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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:

  1. https://en.wikipedia.org/wiki/Riesz_representation_theorem
  2. 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?)

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    $\begingroup$ I believe both are special cases of this theorem: en.wikipedia.org/wiki/… which relates the linear functionals on spaces of continuous functions on a locally compact space to measures (on that space). $\endgroup$ – Justin Benfield Aug 28 '16 at 2:51
  • $\begingroup$ Could you explain how those two representation theorems relate? One is a statement on representing the elements in the dual space in terms of an inner product, and the other relates a subset of the dual space (positive linear functionals) to measures. $\endgroup$ – Merkh Aug 28 '16 at 3:07
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In latter part of the 19th century, the Mathematician Bourlet was interested in linear functions $F : U \rightarrow \mathbb{C}$ on the set of holomorphic functions $U$ on the unit disk. He was able to characterize the functions $F$ of this type, assuming that $F$ was continuous. This was the first characterization of the type. Today we would refer to $F$ as a continuous linear functional.

In 1903, Hadamard attacked the same problem for $F : C[a,b]\rightarrow\mathbb{C}$, assume that $F$ was continuous with respect to uniformly convergent sequences of function in $C[a,b]$. His characterization was not as complete as it is given in Functional Analysis today, but the results gained considerable attention. This led others to consider different function space and other convergence criteria on those functions spaces.

A few years after Hadamard's work, a general inner product space and a Hilbert space were defined for the first time. Mathematicians were immediately interested in knowing the continuous functionals on a Hilbert space. Frechet and F. Riesz independently determined that the continuous linear functionals on a Hilbert $\ell^2$ would have the form $F(x) = \langle x,y\rangle$ for a unique $y$.

In 1909, F. Riesz returned to the problem of continuous linear functionals on $C[a,b]$, and was able to use the Riemann-Stieltjes integral to uniquely represent such a continuous linear function as $$ F(f) = \int_{a}^{b}f(t)d\mu(t), $$ for function $\mu$ of bounded variation on $[a,b]$. Uniqueness requires some normalization for $\mu$, but the normalization is reasonable. The results of F. Riesz were a radical departure from the conventional thought of the time on Linear Algebra. One of the interesting properties of the new infinite-dimensional spaces was how the functionals in such a case could not be identified with the elements of the underlying space $C[a,b]$, which would naturally lead to separating the notions of a vector space and its continuous dual. The spaces $\ell^p$ were later studied by Riesz, and found to illustrate the necessity of separating the space and its dual in a simpler manner. And spaces such as $C[a,b]$ led Mathematicians to abandon Cayley's Linear Algebra view of elements as a sequence of coordinates, and to deal directly with the vectors of the space. A traditional basis for $C[a,b]$ is non-trivial, making the study of $C[a,b]$ by coordinates along a basis almost useless.

A general context for the dual of a vector space $V$ with topology is a space of continuous linear functionals on the space $V$. The typical vector space $V$ that can be studied this way is a locally convex topological vector space, because local convexity is intimately tied to the Hahn-Banach Theorem for extending linear functional from subspaces to the full space. In this case the dual space is rich enough to consider studying the space $V$ by its dual.

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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_b(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 $$\ell(f) = \int_X f d\mu, ~~~~\forall~~f\in C_b(X).$$ Where $C_b(X)$ denote the space of bounded and continuous functions on $X$.

Note that we have $\ell \in (C_b(X))^*$ 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 prove the continuity of $\ell$ and hence $\ell \in (C_b(X))^*$

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  • $\begingroup$ What is $C_b(X)$? $\endgroup$ – user99914 Dec 3 '17 at 3:56
  • $\begingroup$ @John Ma it is the space le bounded functions on X $\endgroup$ – Guy Fsone Dec 3 '17 at 7:42
  • $\begingroup$ ok, so it seems that I cannot find this version in wiki. Can you also cite another reference? $\endgroup$ – user99914 Dec 3 '17 at 7:46
  • $\begingroup$ why this is just a mofication of version on wiki. what is new here ? $\endgroup$ – Guy Fsone Dec 3 '17 at 7:52
  • $\begingroup$ Ok, so can you at least sketch how your assertion follows from one of the Riesz representation theorem in the wiki? $\endgroup$ – user99914 Dec 3 '17 at 7:58

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