18

In Wikipedia, there are three versions of Riesz theorems:

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

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

3 The representation theorem for the dual of $C_0(X)$, where $X$ is a locally compact Hausdorff space.

I was wondering

  1. if none of the three versions is more general than the others, in the sense that no one can be derived from another?
  2. when two or three of them can coincide?

Thanks and regards!

Community
  • 1
Tim
  • 47,382
  • In general, a theorem which describes (some class of) linear functionals on a certain space is sometimes called a Riesz representation theorem. For instance, the fact that the dual of $L^p$ is $L^q$ (under the suitable assumptions) is mentioned as a/the Riesz representation theorem in some real analysis texts. – Mark Jul 07 '11 at 21:25
  • 1
    @Mark: I don't think this general version is correct as you state it. The reason why they are all called Riesz representation theorem is because Riesz identified these dual spaces in his foundational works on functional analysis. The $L^2$ and $L^p$-versions are often called Fréchet-Riesz because of two 1907 Comptes Rendus notes in issue 144 (one by F. and one by R.), and the measure version is often called Riesz-Markov (sometimes Kakutani is added) for similar reasons. – t.b. Jul 08 '11 at 00:58
  • I dunno, googling "a Riesz representation theorem" and "Riesz type theorem" yields all sorts of results of this form (e.g. http://www.sciencedirect.com/science/article/pii/0022247X77901445 and http://www.jstor.org/stable/2037423) – Mark Jul 08 '11 at 03:23

1 Answers1

22

These are three different theorems and there's no relation between 1) and the others except in the case when $X$ is finite, where all three theorems coincide (since $\ell^2(X)$, $C_0(X)$ and $C_c(X)$ then are the same topological vector space). Note however that $C_0(X)$ and $C_c(X)$ are never Hilbert spaces (unless the locally compact space $X$ is empty or a point), so 2) and 3) can't have a direct relation to 1).

Since positivity implies continuity 2) can be interpreted as characterizing continuous linear functionals on $C_c{(X)}$ as well, and I'm addressing this version below.

The results 2) and 3) are closely related and often 3) is proved as a corollary of 2).

Note that the space $C_c(X)$ is dense in $C_{0}(X)$ with respect to the sup-norm. So a continuous linear functional on $C_c(X)$ (= a signed Radon measure) extends (uniquely) to a continuous linear functional $C_{0}(X)$ (= a signed bounded Radon measure) if and only if it is of bounded variation. Moreover, 2) and 3) coincide if $X$ is compact (and there are also proofs of 3) reducing it to that case).

t.b.
  • 78,116
  • @Theo: Thanks! In your first sentence, by "when $X$ is finite", do you mean 1 and the others coincide for $L_2(X)$? – Tim May 03 '11 at 16:20
  • @Tim: If $X$ is a finite set with $n$ elements, then these are all the same (rather trivial) theorem on $\mathbb{R}^n$ or $\mathbb{C}^n$. – t.b. May 03 '11 at 16:22
  • @Theo: Thanks! On $L_2$ space as a Hilbert space with the usual inner product, will the three coincide? – Tim May 03 '11 at 16:24
  • @Tim: No, I'm not saying that. What I'm saying is: If $X$ has $n$ elements, then $\ell^2(X)$, $C_0(X)$ and $C_c(X)$ are all isomorphic to the same topological vector space, namely $\mathbb{R}^n$ or $\mathbb{C}^n$ (since there is only one Hausdorff topological vector space of dimension $n$ over $\mathbb{R}$ or $\mathbb{C}$. In general, $L^2(X)$ depends on the choice of a measure on $X$ and the Riesz theorem for $L^2$ is ultimately an abstract theorem on Hilbert spaces. While the two theorems on $C_0$ and $C_c$ are theorems on these two latter spaces (which are never Hilbert spaces). – t.b. May 03 '11 at 16:31
  • @Tim: Ah, maybe I misunderstood you. Yes, I was speaking of $L^2(X)$ when $X$ is finite (and equipped with counting measure, say). – t.b. May 03 '11 at 16:34
  • @Theo: Thanks! You didn't misunderstand me. – Tim May 03 '11 at 16:36
  • But positivity does not imply continuity. The Lebesgue integral is positive but discontinuous on $C_c(\mathbb{R})$. – scineram Oct 31 '11 at 15:31
  • @scineram: I was allowing myself to be a bit sloppy in my answer. Sorry about the confusion. It depends of course on what topology you put on $C_c(\mathbb{R})$ and I decided to sweep that under the rug. – t.b. Oct 31 '11 at 15:49
  • So when you wrote "positivity implies continuity", what topology did you put on $C_c(\mathbb R)$? The $|\cdot|_\infty$, I assume, given that you use that in your last paragraph. – Rudy the Reindeer Aug 16 '12 at 14:45
  • @t.b. Thanks a lot for the link, off to read it now. I am indeed slightly worried since there are tumeni Riesz theorems, all of which are a bit the same but not really and all of which are full of words that sound a bit the same but aren't. And I need to be able to remember at least two of them. – Rudy the Reindeer Aug 16 '12 at 15:05
  • I'm also curious about the topology on $C_c(\mathbb{R})$ if its not the sup norm. And if it is not the sup norm, how does one use the above density argument? – Christian Bueno Aug 22 '17 at 20:54