|
|
| View previous topic :: View next topic |
| Author |
Message |
Cooper
Guest
|
 Posted: Sun May 27, 2007 10:00 am Post subject: Question on dual space. |
 |
|
Hi!
I want to ask some question in real analysis.
If X consists of 2 points, a, b define a measure m(a)=1 ,m(b)=m(X)=inf
m ( empty set ) =0
Is it true that L_inf (m) is the dual space of L_1 (m) ?
If there exist such isomorphism between these space, then I wonder
that whether or not for some g in L_inf(m),
the corresponding bounded linear functional A_g must have the form A_g
(f) =integral fg dm .
Any help would be very appreciated. |
|
| Back to top |
|
 |
| |
Ads |
Advertising
Sponsor
|
|
José Carlos Santos
Guest
|
| Posted: Sun May 27, 2007 11:01 am Post subject: Re: Question on dual space. |
|
|
On 27-05-2007 6:00, Cooper wrote:
| Quote: |
I want to ask some question in real analysis.
If X consists of 2 points, a, b define a measure m(a)=1 ,m(b)=m(X)=inf
m ( empty set ) =0
|
This doesn't make sense. A measure is a function defined on _subsets_ of
X, not on _elements_ of X. I suppose that you meant that m({a}) = 1 and
that m({b}) = 0. But m(X) cannot be equal to 0. It must be equal to 1.
| Quote: |
Is it true that L_inf (m) is the dual space of L_1 (m) ?
|
Yes. Actually, they are equal. Both spaces are the set of all functions
from X into R U {+oo,-oo} such that f(a) is real and the norm of such a
function is |f(a)|.
Best regards,
Jose Carlos Santos |
|
| Back to top |
|
|
| |
Ads |
Advertising
Sponsor
|
|
|
|
You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum
You cannot vote in polls in this forum
|
|
FAQ
Search
Memberlist
Usergroups
Register
Profile
Log in to check your private messages
Log in
