# Versions of Riesz Representation Theorem until now

I am writing on Riesz Representation Theorem. How this theorem was motivated and what further generalizations were done while it was on its way to where it is now. Starting from the beginning, Frigyes Riesz originally proved the Riesz Representation Theorem on C[0,1] in 1909, which stated that

"Given the linear operation $$A[f(x)]$$, we can determine the function of bounded variation $$α(x)$$, such that, for any continuous function $$f(x)$$, we have $$A[f(x)]=\int_0^1f(x)dα(x)."$$

Then more things were extended in this theorem and there were so many such representations were done from 1909 till now.. Some of them are:

(a.)The Hilbert space representation theorem for the (continuous) dual space of a Hilbert space;

(b.)The representation theorem for positive linear functionals on $$C_c(X)$$, where X is a locally compact Hausdorff space;

(c.) The representation theorem for bounded linear functionals on $$C_c(X)$$, where X is a locally compact Hausdorff space;

Then it is generalized to bounded linear operators:

(d.) Let $$S$$ be a locally compact Hausdorff space, $$X$$ be a Banach space and $$T$$ be a weakly compact operator from $$C_0(S)$$ to $$X$$. Then there exists a unique regular vector valued measure $$\mu: \mathcal{B}(S)\to X$$, such that $$T(f) = \int f d\mu, ~~f\in C_0(S).$$

(e.) $$T:C_0(S,X)\to Y$$ be a bounded linear operator where $$S$$ is locally compact Hausdorff space and $$X,Y$$ are Banach spaces.Then if for every $$x\in X$$, the bounded linear operator $$T_x: C_0(S)\to Y$$ defined by $$T_x(g)=T(g\circ x), \ \ g\in C_0(S)$$ is weakly compact, then there exists a Baire operator valued measure $$m : \mathcal{C}(\mathcal{B}_a(S))\to L(X,Y)$$ countably additive in strong operator topology such that $$T(f)=\int f\ dm , \ \ f\in C_0(S,X)$$

After that this result is extended to locally convex Hausdorff topological vector space also.

Request: But the problem is, I do not know any exact years and many concepts those were added in this theorem. So I'll be really grateful if I could be provided with all the details those I need to write properly and comprehensively on this theorem such as the years and the names of the mathematicians who have been generalizing to this theorem until now. I really apreciate your help regarding what I asked it for.

• Have you looked up references and chased them back to the original papers? – Simon S Jun 15 '15 at 15:33
• Yeah, from references I got this information, but I still doubt that I am missing some concepts. Also I want to know about recent research work done on this topic. – User Jun 15 '15 at 16:14
• You're asking a classic question about the literature in a subject. This is not a quick or easy question to answer. If I were you, I would try and find a recent survey paper on the topic or a PhD thesis somewhere. Good luck. – Simon S Jun 15 '15 at 16:33
• thanks for the suggestion @SimonS . I am trying my best to find some recent survey paper. – User Jun 16 '15 at 8:39
• This is a very nice question, have you had any progress on it? – Conrado Costa Oct 15 '16 at 17:27