Riesz's Sur les opérations fonctionnelles linéaires

F. Riesz announces the Riesz Representation Theorem (1909) — click image for original.

To define what is meant by a linear operation, it is necessary to specify the domain of the functional. We consider the totality of all real continuous functions \( \Omega \) between two fixed numbers, for example between \( 0 \) and \( 1 \); for this class, we define the limit function based on the assumption of uniform convergence. The functional operation \( \text{A}[f(x)] \), which associates to each element of \( \Omega \) a corresponding real number, will be called continuous if when \( f(x) \) is the limit of \( f_i(x) \), then \( \text{A}(f_i) \) tends to \( \text{A}(f) \). Such a distributive and continuous operation is said to be linear. It is easy to show that this operation is bounded, that is to say, there is a constant \( M_A \) such that for every element \( f(x) \) we have

\[ \begin{equation}{|\text{A}[f(x)]| \leq M_A \times max. |f(x)|.} \tag{1} \end{equation} \]

M. Hadamard had demonstrated the remarkable fact that any linear operation \( \text{A}[f(x)] \) is of the form \( {\displaystyle \lim_{n \rightarrow \infty}} \int_0^1 k_n(x) f(x) dx \), where the \( k_n(x) \) are continuous functions^[1]. In the present note, we will develop a new analytical expression for the linear operation, containing only one generating function.

To this end, we consider the generalized integral

\[ \begin{equation}{\int_0^1 f(x)d\alpha(x).} \tag{2} \end{equation} \]

Let us briefly recall the meaning of this expression. We understand the limit of the sum \( \sum_i f(\xi_i)[\alpha(x_{i+1}) - \alpha(x_i)] \) corresponding to a division of the interval (0, 1) into a finite number of subintervals; \( \xi_i \) denotes an element of the interval \( (x_i, x_ {i + i}) \). The passage to the limit is subject only to the sole condition that the length of the subintervals tends uniformly to zero.^[2]

We do not need to develop the most general conditions for the integral (2) to make sense. We just have to note that for a function \( f(x) \) assumed to be continuous, the integral (2) exists for any function \( \alpha(x) \) of bounded variation, continuous or not. In this case, we have the inequality

\[ \begin{equation}{{\left|\int_0^1 f(x)d\alpha(x)\right|} \leq \text{ maximum of } |f(x)| \times \text{ total variation of } \alpha(x).} \tag{3} \end{equation} \]

After these preliminaries, given a linear operation \( \text{A}[f(x)] \), we define the function \( \text{A}(x) \) by the equality \( \text{A}(\xi) = \text{A}[f(x;\xi)] \), where we designate by \( f(x;\xi) \) the function equal to \( x \) for \( 0 \leq x \leq \xi \) and to \( \xi \) for \( \xi \leq x \leq 1 \). Now by applying the inequality (1) to the continuous function \( f(x) \) defined by the conditions \( {f(x_i) = 0}; {f\left({{x_i+x_{i+1}} \over 2}\right)} = {\text{sign}[\text{A}(x_{i+i}) - \text{A}(x_i)]} \), \( f(x) \) is linear in each half-interval, and we get the inequality

\[ {{\LARGE{\sum_i}} {{\left|\text{A}(x_i)-2\text{A}\left({{x_i+x_{i+1}}\over 2}\right)+\text{A}(x_{i+1})\right|} \over {x_{i+1}-x_i}}} \leq {M_A \over 2}. \]

It follows that the derived numbers of the function \( \text{A}(x) \) exist and that these derivatives are functions of bounded variation.

Now we define a function \( \alpha (x) \) with the following conditions: for \( 0 < x < 1 \), \( -\alpha (x) \) = one of these derived numbers, for example the upper right derivative; \( \alpha (0) = -\text{A}[n(x)] \), \( n(x) \) indicating the function with the constant value \(1 \); \( \alpha (1) = 0\). The function \( \alpha (x) \) being of bounded variation, the integral (2) exists for any continuous function \( f(x) \). Particularly if the continuous function \( f(x) \) is formed of a finite number of linear pieces, a slight transformation is already enough to see that the integral (2) is equal to \( \text{A}[f(x)] \). Noting that every continuous function is the limit of such functions based on inequality (3), we conclude that the same fact obtains for any function \( f(x) \). So we have the theorem:

Given the linear operation \( \text{A}[f(x)] \), we can determine the function of bounded variation \( \alpha(x) \), such that, for any continuous function \( f(x) \), we have \[ {\text{A}[f(x)]} = {\int_0^1 f(x)d\alpha(x)}. \]

Note again that the required property in our theorem does not uniquely determine the function \( \alpha(x) \). We see the nature of this indeterminacy in the following theorem: For the integral (2) to be zero for each element \( f(x) \) of \( \Omega \), it is necessary and sufficient that the function of bounded variation \( \alpha(x) \) be constant, except perhaps for a countable set not containing the points \( 0, 1 \). We can also take this lack of uniqueness to make \( a(x) \) such that its total variation is as small as possible.

In this context, we've also solved a very interesting problem, already discussed with more restricted conditions by M. Haar (Dissertation, Göttingen, 1909). Here is our result:

Let \( \text{A}_1(f), \text{A}_2(f) \cdots \text{A}(f) \) be linear operations and \( \alpha_1(x), \alpha_2(x), \cdots \alpha(x) \) their generating functions; we assume that their total variation is the smallest possible; moreover, we assume \( \alpha_1(0) = \alpha_2(0) = \cdots = \alpha(0) = 0 \). Under these conditions, for any element \( f(x) \) of \( \Omega \), it is necessary and sufficient for the infinite sequence \( \text{A}_n(f) \) to tend towards \( \text{A}(f) \): 1° that we have \[ {\int_0^\xi \alpha(x)dx} = {\displaystyle \lim_{n \rightarrow \infty}}{\int_0^\xi \alpha_n(x)dx} \hskip{20pt} (0 < \xi \leq 1), \hskip{20pt} \alpha(1) = {\displaystyle \lim_{n \rightarrow \infty}} \alpha_n(1); \] 2° that the total variation of the functions \( \alpha_n(x) \) does not exceed any finite bound.

Thanks to this theorem, a function of two variables can be completely characterized by an analytical expression of the connected linear functional transformation, by matching up in this manner each distributive and continuous operation on elements of \( \Omega \), with a specific element of the same class or another similar class. By the same theorem it is easy to see how our expression is linked to that of M. Hadamard.

^ 1. Sur les opérations fonctionnelles (Comptes rendus, 9 February 1903). Cf. also M. Fréchet, Sur les opérations linéaires (Transactions American Math. Soc., 5, 1904, p. 493-499).

^ 2. In the literature, this notion of the integral goes back to Stieltjes, Recherches sur les fractions continues (Annales de Toulouse, 8, 1894). M. Jules (Gyula / Julius) Kőnig would inform me that he had used it much earlier in his course, but he only wrote one note on the subject, published in Hungarian : Mathematikai ès Tennészettudomànyi Ertesitő, 1897.

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Translated by Mike Bertrand (Apr 13, 2015)