TITLE: Is Riesz representation theorem general?
QUESTION [6 upvotes]: It is my first course to learn functional analysis.  
I am looking some video about functional analysis and it seems that this theorem is mainly used in Hilbert space:  

https://en.wikipedia.org/wiki/Riesz_representation_theorem 
Book: Functions, spaces, and expansion, By Ole p. 70.   

However, I also found that this theorem can also be applied to $l^p$ space: 
Riesz Representation Theorem for $l_p$
Can this theorem be applied to any vector space? (like topological space?)

REPLY [2 votes]: The more general version of the Riesz represenstation theorem in Topological space is the so called Riesz-Markov theorem. It asserts that:

Thereom(Riesz_Markov) : If $X$ is locally compact Hausdorff space and $\ell : C_0(X)\to \Bbb R. $ is a linear and continuous form satisfying $\ell(f)\ge 0$ whenever $f\ge 0$. Then there exists a unique regular Borel measure $\mu$ on $X$ such that $$\ell(f) = \int_X f d\mu, ~~~~\forall~~f\in C_0(X).$$
Where $C_0(X)$ denote the space of bounded and continuous functions on $X$.


Note that we have $\ell \in (C_0(X))^*$ see below.

Remark: Such operators $\ell$ satisfying: $\ell(f)\ge 0$ whenever $f\ge 0$   is automatically bounded. In fact: we have $$\|f\|_\infty\pm f\ge 0\implies \ell(1)\|f\|_\infty\pm \ell(f) \overset{\text{linearity}}{=} \ell(\|f\|_\infty)\pm \ell(f)\overset{\text{linearity}}{=}\ell(\|f\|_\infty\pm f) \ge 0$$
That is for all $f\in C_b(X)$ we have, $$ |\ell(f)| = \pm\ell(f) \le \ell(1)\|f\|_\infty$$
this proves the continuity of $\ell$ and hence $\ell \in (C_0(X))^*$.