The norm of any Hilbert space functional is an extended real. (Contributed by NM, 9-Feb-2006) (New usage is discouraged.)

Ref Expression
Assertion nmfnxr
|- ( T : ~H --> CC -> ( normfn ` T ) e. RR* )

Proof

Step Hyp Ref Expression
1 nmfnval
 |-  ( T : ~H --> CC -> ( normfn ` T ) = sup ( { x | E. y e. ~H ( ( normh ` y ) <_ 1 /\ x = ( abs ` ( T ` y ) ) ) } , RR* , < ) )
2 nmfnsetre
 |-  ( T : ~H --> CC -> { x | E. y e. ~H ( ( normh ` y ) <_ 1 /\ x = ( abs ` ( T ` y ) ) ) } C_ RR )
3 ressxr
 |-  RR C_ RR*
4 2 3 sstrdi
 |-  ( T : ~H --> CC -> { x | E. y e. ~H ( ( normh ` y ) <_ 1 /\ x = ( abs ` ( T ` y ) ) ) } C_ RR* )
5 supxrcl
 |-  ( { x | E. y e. ~H ( ( normh ` y ) <_ 1 /\ x = ( abs ` ( T ` y ) ) ) } C_ RR* -> sup ( { x | E. y e. ~H ( ( normh ` y ) <_ 1 /\ x = ( abs ` ( T ` y ) ) ) } , RR* , < ) e. RR* )
6 4 5 syl
 |-  ( T : ~H --> CC -> sup ( { x | E. y e. ~H ( ( normh ` y ) <_ 1 /\ x = ( abs ` ( T ` y ) ) ) } , RR* , < ) e. RR* )
7 1 6 eqeltrd
 |-  ( T : ~H --> CC -> ( normfn ` T ) e. RR* )