21. Topological vector spaces.

The general theory of topological vector spaces was founded during the period which goes from 1920 to 1930 approximately. But it had been prepared for a long time before by the study of numerous problems of Functional Analysis; and its history cannot be retraced without indicating, at least summarily, how the study of these problems gradually brought mathematicians (especially from the beginning of the XXth century) to become aware of the common ancestry of the questions under consideration, and of the possibility of formulating them in a much more general way and of applying to them uniform solution procedures.

It can be said that the analogies between Algebra and Analysis, and the idea of considering functional equations (that is to say where the unknown is a function) as "limit cases" of algebraic equations, go back to the beginnings of infinitesimal Calculus, which in a kind of way answers this need to generalise "from the finite to the infinite". But the direct algebraic ancestor of the infinitesimal Calculus is the calculus of finite differences (cf. pp. 185 ff.), and not the solution of general linear systems; and it is not before the middle of the XVIIIth century that the first analogies between this latter and the problems of the differential Calculus are manifested, in connection with the equation of vibrating cords. We will not go into the detail of the history of this problem here; but we must pick up the appearance of two fundamental ideas, which are found again constantly in what follows, and which both appear to be due to D. Bernoulli. The first consists in considering the oscillation of the cord as a "limit case" of the oscillation of a system of $n$ point masses, as $n$ tends to infinity; it is known that, for $n$ finite, this problem would a little bit later give the first example of the search for eigenvalues of a linear transformation (cf. p. 88); there correspond to these numbers, in the "passage to the limit" being considered, the frequencies of the "proper oscillations" of the cord, observed experimentally since long before, and whose theoretic existence had been established (notably by Taylor) at the beginning of the century. This formal analogy, although fairly rarely mentioned in what followed ([302 b], p. 390), seems never to have been lost sight of during the XIXth century; but, as we will see later, it did not acquire its full importance until about 1890-1900.

The other idea of D. Bernoulli (perhaps inspired by experimental facts) is the "principle of superposition", according to which the most general oscillation of the cord must be capable of being "decomposed" into the superposition of "proper oscillations", which, mathematically speaking, means that the general solution of the equation of vibrating cords must be capable of being expanded as a series $\sum_n c_n \phi_n(x,t)$, where the $\phi_n(x,t)$ represent the proper oscillations. It is known that this principle was to let loose a long quarrel on the possibility of expanding an "arbitrary" function as a trigonometric series, a quarrel that was only settled by the work of Fourier and of Dirichlet in the first third of the XIXth century. But even before this result was reached, other examples had been met of expansion in series of "orthogonal"$^1$ functions: spherical functions and Legendre polynomials, as well as different systems of the form $(e^{i \lambda_n x})$, where the $\lambda_n$ are no longer multiples of the same number, and which were introduced from the XVIIIth century in oscillation problems, as well as by Fourier and Poisson during their researches on the theory of heat. Around 1830, all the phenomena observed in these various special cases are systematised, by Sturm [302 a and b] and Liouville [204 a and b] in a general theory of oscillations, for functions of one variable: they consider the differential equation

with boundary conditions

and prove the following fundamental results:

1) the problem only has a solution $\neq 0$ when $\lambda$ takes one of the values from a sequence $(\lambda_n)$ of numbers $> 0$, tending to $+\infty$; 2) for each $\lambda_n$, the solutions are multiples of the same function $v_n$, that can be assumed to be "normalised" by the condition $\int_a^b \rho v_n^2 dx = 1$, and we have $\int_a^b \rho v_m v_n dx = 0$ for $m \neq n$; 3) every function $f$, twice differentiable in $[a, b]$, and satisfying the boundary conditions (21.2), is expandable as a uniformly convergent series $f(x) = \sum_n c_n v_n(x)$, where $c_n = \int_a^b \rho f v_n dx$; 4) we have the equality $\int_a^b \rho f^2 dx = \sum_n c_n^2$ (already proved by Parseval in 1799 — in a purely formal way, besides — for the system of trigonometric functions, and from which flows immediately the "Bessel inequality" stated by this latter (still for trigonometric series) in 1828).

Half a century later, these properties are completed by the work of Gram [133] who, following the researches of Tchebychef, brings to light the relation between the expansions in series of orthogonal functions and the problem

$^1$ This term does not appear however before the works of Hilbert.

of the "best quadratic approximation" (issuing directly from the "method of least squares" of Gauss, in error theory): this latter consists, given a finite sequence of functions $(\psi_i)_{1 \leq i \leq n}$, of finding, for a function $f$, the linear combination $\sum_i a_i \psi_i$ for which the integral $\int_a^b \rho(f - \sum_i a_i \psi_i)^2 dx$ attains its minimum. In principle it is a case there only of a basic linear algebra problem, but Gram solves it in an original way, by applying to the $\psi_i$ the process of "orthonormalization" generally known by the name of Erhard Schmidt. Going on next to the case of an infinite orthogonal system $(\phi_n)$, he sets himself the question of knowing when the "best quadratic approximation" $\mu_n$ of a function $f$ by linear combinations of the first $n$ functions of the sequence, tends to 0 when $n$ tends to infinity;$^2$ he is thus led to define the notion of a complete orthonormal system, and recognises that this property is equivalent to the non-existence of functions $\neq 0$ orthogonal to all the $\phi_n$. He even seeks to elucidate the concept of "mean quadratic convergence", but, before the introduction of the fundamental notions of measure theory, he could hardly obtain any very specific results in this direction.

In the second half of the XIXth century, the main effort of the analysts is mainly towards the extension of the Sturm-Liouville theory to functions of several variables, to which led notably the study of partial differential equations of elliptic type from mathematical Physics, and the boundary problems that are naturally associated with them. Interest is concentrated principally on the "vibrating membranes" equation

where the solutions are sought that are zero on the contour for a fairly regular domain $G$; and it is only little by little that the considerable analytic difficulties presented by this problem were overcome, a problem to which it was not thought possible to apply the methods that had succeeded for functions of a single variable. Let us remind ourselves of the main stages towards the solution: the introduction of the "Green's function" of $G$, whose existence was proved by Schwarz; the proof, also due to Schwarz, of the existence of the smallest eigenvalue; finally, in 1894, in a famous memoir ([251 a], v. IX, pp. 123-196), H. Poincaré succeeded in proving the existence and essential properties of all the eigenvalues, by considering, for a given "second member" $f$, the solution $u_\lambda$ of the equation $L_\lambda(u) = f$ which is zero on the contour, and in proving, by a clever generalisation of the method of Schwarz, that $u_\lambda$ is a meromorphic function of the complex variable $\lambda$, having only real simple poles $\lambda_n$, which are precisely, the sought for eigenvalues.

This research is closely linked to the beginnings of the theory of linear integral equations, which was to contribute no doubt most to the coming of modern ideas. We will limit ourselves here to giving a few short indications

$^2$ It must be noted that, in all this study, Gram does not limit himself to the consideration of continuous functions, but insists on the importance of the condition $\int_a^b \rho f^2 dx < +\infty$.

on the development of this theory. This type of functional equation, having first appeared sporadically in the first half of the XIXth century (Abel, Liouville), had acquired importance since Beer and C. Neumann had reduced the solution of "Dirichlet's problem" for a fairly regular domain $G$, to the solution of an "integral equation of the second type"

for an unknown function $u$; an equation that C. Neumann had succeeded in solving by means of a procedure of "successive approximations" in 1877. Driven no doubt as much by the algebraic analogies already mentioned as by the results that he had just obtained on the vibrating membranes equation, H. Poincaré, in 1896 ([251 a], v. IX, pp. 202-272), had the idea of introducing a variable parameter $\lambda$ in front of the integral in the preceding equation, and affirms that, as for the vibrating membranes equation, the solution is then a meromorphic function of $\lambda$; but he does not succeed in proving this result, which was only established (for a continuous "kernel" $K$ and a finite interval $[a, b]$) by I. Fredholm four years later [116]. This latter, more consciously perhaps even than his predecessors, lets himself be completely guided by the analogy of (21.4) with the linear system

in order to obtain the solution of (21.4) as a quotient of two expressions constructed on the model of determinants that occur in the formulae of Cramer. It was not besides a new idea: from the beginning of the XIXth century, the method of "indeterminate coefficients" (consisting of obtaining an unknown function assumed expandable in series $\sum_n c_n \phi_n$, where the $\phi_n$ are known functions, by calculating the coefficients $c_n$) had led to "linear systems in an infinity of unknowns"

Fourier, who comes across such a system, "solves" it still like a mathematician of the XVIIIth century: he suppresses all the terms having an index $i$ or $j$ greater than $n$, solves explicitly the finite system thus obtained, by means of Cramer's rule, then "passes to the limit" by making $n$ tend to $+\infty$ in the solution! When later it was no longer sufficient to play such lazy games of pass the parcel, it is still by means of the theory of determinants that the first attempts to attack the problem were made; starting in 1886 (following the work of Hill), H. Poincaré, then H. von Koch, had built a theory of "infinite determinants" which allows the solution of certain types of systems (21.6) following the classical model; and if these results were not directly applicable to the problem Fredholm aimed at, at least it is certain that the theory of von Koch, in particular, was of use as a model for the formation of his "determinants".

It is at this moment that Hilbert enters the scene and gives a new impulsion to the theory [163 b]. He starts by completing the work of Fredholm in achieving effectively the passage to the limit that leads from the solution of (21.5) to that of (21.4); but he adds to it immediately the corresponding passage to the limit for the theory of real quadratic forms, to which the types of integral equations with symmetric kernel (that is to say such that $K(y, x) = K(x, y)$) were leading naturally, by far the most frequent in mathematical Physics. He reaches thus the fundamental formula that generalises directly the reduction of a quadratic form to its axes

the $\lambda_n$ being the eigenvalues (necessarily real) of the kernel $K$, the $\phi_n$ forming the orthogonal system of corresponding eigenfunctions, and the second member of the formula (21.7) being a convergent series for $\int_a^b x^2(s)dx \leq 1$. He shows also how every function "representable" in the form $f(x) = \int_a^b K(x, y)g(y)dy$ allows the "expansion" $\sum_{n=1}^\infty \phi_n \int_a^b \phi_n(y)f(y)dy$, and carrying on the analogy with the classical theory of quadratic forms, he indicates a procedure for the determination of the $\lambda_n$ by a variational method, which is none other than the extension of the well-known extremal properties of the axes of a quadric ([163 b], pp. 1-38).

These first results of Hilbert were almost immediately taken up by E. Schmidt, in a simpler and more general form, avoiding the introduction of the "Fredholm determinants" as well as the passage from finite to infinite, and already very near an abstract statement, the fundamental properties of linearity and of positivity of the integral being visibly used only in the proofs [274 a]. But already Hilbert had reached conceptions which are much more general still. All the preceding work brought out the importance of functions that are square integrable, and the Parseval formula established a tight link between these functions and the sequences $(c_n)$ such that $\sum_n c_n^2 < +\infty$. It is no doubt this idea that guides Hilbert in his memoirs of 1906 ([163 b], chap. XI-XIII) where, taking up the old method of "indeterminate coefficients", he shows that the solution of the integral equation (21.4) is equivalent to a system of an infinity of linear equations

for the "Fourier coefficients" $x_p = \int_a^b u(t)\omega_p(t)dt$ of the unknown function $u$ with respect to a given complete orthonormal system $(\omega_n)$ (with $b_p = \int_a^b f(t) \omega_p(t) dt \text{ and } k_{pq} = \int_a^b \int_a^b K(s,t) \omega_p(s) \omega_q(t) ds dt$). As well, the only solutions of (21.8) to be considered from this point of view are those for which $\sum_n x_n^2 < +\infty$; also it is to this type of solution that Hilbert systematically limits himself; but over against that he relaxes the conditions imposed on the "infinite matrix" ($k_{pq}$) (which, in (21.8), is such that $\sum_{p,q} k_{pq}^2 < +\infty$). From that moment, it is clear that the "Hilbert space" of sequences $x = (x_n)$ of real numbers such that $\sum_n x_n^2 < +\infty$, although not explicitly introduced, underlies all the theory, and appears as a "passage to the limit" starting from Euclidean space of finite dimension. Over and above that, what is particularly important for subsequent developments, Hilbert is led to introduce in this space, not only one, but two distinct notions of convergence (corresponding to what has been called the weak topology and the strong topology$^3$), as well as a "principle of choice" which is none other than the property of weak compactness of the unit ball. The new linear algebra that he develops in connection with the solution of the systems (21.8) rests entirely on these topological notions: linear maps, linear forms and bilinear forms (associated with linear maps) are classified and studied according to their "continuity"$^4$ properties. In particular, Hilbert discovers that the success of Fredholm's method rests on the notion of "complete continuity", that he brings out by formulating it for bilinear forms$^5$ and studying it deeply; we can not give here more details about this important notion, nor on the admirable and deep work where Hilbert inaugurates the spectral theory of symmetric bilinear forms (bounded or not).

The language of Hilbert still remains classical, and, throughout the "Grundzüge", he never ceases to have in view the applications of the theory, of which he develops numerous examples (taking up about half the volume). The following generation will already adopt a more abstract point of view. Under the influence of ideas of Fréchet and of F. Riesz on general topology (see p. 142), E. Schmidt [274 b] and Fréchet himself introduce deliberately, in 1907-1908, the language of Euclidean geometry into "Hilbert space" (real or complex); it is in these works that is found the first mention of the norm (with the actual notation $||x||$), the triangle inequality that it satisfies, the fact that Hilbert space is "separable" and complete; moreover, E. Schmidt proves the existence of the orthogonal projection on a closed linear variety, which allows him to give a simpler and more general form to Hilbert's theory of linear systems. In 1907 also, Fréchet and F. Riesz remark that the space of

$^3$ The Calculus of variations had already led in a natural way to considering different notions of convergence on the same set of functions (depending on whether one insisted only on the uniform convergence of functions, or the uniform convergence of functions and a certain number of their derivatives); but the modes of convergence defined by Hilbert were of a type quite new at that time.

$^4$ It must be noted that, until about 1935, by "continuous" function is always meant in practice a map mapping every convergent series to a convergent series.

$^5$ For Hilbert, a bilinear form $B(x,y)$ is completely continuous if, when the sequences $(x_n), (y_n)$ tend weakly to $x$ and $y$ respectively, $B(x_n, y_n)$ tends to $B(x, y)$.

square summable series has a fully analogous "geometry"; an analogy that is explained perfectly when, some months later, F. Riesz and E. Fischer prove that this space is complete and isomorphic to "Hilbert space", at the same time bringing out in a startling way the value of the tool newly created by Lebesgue. From that moment the essential points of the theory of Hilbert spaces can be considered as acquired; among more recent progress, mention should most notably be made of the axiomatic presentation of the theory given around 1930 by M. H. Stone and J. von Neumann, as well as the casting off of the restrictions of "separability", which is effected around 1934, in the works of Rellich, Löwig, and F. Riesz [260 g].

However other streams of ideas were coming, in the first years of the XXth century, to reinforce the tendency that was leading to the theory of normed spaces. The general idea of a "functional" (that is to say a function with numerical values defined on a set whose elements are themselves numerical functions of one or several real variables) was brought out in the last decades of the XIXth century, linked to the calculus of variations, on the one hand, the theory of integral equations on the other. But if it is mainly to the Italian school, around Pincherle and especially Volterra, that is due the bringing to light of this notion, as well as the more general idea of "operator", the work of this school remained often of a passably formal nature and attached to problems of a particular type, lacking a sufficiently deep analysis of the underlying topological concepts. In 1903, Hadamard inaugurates the modern theory of "topological" duality, by seeking the most general continuous linear "functionals" on the space $C(I)$ of numerical functions on a compact interval $I$ (a space furnished with a topology of uniform convergence), and in characterising them as limits of sequences of integrals $x \mapsto \int_I k_n(t)x(t)dt$. In 1907, Fréchet and F. Riesz show in the same way that, on a Hilbert space, continuous linear forms are the "bounded" forms introduced by Hilbert; then, in 1909, F. Riesz puts into definitive form Hadamard's theorem, by expressing all continuous linear functionals on $C(I)$ by a Stieltjes integral, a theorem that would later become the point of departure for the modern theory of integration (see p. 226).

The following year, it is still F. Riesz [260 c] who makes new and important progress in the theory by introducing and studying (copied from the theory of Hilbert space) the spaces $L^p(I)$ of functions that are $p$-power summable in an interval $I$ (for an exponent $p$ such that $1 < p < +\infty$), a study that he follows three years later [260 e] with analogous work on the spaces of sequences $L^p(\mathbf{N})$; these researches, as we will see later, were greatly to contribute to clarifying the ideas on duality, because of the fact that here were met for the first time two spaces which were dual and not naturally isomorphic.$^6$

$^6$ Although the duality between $L^1$ and $L^\infty$ is implicit in most of the works on the Lebesgue integral at this time, it is only in 1918 that H. Steinhaus proved that every continuous linear form on $L^1(I)$ ($I$ a finite interval) is of the form $x \mapsto \int_I f(t)x(t)dt$, where $f \in L^\infty(I)$.

From that moment, F. Riesz was thinking about an axiomatic study encompassing all these results ([260 c], p. 452); and it seems that only the scruples of an analyst concerned not to distance himself too much from classical mathematics kept him from writing his famous memoir of 1918 on Fredholm theory in this form [260 f]. He considers there in principle the space $C(I)$ of functions continuous on a compact interval; but, after having defined the norm of this space, and having remarked that $C(I)$, together with this norm, is complete, he never uses again in his arguments any thing other than the axioms of normed complete spaces.$^7$ Without going into a detailed examination of this work here, let us mention that it is there that is to be found for the first time defined in a general way the notion of a completely continuous linear map (by the property of transforming a neighbourhood into a relatively compact set);$^8$ and, by a chef-d’oeuvre of axiomatic analysis, the whole of Fredholm’s theory (in its qualitative aspect) is reduced to a single fundamental theorem, namely that every normed locally compact space is finite dimensional.

The general definition of normed spaces was given in 1920-22 by S. Banach, H. Hahn and E. Helly (this latter only considering spaces of sequences of real or complex numbers). In the ten years that followed, the theory of spaces is developed mainly around two questions of fundamental importance for applications: the theory of duality and the theorems that are attached to the notion of Baire “category”.

We have seen that the idea of duality (in the topological sense) goes back to the beginning of the XXth century; it underlies Hilbert’s theory and occupies a central place in the work of F. Riesz. This latter observes, for example, already in 1911 ([260 d], pp. 41-42), that the relation $|f(x)| \leq M \|x\|$ (taken as the definition of “bounded” linear functionals in Hilbert space) is equivalent to the continuity of $f$ when one is in the space $C(I)$, and that by an argument of fully general character. In connection with the characterisation of continuous linear functionals on $C(I)$, he remarks also that the condition for a set $A$ to be dense in $C(I)$ is that there does not exist any Stieltjes measure $\mu \neq 0$ on $I$ that is “orthogonal” to every function of $A$ (generalising thus the Gram condition for complete orthonormal systems); he notes finally, in the same work, that the dual of the space $L^\infty$ is “bigger” than the space of Stieltjes measures ([260 d], p. 62).

$^7$ F. Riesz besides remarks explicitly that the application of his theorems to continuous functions is only there as a “touch stone” for much more general concepts ([260 f], p. 71).

$^8$ In his works on the $L^p$ spaces, F. Riesz had defined completely continuous maps as being those that transform every weakly convergent sequence into a strongly convergent one; which (taking into account the weak compactness of the unit ball in $L^p$ for $1 < p < +\infty$) is equivalent, in this case, to the preceding definition; furthermore, F. Riesz had indicated that, for the space $L^2$, his definition was equivalent to that of Hilbert (by translating it from the language of linear maps into that of bilinear forms ([260 c], p. 487)).

On the other hand, in his work on the spaces $L^p(I)$ and $L^p(\mathbf{N})$, F. Riesz succeeds in modifying the method for the solution of linear systems in Hilbert space, given by E. Schmidt [274 b], in such a way as to make it applicable to more general cases. The idea of E. Schmidt consisted in determining an "extremal" solution of (21.6) by seeking the point of the closed linear variety represented by the equations (21.6), whose distance to the origin is minimal. By using the same idea, F. Riesz shows that a necessary and sufficient condition that there exists a function $x \in L^p(a, b)$ satisfying the equations

(where the $\alpha_i$ belong to $L^q$ (with $\frac{1}{p} + \frac{1}{q} = 1$)), and such that as well $\int_a^b |x(t)|^p dt \leq M^p$, is that, for all finite sequences $(\lambda_i)_{1 \leq i \leq n}$ of real numbers, one has

In 1911 [260 d], he treats in an analogous way the "problem of generalised moments", consisting of solving the system

where the $\alpha_i$ are continuous and the unknown is a Stieltjes measure $\xi$;$^9$ and it is visible here that the problem can be stated by saying that it is a case of determining a continuous linear functional on $C(I)$ by its values on a sequence of given points in this space. It is also in this form that Helly treats the problem in 1912, — obtaining the conditions of F. Riesz by a fairly different method and of wider applicability$^{10}$ — and that he takes it up again in 1921, under much more general conditions. Introducing the notion of norm (on spaces of sequences) as we have seen above, he remarks that this notion generalises that of the "gauge" of a convex body in space of $n$ dimensions, used by Minkowski in his famous work on the "geometry of numbers" [221 a and b]. In the course of this work, Minkowski had also defined (in $\mathbf{R}^n$) the

$^9$ The classical "problem of moments" corresponds to the case when the interval ]$a, b$ is ]$0, +\infty$ or ]$-\infty, +\infty$, and where $\alpha_i(t) = t^i$; as well there is imposed on the measure $\xi$ the condition that it is positive (F. Riesz indicates in his memoir of 1911 how his general conditions must be modified when solutions of this kind are sought). Among the various methods of solution of the classical problem of moments we must notice particularly that of M. Riesz, which combines elegantly the general ideas of functional Calculus and the theory of functions of a complex variable to obtain explicit conditions on the $b_i$ [261]. $^{10}$ Like F. Riesz ([260 d], pp. 49-50), Helly uses in this proof a "choice principle" which is none other, of course, than the weak compactness of the unit ball in the space of Stieltjes measures; F. Riesz had also made use of the analogous property in the $L^p$ ($1 < p < +\infty$).

notions of support hyperplane and of "support function" [221 b], and proved the existence of a support hyperplane at all boundary points of a convex body ([221 a], pp. 33-35). Helly extends these notions to a space of sequences $E$, furnished with an arbitrary norm; he establishes a duality between $E$ and the space $E'$ of sequences $u = (u_n)$ such that, for all $x = (x_n) \in E$, the series $(u_n x_n)$ should be convergent; $\langle u, x \rangle$ designating the sum of this series, he defines in $E'$ a norm by the formula $\sup_{x \neq 0} |\langle u, x \rangle| / ||x||$, which gives the support function in finite dimensional spaces.$^{11}$

The solution of a system (21.6) in $E$, where each of the sequences $u_i = (a_{ij})_{j \geq 1}$ is assumed to belong to $E'$, reduces, as Helly shows then, to solving successively the following two problems: $1^\circ$ find a continuous linear form $L$ on the normed space $E'$, such that $L(u_i) = b_i$ for every index $i$, which, as he indicates, leads to conditions of type (21.10); $2^\circ$ seek if such a linear form can be written $u \mapsto \langle u, x \rangle$ for an $x \in E$. This last problem, as Helly observes, does not necessarily have a solution even when $L$ exists, and he limits himself to giving some sufficient conditions that imply the existence of the solution $x \in E$ in certain particular cases [156].

These ideas acquire their definitive form in 1927, in a fundamental memoir of H. Hahn [142] whose results are found again (independently) by S. Banach two years later [14 c]. The procedure of Minkowski-Helly is applied by Hahn to an arbitrary normed space, and therefore gives to the dual the structure of a (complete) normed space; which allows Hahn immediately to consider successive duals of a normed space, and to state in a general way the problem of reflexive spaces, foreseen by Helly. But above all the crowning problem of the extension of a continuous linear functional while preserving its norm is definitively solved by Hahn in a completely general way, by a transfinite induction argument on the dimension — thus giving one of the first examples of an important application of the axiom of choice to Functional Analysis.$^{12}$ To these results, Banach adds a thorough study of the relations between a continuous linear map and its transpose, extending to general normed spaces results known until then only for the spaces $L^p$ [260 c], by means of a deep theorem on the weakly closed subsets of a dual; these results are expressed besides in a more striking way by using the notion of quotient space of a normed space, introduced a few years later by Hausdorff and Banach himself. Finally, it is again Banach who discovers the link between the weak compactness of the unit ball (observed in many particular cases, as we have pointed out above) and reflexivity, at least for spaces of countable type ([14 a], p. 189). The theory of the duality of normed spaces can, from this moment, be considered as settled in its major aspects.

$^{11}$ To obtain a norm in this way, it must be assumed that the relation $\langle u, x \rangle = 0$ for all $x \in E$ implies $u = 0$, as besides Helly remarks explicitly.

$^{12}$ Banach had already used an analogous argument in 1923 [14 b], in order to define an invariant measure on the plane (defined for all bounded subsets).

At the same time some theorems of paradoxical aspect were clarified, the first examples of which go back to the neighbourhood of 1910. Hellinger and Toeplitz had in effect proved in substance that year, that a sequence of bounded bilinear forms $B_n(x, y)$ on a Hilbert space, whose values $B_n(a, b)$ for every given pair $(a, b)$ are bounded (by a number dependant a priori on $a$ and $b$), is in fact uniformly bounded on every ball. Their proof proceeds by contradiction and consists in constructing a particular pair $(a, b)$ violating the hypothesis, by an induction method known since by the name of "method of the sliding hump", and which is still in use in many analogous questions. Already in 1905, Lebesgue had besides used an analogous procedure to prove the existence of continuous functions whose Fourier series diverges at certain points; and, the same year as Hellinger and Toeplitz, he applied the same method to prove that a weakly convergent series in $L^1$ is bounded in the norm.$^{13}$ These examples are multiplied in the years that follow, but without the introduction of any new ideas until 1927, the date when Banach and Steinhaus (with the partial collaboration of S. Saks) link these phenomena to the notion of thin sets and to the theorem of Baire on complete metric spaces, obtaining a general statement that covers all the previous particular cases [15]. The study of questions of "category" in normed complete spaces besides leads Banach, at the same time, to numerous other results on continuous linear maps; the most remarkable and no doubt the deepest is the theorem of the "closed graph" which, like the Banach-Steinhaus theorem, showed itself to be a tool of the first order in modern Functional Analysis [14 c].

The publication of the treatise of Banach on "Linear Operators" [14 a] marks, one could say, the beginning of the adult age for the theory of normed spaces. All the results of which we have been speaking, as well as many others, are set out in this volume, in a way that is still a bit disorganised, but accompanied with many striking examples taken from varied domains of Analysis, and which seemed to forecast a brilliant future for the theory. As it happened, the work had considerable success, and one of its most immediate effects was the quasi-universal adoption of the language and notations used by Banach. But, in spite of a large amount of research undertaken for 40 years on Banach spaces ([203 bis]), if one makes an exception of the theory of Banach algebras and its applications to commutative and non-commutative harmonic analysis, the almost total absence of new applications of the theory to the big problems of classical Analysis has somewhat disappointed the hopes based upon it.

It is rather in the sense of an expanding and a more thrusting axiomatic analysis of the concepts relative to normed spaces that the most fruitful developments have occurred. Although the functional spaces met since the begin-

$^{13}$ Let us also note the analogous (easier) theorem proved by Landau in 1907 and which served as the point of departure for F. Riesz in his theory of $L^p$ spaces: if the series with general term $u_n x_n$ converges for all sequences $(x_n) \in L^q(\mathbf{N})$, the sequence $(u_n)$ belongs to $L^p(\mathbf{N})$ (with $\frac{1}{p} + \frac{1}{q} = 1$).

ning of the XXth century arose for the most part furnished with a "natural" norm, it had not happened without some exceptions being noticed. Around 1910, E. H. Moore had proposed generalising the notion of uniform convergence by replacing it with the notion of "relative uniform convergence", where a neighbourhood of 0 is made up of the functions $f$ satisfying a relation $|f(t)| \leq \epsilon g(t)$, $g$ being a function everywhere $> 0$, which could vary with the neighbourhood. It had on the other hand been observed, before 1930, that notions such as simple convergence, convergence in measure for measurable functions, or compact convergence for entire functions, are not capable of being defined by means of a norm; and in 1926, Fréchet had noted that vector spaces of this nature can be metrisable and complete. But the theory of these more general spaces was only to develop in a fruitful way in combination with the idea of convexity. This latter (that we saw appearing with Helly) was an object of study for Banach and his pupils, who recognised the possibility of interpreting thus in a more geometric way numerous statements of the theory of normed spaces, preparing the way for the general definition of locally convex spaces, given by J. von Neumann in 1935. The theory of these spaces, and notably the questions dealing with duality, have been especially developed during the last ten years. What must be noted in this context, is on the one hand, the progress in simplicity and in generality made possible by the setting up of the fundamental notions of general Topology, carried out between 1930 and 1940; on the other hand, the importance taken by the notion of bounded set, introduced by Kolmogoroff and von Neumann in 1935, and whose fundamental role in duality theory has been brought to light by the work of Mackey [213 a and b] and of Grothendieck [138 b and c]. Finally and especially, it is certain that the main impulsion that motivated this research came from new possibilities for applications to Analysis, in domains where Banach theory was inoperative: the theory of sequence spaces must be mentioned in this context, developed by Köthe, Toeplitz and their pupils since 1934 in a series of memoirs [184], the recent setting up of the theory of "analytic functionals" of Fantappié, and above all the theory of distributions of L. Schwartz [280], where the modern theory of locally convex spaces found a field of applications that is without doubt a long way from being exhausted.