The Kolmogorov-Riesz compactness theorem. (English) Zbl 1208.46027
Expo. Math. 28, No. 4, 385-394 (2010); addendum ibid. 34, No. 2, 243-245 (2016).
Summary: We show that the Arzelà-Ascoli theorem and the Kolmogorov compactness theorem both are consequences of a simple lemma on compactness in metric spaces. Their relation to Helly’s theorem is discussed. The paper contains a detailed discussion on the historical background of the Kolmogorov compactness theorem.
MSC:
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 46B50 | Compactness in Banach (or normed) spaces |
| 46-03 | History of functional analysis |
| 01A60 | History of mathematics in the 20th century |
References:
| [1] | Adams, A.; Fournier, J. F., Sobolev Spaces (2003), Academic Press: Academic Press Amsterdam · Zbl 1098.46001 |
| [2] | Arzelà, C., Sulle funzione de linee, Mem. Accad. Sci. Bologna, 5-5, 225-244 (1894/95), (in Italian) |
| [3] | Ascoli, G., Le curve limiti di una varietá data di curve, Rend. Accad. Lincei, 18, 521-586 (1884), (in Italian) · JFM 16.0342.02 |
| [4] | DiBenedetto, E., Real Analysis (2002), Birkhäuser: Birkhäuser Boston · Zbl 1012.26001 |
| [5] | Bruno, G.; Grande, R., A compactness criterion in \(B^q_{ ap }\) spaces, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 20, 95-121 (1996) · Zbl 0942.42004 |
| [6] | Bruno, G.; Grande, R., Compact embedding theorems for Sobolev-Besicovitch spaces of almost periodic functions, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 20, 157-173 (1996) · Zbl 0942.42003 |
| [7] | Dörfler, M.; Feichtinger, H. G.; Gröchenig, K., Compactness criteria in function spaces, Coll. Math., 94, 37-50 (2002) · Zbl 1017.46014 |
| [8] | Dunford, N.; Schwartz, J. T., Linear Operators. Part I: General Theory (1988), Wiley: Wiley New York |
| [9] | Edwards, R. E., Functional Analysis (1965), Holt, Rinehart and Winston: Holt, Rinehart and Winston Chicago · Zbl 0182.16101 |
| [10] | Evans, L. C.; Gariepy, R. F., Measure Theory and Fine Properties of Functions (1992), CRC Press: CRC Press Boca Raton · Zbl 0804.28001 |
| [11] | Feichtinger, H. G., Compactness in translation invariant Banach spaces of distributions and compact multipliers, J. Math. Anal. Appl., 102, 289-327 (1984) · Zbl 0515.46044 |
| [12] | Fréchet, M., Essai de geometrie analytique, Nouv. Ann. Math., 4, 97-116 (1908), 289-317 (in French) · JFM 39.0536.02 |
| [13] | Fréchet, M., Sur les ensembles compacts de fonctions de carrés sommables, Acta Litt. Sci. Szeged., 8, 116-126 (1937), (in French) · JFM 63.0200.02 |
| [14] | Glimm, J., Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math., 18, 697-715 (1965) · Zbl 0141.28902 |
| [15] | Hanson, E. H., A note on compactness, Bull. Am. Math. Soc., 39, 397-400 (1933) · JFM 59.0408.03 |
| [16] | Helly, E., Über lineare Funktionaloperationen, Österreich. Akad. Wiss. Math.-Natur. Kl. S.-B. IIa, 121, 265-297 (1912), (in German) · JFM 43.0418.02 |
| [17] | Izumi, S., On the compactness of a class of functions, Proc. Imp. Acad. Tokyo, 15, 111-113 (1939) · JFM 65.1192.04 |
| [18] | Kolmogorov, A. N., Über Kompaktheit der Funktionenmengen bei der Konvergenz im Mittel, Nachr. Ges. Wiss. Göttingen, 9, 60-63 (1931), (English translation: On the compactness of sets of functions in the case of convergence in the mean, in: V.M. Tikhomirov (Ed.), Selected Works of A.N. Kolmogorov, vol. I, Kluwer, Dordrecht, 1991, pp. 147-150). · JFM 57.0271.03 |
| [19] | Kondrachov, W., Sur certaines propriétés des fonctions dans l’espace, C. R. Acad. Sci. URSS., n. Sér, 48, 535-538 (1945) · Zbl 0061.26202 |
| [20] | Maitre, E., On a nonlinear compactness lemma in \(L^p(0,T;B)\), Int. J. Math. Math. Sci., 27, 1725-1730 (2003) · Zbl 1032.46032 |
| [21] | Nicolescu, M., On the criterion of compactness of A. Kolmogorov, Acad. Repub. Pop. Române. Bul. Şti. Ser. Math. Fiz. Chim., 2, 407-415 (1950), (in Romanian) |
| [22] | Pego, R. L., Compactness in \(L^2\) and the Fourier transform, Proc. Am. Math. Soc., 95, 252-254 (1985) · Zbl 0589.46020 |
| [23] | Phillips, R. S., On linear transforms, Trans. Am. Math. Soc., 48, 516-541 (1940) · Zbl 0025.34202 |
| [24] | F. Rellich, Ein Satz über mittlere Konvergenz, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (1930) 30-35.; F. Rellich, Ein Satz über mittlere Konvergenz, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (1930) 30-35. · JFM 56.0224.02 |
| [25] | Riesz, M., Sur les ensembles compacts de fonctions sommables, Acta Szeged Sect. Math., 6, 136-142 (1933), (in French). Also in L. Gårding, L. Hörmander (Eds.), Marcel Riesz—Collected Papers, Springer, Berlin (1988) · JFM 59.0276.01 |
| [26] | Simon, J., Compact sets in the space \(L^p(0,T;B)\), Ann. Mat. Pura Appl., 146, 65-96 (1987) · Zbl 0629.46031 |
| [27] | Takahashi, T., On the compactness of the function-set by the convergence in mean of general type, Studio Math., 5, 141-150 (1934) · JFM 60.0982.01 |
| [28] | Tamarkin, J. D., On the compactness of the space \(L_p\), Bull. Am. Math. Soc., 32, 79-84 (1932) · Zbl 0004.05801 |
| [29] | Tret’yachenko, Yu. V.; Chistyakov, V. V., Selection principle for pointwise bounded sequences of functions, Math. Notes, 84, 396-406 (2008) · Zbl 1167.40002 |
| [30] | Tsuji, M., On the compactness of space \(L^p(p > 0)\) and its application to integral operators, Kodai Math. J., 33-36 (1951) · Zbl 0044.10701 |
| [31] | A. Tulajkov, Zur Kompaktheit im Raum \(L_pp\); A. Tulajkov, Zur Kompaktheit im Raum \(L_pp\) · JFM 59.0275.02 |
| [32] | Veress, P., Über Functionenmengen, Acta Sci. Math. (Szeged), 3, 177-192 (1927), (in German) · JFM 53.0233.03 |
| [33] | Weil, A., L’intégration dans les groupes topologiques et ses applications (1940), Hermann et Cie.: Hermann et Cie. Paris · Zbl 0063.08195 |
| [34] | Yosida, K., Functional Analysis (1980), Springer: Springer Berlin · Zbl 0217.16001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
