Hi Nathalie,

 I think is very important to look through the link that George sent. It gives good answer. Also, you can please see the link below if it's of help.

https://regularize.wordpress.com/2011/11/11/dual-spaces-of-continuous-functions/

Best Regards,

Hannah

Hi, I have found a possible answer on the following link:

http://mathoverflow.net/questions/83593/dual-space-of-continuous-functions

Thank you.

Dear Natalie,

I have just seen your question and the answer by Richard Epenoy is a good one

Thank you Fernando for your answer!

I appreciate your remark George. I looked on the Stone-Čech compactification theorem and the construction seems to not be so evident.

Regards

Hi Nathalie,

 I think is very important to look through the link that George sent. It gives good answer. Also, you can please see the link below if it's of help.

https://regularize.wordpress.com/2011/11/11/dual-spaces-of-continuous-functions/

Best Regards,

Hannah

Thank you for the interesting discussions on the dual of the space/subspace of continuous function.  I also agree with Richard and George's comments. It is a Good question that might deserve more studies dependent on the type of project one is currently working on.   

 http://mathoverflow.net/questions/44183/dual-of-bounded-uniformly-continuous-functions

I wish to add that Hannah's source provides some more formal, detailed theorem and proofs which is very nice in the setting of locally compact metric spaces. 

https://regularize.wordpress.com/2011/11/11/dual-spaces-of-continuous-functions/

Best Regards,  Shi-Jun

Thank you for your helpful remarks Hannah and Shijun.

Other good references regarding this subject could be the following two books:

J. B. Cooper (1987) Saks-spaces-and-applications-to-functional-analysis

and

Dunford, N.; Schwartz, J.T. (1958), Linear operators, Part I, Wiley-Interscience

Hi everybody,

The (algebraic or topological) dual space of the space $C_b(X)$ for any completely regular space  $X$, mertizable or not, is the same as that of $C(\beta X)$, since the two spaces are isometrically isomorphic, where $\beta X$ is the stone ceck compactification of $X$. The isomorphism being obviously given by the extension $f \mapsto f^{\beta}$. Therefore the situation is the same as for compact spaces.