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On linear functional equations. (Über lineare Funktionalgleichungen.) (German) JFM 46.0635.01
There are two reviews of this item, the original review from 1916 by Ernst David Hellinger and a “looking back” review from 2018 by Albrecht Pietsch.
A Russian translation was published in Usp. Mat. Nauk, 1936, No. 1, 175–199 (1936; mathnetru:eng/umn/y1936/i1/p175).
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First review by E. D. Hellinger (Frankfurt a. M.) (1916):
Die Arbeit behandelt eine umfassende Klasse der in ihren Eigenschaften den linearen Integralgleichungen 2. Art mit stetigem Kern analogen linearen Funktionalgleichungen $B[\varphi]=\varphi-A[\varphi]=f \eqno(1)$ nach einer Methode, die lediglich die allgemeinen charakteristischen Eigenschaften der linearen Funktionaltransformationen benutzt und von der Art der Objekte dieser Transformationen (der Elemente des zugrunde gelegten unendlichdimensionalen Raumes) im wesentlichen unabhängig ist; die Entwicklungen werden durchgeführt für den Raum der stetigen Funktionen $$f(x)$$ von $$x$$ im Intervalle $$a\leqq x\leqq b$$. Dabei wird als Norm $$\| f(x)\|$$ verwendet das Maximum von $$| f(x)|$$, als Distanz zweier Funktionen die Norm ihrer Differenz und demgemäß als Konvergenzbegriff für eine Funktionenfolge der der gleichmäßigen Konvergenz; als kompakte Funktionenfolge im Sinne von Fréchet hat daher eine solche zu gelten, bei der jede Teilfolge eine gleichmäßig konvergente Teilfolge enthält. Das Haupthilfsmittel bilden einige Untersuchungen über lineare Mannigfaltigkeiten von Funktionen (die im Sinne gleichmäßiger Konvergenz abgeschlossen sind), die sich auf Distanzabschätzungen von verschiedenen linearen Mannigfaltigkeiten angehörenden Funktionen beziehen, und die die bei dem vorliegenden Normbegriff nicht anwendbaren Verfahren der orthogonalen Projektion (Besselsche Ungleichung oder dgl.) ersetzen; wesentlich ist ferner die Bemerkung, daß nur in einer Mannigfaltigkeit endlicher Dimensionszahl jede beschränkte Funktionenfolge (mit beschränkter Norm) kompakt ist.
Eine Funktionaltransformation $$f_1=A[f]$$ heißt linear, wenn sie distributiv und beschränkt (d.h. $$\| A[f]\| \leqq M\| f\|$$ mit festem $$M$$ für alle $$f$$) ist; sie heißt vollstetig, wenn sie jede beschränkte Funktionenfolge in eine kompakte überführt. Für solche vollstetigen Transformationen $$A$$ wird die Funktionalgleichung (1) untersucht. Die letzte Bemerkung des vorigen Absatzes führt zu der Erkenntnis, daß die Lösungen der homogenen Gleichung $$B[\varphi]=0$$ eine lineare Mannigfaltigkeit endlicher Dimensionszahl bilden; das gleiche gilt für die iterierten Gleichungen $$B^n[\varphi]=0$$, und da sich zeigt, daß von einem hinreichend großen $$n=\nu$$ an hier keine neuen Funktionen mehr hinzutreten können, ergibt sich so die endlichdimensionale Mannigfaltigkeit der Hauptfunktionen – wie sie in der Theorie der Integralgleichungen heißen –, die Verf. als “Nullelemente” bezeichnet. Analog läuft die Untersuchung der durch die Transformationen $$g=B^n[\varphi]$$ aus dem Raum aller $$\varphi$$ entstehenden linearen Mannigfaltigkeiten, die für $$n\geqq\nu$$ miteinander identisch sind (“Kernelemente”) und für die Lösung der inhomogenen Gleichung charakteristisch werden. Durch die Unterscheidung $$\nu\geqq 0$$ ergibt sich dann die Fredholmsche Alternative; allgemein läßt sich jede Funktion eindeutig als Summe eines Kern- und eines Nullelementes und entsprechend $$A$$ eindeutig als die Summe zweier Transformationen darstellen, von denen die eine $$A_1$$ alle Nullelemente, die andere $$A_2$$ alle Kernelemente in Null überführt. Dann ist $$E-A_1$$ eindeutig umkehrbar, während die zu $$E-A_2$$ gehörigen Gleichungen dieselben Lösungsverhältnisse aufweisen, wie (1). Da $$A_2$$ eine Transformation der endlichdimensionalen Mannigfaltigkeit der Nullelemente in sich darstellt, so ist damit die Untersuchung von (1) auf ein algebraisches Problem in endlichvielen Dimensionen zurückgeführt.
Eine Transformation “vom Integraltypus” $$K[f]=\int_a^b k(x,y)f(y)dy$$ mit stetigem $$k(x,y)$$ erweist sich leicht als vollstetige Transformation; die entwickelten Sätze ergeben dann ohne Schwierigkeit die gesamte Auflösungstheorie der Integralgleichung 2. Art sowie die Abspaltung des Hauptbestandteiles des Kernes und seinen Ausdruck durch die Hauptfunktionen.
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Looking-back review by Albrecht Pietsch (Jena) (2018):
The centennial anniversary of the appearance of the Riesz spectral theory of compact operators presents a welcome occasion for writing a detailed appreciation. The paper under review certainly ranks among the most important publications of classical Banach space theory, next to S. Banach’s thesis [“Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales”, Fundam. Math. 3, 133–181 (1922; JFM 48.0201.01)] and its outgrowth, the monograph [Théorie des opérations linéaires. Warszawa: PAN (1932; Zbl 0005.20901)], the latter having received a “Looking Back” review in 2017. The presentation by both these authors is so cogent that most of their proofs can be used in today’s lectures.
While Riesz’s paper was concluded in Győr (Hungary) on January 19, 1916 and is listed as “imprimé le [3–5] décembre 1916” in the journal, in most bibliographies it is dated to 1918 when, delayed by the circumstances of World War I, Volume 41 of Acta Mathematica was eventually published. As was his custom, the author separately published a Hungarian version, dated February 14, 1916 and entitled “Lineáris függvényegyenletekről”, which can be found in Riesz’s Collected Works [Á. Császár (ed.), Gesammelte Arbeiten. Band I, II. Budapest: Verlag der Ungarischen Akademie der Wissenschaften. 1017–1052 (1960; Zbl 0101.00201)].
We stress the fact that F. Riesz wrote his contribution at a time when the concept of an abstract Banach space did not exist. Indeed, claiming
Der in den neueren Untersuchungen über diverse Funktionalräume bewanderte Leser wird die allgemeinere Verwendbarkeit der Methode sofort erkennen,
he exclusively used the space $$C[a,b]$$ of continuous functions on an interval $$[a,b]$$, equipped with the sup-norm. In other words, he stated that almost all of his results remain true in abstract Banach spaces.
Around the year 1905, D. Hilbert and his pupil E. Schmidt had developed a determinant-free approach to Fredholm’s theory of integral equations of the second kind, which is based on $$\ell_2$$ and $$L_2[a,b]$$ (in a hidden form); see [D. Hilbert and E. Schmidt, Integralgleichungen und Gleichungen mit unendlich vielen Unbekannten. Wien etc.: Springer-Verlag; Leipzig: BSB B.G. Teubner Verlagsgesellschaft (1989; Zbl 0689.01012)]. Subsequently, F. Riesz introduced the spaces $$\ell_p$$ and $$L_p[a,b]$$ with $$1 \leq p < \infty$$; see [F. Riesz, Les systèmes d’équations linéaires à une infinité d’inconnues. Paris: Gauthier-Villars (1913; JFM 44.0401.01)] and [F. Riesz, “Untersuchungen über Systeme integrierbarer Funktionen”, Math. Ann. 69, 449–497 (1910; JFM 41.0383.01)]. Since orthogonality was no longer available for $$p \not= 2$$, he tried to overcome this trouble in the most simple case, namely, $$C[a,b]$$.
To understand the situation in which F. Riesz started his investigations and to see their most important applications, the reader may consult the beautiful survey [E. Hellinger and O. Toeplitz, Integralgleichungen und Gleichungen mit unendlichvielen Unbekannten. Leipzig: B. G. Teubner (1927; JFM 53.0350.01)].
D. Hilbert considered completely continuous bilinear forms on $$\ell_2 \times \ell_2$$, and F. Riesz observed that the corresponding operators are characterized by the property that weakly convergent sequences are mapped to norm convergent sequences. On the other hand, in the paper under review, he refers to an operator $$T$$ on $$C[a,b]$$ as completely continuous if every bounded sequence $$(f_n)$$ contains a subsequence whose image $$(T f_{n_i})$$ is norm convergent. For the moment, we will distinguish these concepts by saying that an operator is completely continuous ‘in the sense of Hilbert’ or ‘in the sense of Riesz’. Certainly, F. Riesz knew that both kinds of complete continuity coincide for operators on $$\ell_2$$. For general spaces, completely continuous operators in the sense of Riesz are also completely continuous in the sense of Hilbert, whereas the converse implication fails. However, the following counterexamples were available only later. In §4 of [J. Schur, “Über lineare Transformationen in der Theorie der unendlichen Reihen”, J. Reine Angew. Math. 151, 79–111 (1920; JFM 47.0197.01)], Issai (also called Schaia, which seems to justify the inital J.) Schur showed that the identity map of $$\ell_1$$ is completely continuous in the sense of Hilbert and, of course, it fails to be completely continuous in the sense of Riesz. Moreover, let $$C(\mathbb T)$$ be the space of all continuous $$2\pi$$-periodic functions. Then the rule
$P : f(t) = \sum_{n=-\infty}^\infty \gamma_n {\mathrm e}^{{\mathrm i} n t } \;\mapsto\; \big(\gamma_{2^k}\big)_{k=0}^\infty$
defines a $$2$$-summing operator from $$C(\mathbb T)$$ onto $$\ell_2$$, which is completely continuous in the sense of Hilbert but not in the sense of Riesz; see Section III.F of [P. Wojtaszczyk, Banach spaces for analysts. Cambridge: University Press (1991; Zbl 0724.46012)].
On page 49 of [E. Hille, Functional analysis and semi-groups. AMS Colloquium Publ. 31, New York (1948; Zbl 0033.06501)], the term ‘compact’ was used instead of completely continuous in the sense of Riesz. Luckily, this proposal has prevailed and our temporary suffix ‘in the sense of Hilbert’ becomes unnecessary. From now on, we will use the attributes ‘compact’ and ‘completely continuous’ in this way, which has become standard.
Let $$\mathfrak L (X,Y)$$ denote the Banach space of all (bounded, linear) operators from the Banach space $$X$$ into the Banach space $$Y$$. If $$X=Y$$, then we simply write $$\mathfrak L (X)$$ instead of $$\mathfrak L (X,X)$$. The identity map of $$X$$ is denoted by $$I$$ or, more precisely, by $$I_X$$. Every $$A \in \mathfrak L (X,Y)$$ has the range $$\mathcal M (A) := \{ Ax : x \in X \}$$ and the null space $$\mathcal N (A) := \{ x \in X : Ax = \mathsf{o}\}$$.
The main results of F. Riesz say that the following properties hold for any compact operator $$T \in \mathfrak L (X)$$:
If $$\mathcal M \big( (I-T)^m \big) =\mathcal M \big( (I-T)^{m+1} \big)$$, then $$\mathcal M \big( (I-T)^{m+1} \big) =\mathcal M \big( (I-T)^{m+2} \big)$$.
The ranges $$\mathcal M \big( (I-T)^m \big)$$ are closed and form a non-increasing sequence, which stabilizes for some index $$m_0$$.
If $$\mathcal N \big( (I-T)^n \big) =\mathcal N \big( (I-T)^{n+1} \big)$$, then $$\mathcal N \big( (I-T)^{n+1} \big) =\mathcal N \big( (I-T)^{n+2} \big)$$.
The null spaces $$\mathcal N \big( (I-T)^n \big)$$ are finite-dimensional and form a non-decreasing sequence, which stabilizes for some index $$n_0$$.
The indices {$$m_0$$ and $$n_0$$ coincide when they are chosen as small as possible; their joint value is denoted by $$p$$. Then $$X$$ is the direct sum of the $$T$$-invariant subspaces $$\mathcal M \big( (I-T)^p \big)$$ and $$\mathcal N \big( (I-T)^p \big)$$.
In the regular case $$p=0$$, the operator $$I-T$$ is an isomorphism. In other words, the equation $$x-T x =a$$ admits a unique solution $$x \in X$$ for every $$a \in X$$.
In the singular case $$p>0$$, the restriction of $$I-T$$ to $$\mathcal M \big( (I-T)^p \big)$$ is an isomorphism and the restriction to the finite-dimensional space $$\mathcal N \big( (I-T)^p \big)$$ is nilpotent. This means that solving the equation $$x-T x =a$$ is reduced to a problem of classical linear algebra, at least in principle.
The general concept of a dual (adjoint, conjugate) operator, which is based on the Hahn-Banach extension theorem, was introduced only at the end of the 1920’s; see [S. Banach, “Sur les fonctionnelles linéaires. II”, Stud. Math. 1, 223–239 (1929; JFM 55.0240.01)]. Therefore F. Riesz had to restrict his considerations to the very special case of transposed integral equations, which are generated by continuous kernels $$K(x,y)$$ and $$K(y,x)$$. The missing keystone was laid by J. Schauder [“Über lineare, vollstetige Funktionaloperationen”, Stud. Math. 2, 183–196 (1930; JFM 56.0354.01)]. His main result says that the dual operator $$T^\ast \!:\! Y^\ast \!\to\! X^\ast$$ is compact if and only if so is the original operator $$T \!:\! X \!\to\! Y$$. Moreover, $$\dim\big( \mathcal N (I^\ast - T^\ast) \big) = \dim\big( \mathcal N (I - T) \big)$$. In view of this important contribution, the joint outcome is commonly called the Riesz-Schauder theory.
An intermediate result is due to T.H. Hildebrandt [“Über vollstetige lineare Transformationen”, Acta Math. 51, 311–318 (1928; JFM 54.0427.03)], who used – in a hidden form – the codimension of $$\mathcal M (I-T)$$. Indeed, since $$\mathcal N(I^\ast-T^\ast)$$ and $$\big[X/\mathcal M(I-T)\big]^\ast$$ are isometric, we get $$\dim\big[\mathcal N(I^\ast-T^\ast)\big]= \mathrm{cod} \big[\mathcal M(I-T)\big]$$. Therefore $$\dim\big[\mathcal N(I^\ast-T^\ast)\big]= \dim\big[\mathcal N(I-T)\big]$$ is equivalent to the formula $$\mathrm{cod} \big[\mathcal M(I-T)\big] = \dim\big[\mathcal N(I-T)\big]$$, which does not require the knowledge of any dual operator.
Let $$A \in \mathfrak L (X,Y)$$. Following F. Hausdorff [“Zur Theorie der linearen metrischen Räume”, J. Reine Angew. Math. 167, 294–311 (1932; JFM 58.1113.05; Zbl 0003.33104)], the equation $$Ax = b$$ is said to be normally solvable provided that, for given $$b \in Y$$, there exists a solution $$x \in X$$ if and only if $$\langle b , y^\ast \rangle =0$$ whenever $$A^\ast y^\ast = \mathsf{o}$$. Remarkably, this happens just in the case when the range $$\mathcal M (A)$$ is closed. Hence all operators $$I-T$$ with compact $$T \in \mathfrak L (X)$$ are normally solvable.
The preceding results remain true when $$I-T$$ is replaced by $$I - \zeta T$$ with any complex parameter $$\zeta$$. If $$I - \zeta T$$ is singular, then we call $$\zeta$$ a characteristic value. F. Riesz proved that the characteristic values have no finite accumulation point. Note that he referred to those numbers as eigenvalues. This term is now commonly used for $$\lambda \in \mathbb C$$ when working with the scale $$\lambda I - T$$.
Riesz’s paper has stimulated many remarkable developments. Some of them will be sketched in the remainder of this review. For more detailed information the reader is referred to [A. Pietsch, History of Banach spaces and linear operators. Boston: Birkhäuser (2007; Zbl 1121.46002), in particular Sect. 2.6, Subsect. 5.2.2, 5.2.3, and 8.3.1 (short biography)].
Already F. Riesz observed that the class all compact operators is an ideal, now denoted by $$\mathfrak K$$. Further related ideals are $$\mathfrak F$$, the class of finite rank operators, and $$\mathfrak V$$, the class of completely continuous operators. Note that $$\mathfrak F \subset \mathfrak K \subset \mathfrak V$$. From Chap. VI, Théorème 2, of Banach’s monograph, we know that $$\mathfrak K$$ is closed in the norm topology of $$\mathfrak L$$. Therefore the closure $$\overline{\mathfrak F}$$, whose members are the approximable operators, is contained in $$\mathfrak K$$. A long standing open problem asked whether even equality holds. The famous negative answer was finally given by P. Enflo [“A counterexample to the approximation problem in Banach spaces”, Acta Math. 130, 309–317, (1973; Zbl 0067.46012)] when he constructed a Banach space without the approximation property.
According to the Russian terminology, one refers to $$A \in \mathfrak L(X,Y)$$ as a $$\Phi$$-operator ($$\Phi$$ stands for Fredholm) if there exist operators $$U, V \in \mathfrak L(Y,X)$$, $$S \in \mathfrak K (X)$$, and $$T \in \mathfrak K (Y)$$ such that $$UA = I_X -S$$ and $$AV = I_Y -T$$; see [I. Ts. Gohberg and M. G. Kreĭn, “The basic propositions on defect numbers, root numbers and indices of linear operators” (Russian), Usp. Mat. Nauk 12, 43–118 (1957; Zbl 0088.32101); Am. Math. Soc., Transl., II, Ser. 13, 185–264 (1960; Zbl 0089.32201)]. The preceding definition means that $$A$$ is invertible modulo the ideal $$\mathfrak K$$, which can even be replaced by $$\mathfrak F$$. We know from [F. V. Atkinson, “Die normale Auflösbarkeit linearer Gleichungen in normierten Räumen” (Russian), Mat. Sb., N. Ser. 28, 3–14 (1951; Zbl 0042.12001)] that $$\Phi$$-operators are characterized by the property of having finite-dimensional null spaces and finite-codimensional closed ranges. By the way, Lemma 332 of [T. Kato, “Perturbation theory for nullity, deficiency and other quantities of linear operators”, J. Anal. Math. 6, 261–322 (1958; Zbl 0090.09003)] says that $$\mathrm{cod} \big[\mathcal M(A)\big] < \infty$$ automatically implies that $$\mathcal M(A)$$ is closed.
Riesz has shown that, for compact $$T$$, all operators $$I- \zeta T$$ with $$\zeta \in \mathbb C$$ have very nice properties. Therefore the question arose whether his results hold for more general operators. In a first step, [S. M. Nikolskij, “Lineare Gleichungen in metrischen Räumen” (Russian), Doklady Akad. Nauk SSSR 2, 315–319 (1936; JFM 62.0452.03)] confirmed this expectation for operators that admit a compact power. To formulate a complete answer, we refer to $$T \in \mathfrak L(X)$$ as a Riesz operator if every $$I - \zeta T$$ with $$\zeta \in \mathbb C$$ behaves in the desired way. To treat real operators, one must pass to their complexifications. Riesz operators can be characterized by various conditions of quite different flavour.
(1) Every $$I- \zeta T$$ with $$\zeta \in \mathbb C$$ is a $$\Phi$$-operator.
(2) $$T$$ is quasi-nilpotent modulo the ideal $$\mathfrak K (X)$$, which means that
$\lim_{n\to\infty} |\mkern-2mu|\mkern-2mu| T^n |\mkern-2mu|\mkern-2mu|^{1/n} = 0, \quad \text{where} \quad |\mkern-2mu|\mkern-2mu| T^n|\mkern-2mu|\mkern-2mu| = \inf \big\{ \|T^n-K\| : K \in \mathfrak K (X) \big\};$
see [A. F. Ruston, “Operators with a Fredholm theory”, J. Lond. Math. Soc. 29, 318–326 (1954; Zbl 0055.10902)].
(3) The resolvent $$(I- \zeta T)^{-1}$$ is a meromorphic $$\mathfrak L(X)$$-valued function on the complex plane such that the singular part of the Laurent expansion at every pole has finite rank coefficients; see Schauder’s paper quoted previously, [N. Dunford, “Spectral theory. I: Convergence to projections”, Trans. Am. Math. Soc. 54, 185–217 (1951; Zbl 0063.01185), p. 198], and [A. E. Taylor, “Analysis in complex Banach spaces”, Bull. Am. Math. Soc. 49, 652–669 (1943; Zbl 0063.07311), p. 660]. Note that the characteristic values coincide with the poles, whose order is just the index $$p$$ at which the sequences $$\big\{\mathcal M \big( (I-\zeta T)^m \big) \big\}$$ and $$\big\{\mathcal N \big( (I-\zeta T)^n \big) \big\}$$ stabilize.
(4) For every $$\varepsilon >0$$ there exists an exponent $$n$$ such that $$T^n (B_X)$$ can be covered by a finite number of balls $$y + \varepsilon ^n B_X$$. Here $$B_X$$ denotes the closed unit ball of $$X$$. A presentation of the Riesz theory based on a slightly modified geometric property is given in [A. Pietsch, Eigenvalues and $$s$$-numbers. Cambridge: University Press (1987; Zbl 0615.47019), Section 3.2 and §7.4.1].
T. T. West [“Riesz operators in Banach spaces”, Proc. Lond. Math. Soc., III. Ser. 16, 131–140 (1966; Zbl 0139.08401)] observed that the set of all Riesz operators on some Banach space may fail to be closed under addition, multiplication, and passing to the limit with respect to the operator norm. So it makes sense to look for closed ideals $$\mathfrak A$$ such that all components $$\mathfrak A (X)$$ consist of Riesz operators. The most obvious example is $$\mathfrak K$$.
A much larger example, introduced by T. Kato on pp. 284–288 of his paper cited above, is $$\mathfrak S_{in}$$ consisting of the strictly singular (semicompact) operators $$T \in \mathfrak L (X,Y)$$ defined by the following property: If there exists a constant $$c > 0$$ such that $$\|T x\| \geq c \|x\|$$ for all $$x$$ in a closed subspace $$M$$, then $$M$$ is finite-dimensional. Dualisation yields the closed ideal of stricly cosingular (co-semicompact) operators. Both ideals are extensively treated in [D. Przeworska-Rolewicz and S. Rolewicz, Equations in linear spaces. Warszawa: PAN (1968; Zbl 0181.40501), pp. 252–263 and pp. 315–317]. An increasing scale of closed ideals lying between $$\mathfrak K$$ and $$\mathfrak S_{in}$$ was constructed by [A. Pietsch, “A $$1$$-parameter scale of closed ideals formed by strictly singular operators”, in: Toeplitz matrices and singular integral equations. The Bernd Silbermann anniversary volume. Basel: Birkhäuser. 261–265 (2002; Zbl 1036.47010)].
[A. Pietsch, Theorie der Operatorenideale (Zusammenfassung). Jena: Eigenverlag der Friedrich-Schiller-Universität (1972; Zbl 0238.46067), §5.3.1] introduced the largest ideal of this kind, now denoted by $$\mathfrak R_{ad}$$. Its components $$\mathfrak R_{ad} (X,Y)$$ are formed by all operators $$T \in \mathfrak L (X,Y)$$ such that $$I_X + AT$$, or equivalently $$I_Y + TA$$, is a $$\Phi$$-operator for every $$A \in \mathfrak A (Y,X)$$. Since this definition is based on earlier results of [I. Ts. Gokhberg, A. S. Markus and I. A. Feldman, “Normally solvable operators and ideals associated with them”, Am. Math. Soc., Transl., II, Ser. 61, 63–84 (1967; Zbl 0181.40601); translation from Izv. Mold. Fil. Akad. Nauk SSSR, 10 (76), 51–69 (1960)] and [D. Kleinecke, “Almost-finite, compact, and inessential operators”, Proc. Am. Math. Soc. 14, 863–868 (1963; Zbl 0117.34201)], the members of $$\mathfrak R_{ad}$$ were called Gokhberg operators or inessential (which does not mean that Gokhberg $$=$$ inessential}).

MSC:
 47B06 Riesz operators; eigenvalue distributions; approximation numbers, $$s$$-numbers, Kolmogorov numbers, entropy numbers, etc. of operators 47B07 Linear operators defined by compactness properties
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References:
 [1] C. Arzelà, “Sulle funzioni di linee{”, Memorie d. R. Accad. d. Scienze di Bologna, serie 5, t. V (1895), S. 225–244.} [2] DaL k von endlicher Dimensionszahl ist, liesse sich 1/2 durch 1 ersetzen; doch kommt es uns hier auf diese tiefer liegende Tatsache nicht an; der entsprechende Hilfssatz 3. gelangt erst später zur Verwendung. [3] Die Grenzgleichung ||f (n)||f *|| für jede gleichmässig konvergente Folgef (n) (*). erhält man am einfachsten aus den beiden Ungleichungen ||f *||f *-f (n) ||+||f (n)||,||f (n)||f *-f (n)|| +||f *|| und aus der Grenzgleichung, ||f *-f (n)||. [4] Auch hier lässt sich die Zahl 1/2 durch 1 ersetzen, da ja alle in Betracht kommenden Mannigfaltigkeiten von endlicher Dimensionszahl sind. [5] Die BezeichnungB (0) soll daran erinnern, dass für {$$\nu$$}=0 die TransformationB (0) mit der identischenE=B 0 zusammenfällt. [6] W. A. Hurwitz,On the pseudo-resolvent to the kernel of an integral equation, Transactions of the American Math. Soc., Vol. 13 (1912), S. 405–418. · JFM 43.0425.04 [7] F. Riesz,Sur les opérations fonctionnelles linéaires, Comptes rendus de l’Acad. d. Sc., Paris, 29 novembre 1909.
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