The Hahn-Banach Theorem surveyed

Buskes, Gerard

Książka - szczegóły

Tytuł książki

The Hahn-Banach Theorem surveyed

Autorzy

Gerard Buskes

Seria

Rozprawy Matematyczne tom/nr w serii: 327 wydano: 1993

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CONTENTS
1. Prerequisites....................................................................................5
2. The history.......................................................................................6
3. Helly's Part.......................................................................................6
4. Banach's proof.................................................................................7
5. The shortest proof...........................................................................8
6. Luxemburg's proof.........................................................................10
7. Nachbin's proof..............................................................................11
8. Mazur's geometric Hahn-Banach Theorem....................................12
9. The complex numbers....................................................................13
10. Ingleton's Theorem......................................................................14
11. Constructive analysis and unique extensions...............................16
12. The Axiom of Choice and the Ultrafilter Theorem.........................18
13. The Mazur-Orlicz Theorem...........................................................21
14. Simultaneous Hahn-Banach extensions.......................................23
15. Injective Banach spaces and injective Banach lattices.................24
16. The interpolation property............................................................26
17. Invariant extensions.....................................................................28
18. Locally convex spaces.................................................................29
19. Non-commutative Hahn-Banach Theorems..................................30
20. The strength of the Hahn-Banach Theorem.................................31
21. Other categories..........................................................................32
  21.1. Groups and semigroups..........................................................32
  21.2. Vector lattices..........................................................................33
  21.3. Algebras..................................................................................34
  21.4. Distributive lattices and Boolean algebras..............................34
  21.5. Module versions of the Hahn-Banach Theorem......................35
References........................................................................................36

Słowa kluczowe

Tematy

Kategoryzacja MSC:

Wydawca

Instytut Matematyczny Polskiej Akademii Nauk

Miejsce publikacji

Warszawa

Copyright

Seria

Rozprawy Matematyczne tom/nr w serii: 327

Liczba stron

49

Liczba rozdzia³ów

Opis fizyczny

Dissertationes Mathematicae, Tom CCCXXVII

Daty

wydano

1993

otrzymano

1993-01-09

poprawiono

1993-03-15

Twórcy

autor

Gerard Buskes

Department of Mathematics, The University of Mississippi, University, MS 38677, U.S.A.

Bibliografia

[1] R. P. Agnew and A. P. Morse, Extension of linear functionals with applications to limits, integrals, measures and densities, Ann. of Math. 39 (1938), 20-30.
[2] R. P. Agnew, Linear functionals satisfying prescribed conditions, Duke Math. J. 4 (1938), 55-77.
[3] C. A. Akeman and J. Anderson, Lyapunov theorems for operator algebras, Mem. Amer. Math. Soc. 458 (1991).
[4] G. P. Akilov, On extension of linear operations, Leningrad. Gos. Univ. Uchen. Zap. 144 Ser. Mat. Nauk 23 (1952), 47-84 (in Russian); MR 17-1113.
[5] G. P. Akilov, Necessary conditions for the extension of linear operations, Dokl. Akad. Nauk SSSR 59 (1948), 417-418 (in Russian); MR 9-358.
[6] G. P. Akilov, On the extension of linear operations, ibid. 57 (1947), 643-646 (in Russian); MR 9-241.
[7] E. M. Alfsen and B. Hirsberg, On dominated extensions in linear subspaces of $C^∞(X)$, Pacific J. Math. 36 (1971), 567-584; MR 44 #784.
[8] D. Amir, Projections onto continuous function spaces, Proc. Amer. Math. Soc. 15 (1964), 396-402; MR 29 #2634.
[9] D. Amir, On projections and simultaneous extensions, Israel J. Math. 2 (1964), 245-248; MR 31 #5075.
[10] D. Amir, Continuous function spaces with the separable projection property, Bull. Res. Council Israel Sect. 10F (1962), 163-164; MR 27 #566.
[11] D. Amir, Continuous function spaces with the bounded extension property, ibid., 133-138; MR 26 #592.
[12] U. an der Heiden, Dominated extension of nonnegative linear functionals, Arch. Math. (Basel) 26 (1975), 402-406; MR 51 #11054.
[13] P. R. Andenaes, Hahn-Banach extensions which are maximal on a given cone, Math. Ann. 188 (1970), 90-96; MR 41 #8954.
[14] T. B. Andersen, Linear extensions, projections and split faces, J. Funct. Anal. 17 (1974), 163-173; MR 50 #8034.
[15] B. Anger and J. Lembcke, Hahn-Banach type theorems for hypolinear functionals on preordered topological vectorspaces, Pacific J. Math. 53 (1974), 110-121; MR 51 #1334.
[16] B. Anger and J. Lembcke, Extension of linear forms with strict domination on locally compact cones, preprint.
[17] B. Anger and J. Lembcke, Hahn-Banach theorems and hypolinear functionals, Math. Ann. 209 (1974), 127-151; MR 50 #934.
[18] B. Anger and J. Lembcke, Une généralisation du théorème de Hahn-Banach aux fonctions sous-linéaires à valeurs numériques, C. R. Acad. Sci. Paris Sér. A-B 277 (1973), A509-A511; MR 48 #9337.
[19] E. R. Aron, A. W. Hager and J. J. Madden, Extensions of l-homomorphisms, Rocky Mountain J. Math. 12 (1982), 481-490; MR 83k: 06019.
[20] W. B. Arveson, Subalgebras of C*-algebras, Acta Math. 123 (1969), 141-224.
[21] G. Ascoli, Sugli spazi lineari metrici e le loro varietà, Ann. Mat. Pura Appl. (4) 10 (1932), 33-81.
[22] L. Asimow and H. Atkinson, Dominated extensions of continuous affine functions with range an ordered Banach space, Quart. J. Math. Oxford Ser. (2) 23 (1972), 383-389; MR 47 #7376.
[23] G. Aumann, Über die Erweiterung von additiven monotonen Funktionen auf regulär geordneten Halbgruppen, Arch. Math. (Basel) 8 (1957), 422-427; MR 20 #6468.
[24] P. D. Bacsich, Extension of Boolean homomorphisms with bounding semimorphisms, J. Reine Angew. Math. 253 (1972), 24-27; MR 46 #111.
[25] I. A. Bahtin, The extension of a certain class of positive linear functionals, Sibirsk. Mat. Zh. 10 (1969), 1197-1205 (in Russian); MR 40 #4729.
[26] I. A. Bahtin, The extension of positive linear functionals, ibid. 9 (1968), 475-484 (in Russian); MR 37 #3314.
[27] I. A. Bahtin, On the problem of extending positive linear functionals, Dokl. Akad. Nauk SSSR 179 (1968), 759-761 (in Russian); MR 37 #4555.
[28] S. Banach, Théorie des opérations linéaires, Warszawa 1932; English translation by F. Jellett, North-Holland, Amsterdam 1987.
[29] S. Banach und S. Mazur, Zur Theorie der linearen Dimension, Studia Math. 4 (1933), 100-112.
[30] S. Banach, Sur les fonctionnelles linéaires I, ibid. 1 (1929), 211-216.
[31] S. Banach, Sur les fonctionnelles linéaires II, ibid. 1 (1929), 223-229.
[32] S. Banach, Sur le problème de la mesure, Fund. Math. 4 (1923), 7-33.
[33] S. Banach et A. Tarski, Sur la décomposition des ensembles de points en parties respectivement congruentes, ibid. 6 (1924), 244-277.
[34] B. Banaschewski, Extension of invariant linear functionals : Hahn-Banach in the topos of M-sets, J. Pure Appl. Algebra 17 (1980), 227-248; MR 82c: 46081.
[35] B. Banaschewski, The power of the Ultrafilter Theorem, J. London Math. Soc. (2) 27 (1983), 193-202; MR 84f: 03043.
[36] H. Bauer, Sur le prolongement des formes linéaires positives dans un espace vectoriel ordonné, C. R. Acad. Sci. Paris 244 (1957), 289-292.
[37] H. Bauer, Über die Fortsetzung positiver Linearformen, Bayer. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. 1957/1958, 177-190; MR 20 #6646.
[38] J. L. Bell, Some propositions equivalent to the Sikorski Extension Theorem for Boolean algebras, Fund. Math. 130 (1988), 51-55; MR 89k: 03059.
[39] J. L. Bell and D. H. Fremlin, A geometric form of the axiom of choice, ibid. 77 (1972), 167-170; MR 48 #5865.
[40] J. L. Bell, On the strength of the Sikorski Extension Theorem for Boolean algebras, J. Symbolic Logic 48 (1983), 841-846; MR 86b: 03061.
[41] S. Bernau, Extension of vector lattice homomorphisms, J. London Math. Soc. (2) 33 (1986), 516-524; MR 87h: 47088.
[42] L. Bittner, A remark concerning Hahn-Banach's extension theorem on the quasilinearisation of convex functionals, Math. Nachr. 51 (1971), 367-372; MR 47 #2308.
[43] J. Blattner and G. L. Seever, Interpositions and lattice cones of functions, Trans. Amer. Math. Soc. 222 (1976), 65-96.
[44] H. F. Bohnenblust, Convex regions and projections in Minkowski spaces, Ann. of Math. 39 (1938), 301-308.
[45] H. F. Bohnenblust, A characterization of complex Hilbert spaces, Portugal. Math. 3 (1942), 103-109; MR 4-247.
[46] H. F. Bohnenblust and A. Sobczyk, Extensions of functionals on complex linear spaces, Bull. Amer. Math. Soc. 44 (1938), 91-93.
[47] W. E. Bonnice and R. J. Silverman, The Hahn-Banach extension and the least upper bound properties are equivalent, Proc. Amer. Math. Soc. 18 (1967), 843-849; MR 35 #5895.
[48] W. E. Bonnice and R. J. Silverman, The Hahn-Banach Theorem for finite dimensional spaces, Trans. Amer. Math. Soc. 121 (1966), 210-222; MR 32 #2879.
[49] F. F. Bonsall, The decomposition of continuous linear functionals into non-negative components, Proc. Univ. Durham Philos. Soc. Ser. A 13 (1957), 6-11; MR 20 #1192.
[50] F. F. Bonsall and A. W. Goddie, Algebras which represent their linear functionals, Proc. Cambridge Philos. Soc. 49 (1953), 1-14; MR 16-936.
[51] F. F. Bonsall, Sublinear functionals and ideals in partially ordered vector spaces, Proc. London Math. Soc. (3) 4 (1954), 402-418.
[52] J. M. Borwein, On the Hahn-Banach extension property, Proc. Amer. Math. Soc. 86 (1982), 42-46; MR 83i: 46010.
[53] J. Bourgain and L. Tzafriri, On a problem of Kadison and Singer, J. Reine Angew. Math. 420 (1991), 1-43.
[54] A. J. Brandão Lopes Pinto, Banach extension theorem for ordered-complete linear spaces, Boll. Un. Mat. Ital. (4) 6 (1972), 181-184; MR 48 #2719.
[55] W. W. Breckner and E. Scheiber, A Hahn-Banach extension theorem for linear mappings into ordered modules, Mathmatica (Cluj) 19 (1977), 13-27.
[56] D. Bridges, Constructive Functional Analysis, Pitman, London 1979.
[57] E. Briem, Linear extensions and linear liftings in subspaces of C(X), Proc. Amer. Math. Soc. 58 (1976), 85-93.
[58] L. Brown, The Hahn-Banach Theorem and a class of nonlocally convex spaces, Math. Ann. 178 (1968), 295-298.
[59] A. Bruckner, Minimal superadditive extensions of superadditive functions, Pacific J. Math. 10 (1960), 1155-1163.
[60] A. Bruckner, Test for the superadditivity of functions, Proc. Amer. Math. Soc. 13 (1962), 126-130.
[61] C. W. Burden, The Hahn-Banach theorem in a category of sheaves, J. Pure Appl. Algebra 17 (1980), 25-34.
[62] G. Buskes, On an extension theorem by Sikorski, Indag. Math. 50 (1988), 395-404; MR 90a: 46010.
[63] G. Buskes, Separably-injective Banach lattices are injective, Proc. Roy. Irish Acad. Sect. A 85 (1985), 185-186; MR 87h: 46062.
[64] G. Buskes and A. van Rooij, Riesz spaces and the Ultrafilter Theorem I, Compositio Math. 83 (1992), 311-327.
[65] G. Buskes and A. van Rooij, Hahn-Banach for Riesz homomorphisms, Indag. Math. 51 (1989), 25-34; MR 90g: 46011.
[66] R. P. Čakina
[R. P. Chakina], On the question of the extension of linear functionals, Izv. Vyssh. Uchebn. Zaved. Mat. 1974 (6) (145), 91-98 (in Russian); MR 50 #10742.
[67] D. I. Cartwright, Extension of positive operators between Banach lattices, Mem. Amer. Math. Soc. 164 (1975); MR 52 #3913.
[68] F. Charvát, The problem of the extension of linear operations on modules, Časopis Pěst. Mat. 93 (1968), 371-377 (in Czech; English summary); MR 40 #7764.
[69] M.-D. Choi and E. G. Effros, Injectivity and operator spaces J. Funct. Anal. 24 (1977), 156-209.
[70] W. Chojnacki, Sur un théorème de Day, un théorème de Mazur-Orlicz et une généralisation de quelques théorèmes de Silverman, Colloq. Math. 50 (1986), 257-262; MR 88i: 43001; Erratum, ibid. 63 (1992), 139.
[71] A. Cignoli, A Hahn-Banach theorem for distributive lattices, Rev. Un. Mat. Argentina 25 (1971), 335-342; MR 48 #3827.
[72] H. B. Cohen, The k-norm extension property for Banach spaces, Proc. Amer. Math. Soc. 15 (1964), 797-802.
[73] H. B. Cohen and H. E. Lacey, On injective envelopes of Banach spaces, J. Funct. Anal. 4 (1969), 11-30; MR 39 #4644.
[74] H. B. Cohen, Injective envelopes of Banach spaces, Bull. Amer. Math. Soc. 70 (1964), 723-726; MR 32 #1536.
[75] M. G. Crandall and R. S. Phillips, On the extension problem for dissipative operators, J. Funct. Anal. 2 (1968), 147-176; MR 37 #6775.
[76] J. A. Crenshaw, Extreme positive linear operators, Math. Scand. 25 (1969), 195-217.
[77] A. Daigenault, Injective envelopes, Amer. Math. Monthly 76 (1969), 766-774; MR 40 #5690.
[78] L. Danzer, B. Grünbaum and V. Klee, Helly's theorem and its relatives, in: Convexity, Proc. Sympos. Pure Math. 7, Amer. Math. Soc., Providence 1963, 101-180.
[79] M. M. Day, Normed Linear Spaces, Academic Press, New York 1962; MR 20 #1187 and MR 49 #9588.
[80] M. M. Day, Semigroups and amenability, in: Semigroups, Proc. Sympos. on Semigroups Held at Wayne State Univ., Detroit, Michigan, June 27-29, 1968, K. W. Folley (ed.), Academic Press, New York 1969.
[81] N. De Grande - De Kimpe and C. Pérez-García, Weakly closed subspaces and the Hahn-Banach extension property in p-adic analysis, Indag. Math. 50 (1988), 253-261; MR 90a: 46196.
[82] M. de Guzmán, Note on uniqueness in the Hahn-Banach extension theorem, Rev. Acad. Cienc. Madrid 60 (1966), 577-584 (in Spanish; English summary); MR 36 #656.
[83] F. Deutsch, W. Pollul and I. Singer, On set-valued metric projections, Hahn-Banach extension maps and spherical image maps, Duke Math. J. 40 (1973), 355-370; MR 47 #2313.
[84] J. Dieudonné, Sur le théorème de Hahn-Banach, Rev. Sci. 79 (1941), 624-633; MR 7-124.
[85] J. Dieudonné, The History of Functional Analysis, North-Holland Math. Stud. 49, Amsterdam 1981.
[86] H. Dinges, Decomposition in ordered semigroups, J. Funct. Anal. 5 (1970), 436-483.
[87] K. Donner, Extension of Positive Operators and Korovkin Theorems, Lecture Notes in Math. 904, Springer, Berlin 1982; MR 83i: 46008.
[88] S. Dubuc, Fonctionnelles linéaires positives extrémales, C. R. Acad. Sci. Paris Sér. A 270 (1970), 1502-1504; MR 42 #2280.
[89] N. Dunford, Direct decompositions of Banach spaces, Bol. Soc. Mat. Mexicana 3 (1946), 1-12.
[90] P. L. Duren, B. W. Romberg and A. L. Shields, Linear functionals on $H_p$-spaces with 0
[91] M. Eidelheit, Zur Theorie der konvexen Mengen in linearen normierten Räumen, Studia Math. 6 (1936), 104-111.
[92] G. Elliott and I. Halperin, Linear normed spaces with extension property, Canad. Math. Bull. 9 (1966), 433-441; MR 34 #8147.
[93] H. Fakhoury, Sélections linéaires associées au théorème de Hahn-Banach, J. Funct. Anal. 11 (1972), 436-452; MR 50 #955.
[94] H. Fakhoury, Existence d'une projection continue de meilleure approximation dans certains espaces de Banach, J. Math. Pures Appl. 53 (1974), 1-16; MR 47 #5499.
[95] K. Fan, On infinite systems of linear inequalities, J. Math. Anal. Appl. 21 (1968), 475-478; MR 37 #3305.
[96] K. Fan, Extension of invariant linear functionals, Proc. Amer. Math. Soc. 66 (1977), 23-29; MR 56 #16314.
[97] D. Feyel, Deux applications d'une extension du théorème de Hahn-Banach, C. R. Acad. Sci. Paris Sér. A 280 (1975), A193-A196; MR 50 #10743.
[98] K. Floret, Weakly Compact Sets, Lecture Notes in Math. 801, Springer, Berlin 1980.
[99] H. O. Flösser, Fortsetzungen extremaler Funktionale, Math. Ann. 214 (1975), 195-198; MR 51 #6353.
[100] S. R. Foguel, On a theorem of A. E. Taylor, Proc. Amer. Math. Soc. 9 (1958), 325.
[101] M. Foreman and F. Wehrung, The Hahn-Banach Theorem implies existence of a non-Lebesgue measurable set, Fund. Math. 138 (1991), 13-19.
[102] L. Forgac, On the Hahn-Banach Theorem, Math. Slovaca 26 (1976), 39-45; MR 55 #13212.
[103] M. Fréchet, Sur les ensembles de fonctions et les opérations linéaires, C. R. Acad. Sci. Paris 144 (1907), 1414-1416.
[104] B. Fuchssteiner and W. Lusky, Convex Cones, North-Holland Math. Stud. 56, Amsterdam 1981; MR 82m: 46018.
[105] B. Fuchssteiner and H. König, New versions of the Hahn-Banach theorem, in: General Inequalities 2, E. F. Beckenbach (ed.), Internat. Ser. Numer. Math. 47, Birkhäuser, Basel 1980, 255-266.
[106] B. Fuchssteiner, Sur les faces exposées, C. R. Acad. Sci. Paris Sér. A 274 (1972), 38-40; MR 45 #2435.
[107] B. Fuchssteiner, On exposed semigroup homomorphisms, Semigroup Forum 13 (1977), 189-204.
[108] B. Fuchssteiner, Sandwich theorems and lattice semigroups, J. Funct. Anal. 16 (1974), 1-14.
[109] R. E. Fullerton and C. C. Braunschweiger, Quasi-interior points and the extension of linear functionals, Math. Ann. 162 (1966), 214-224; MR 33 #6339.
[110] D. J. H. Garling, Some remarks on injective envelopes, Proc. Amer. Math. Soc. 27 (1971), 303-305.
[111] H. Garnir, M. de Wilde and J. Schmetz, Analyse Fonctionnelle I, Birkhäuser, Basel 1968; MR 40 #6222.
[112] A. Ghika, The extension of general linear functionals in semi-normed modules, Acad. Repub. Pop. Române Bul. Şti. Ser. Mat. Fiz. Chim. 2 (1950), 399-405 (in Romanian; French summary); MR 13-565.
[113] H. A. Gindler, Extensions of linear transformations, Amer. Math. Monthly 71 (1964), 525-529.
[114] A. M. Gleason, Projective topological spaces, Illinois J. Math. 2 (1958), 482-489; MR 22 #12509.
[115] A. Gleit, Corrigendum: Topologies on cones, J. London Math. Soc. (2) 11 (1975), 12; MR 52 #3914.
[116] A. Gleit, Topologies on cones, ibid. 8 (1974), 1-7; MR 49 #5796.
[117] D. B. Goodner, Projections in normed linear spaces, Trans. Amer. Math. Soc. 69 (1950), 89-108; MR 12-266
[118] D. B. Goodner, Separable spaces with the extension property, J. London Math. Soc. 35 (1960), 239-240; MR 22 #11273.
[119] F. P. Greenleaf, Invariant Means on Topological Groups, Van Nostrand, New York 1969; MR 40 #4776.
[120] T. Grilliot, Extensions of algebra homomorphisms, Michigan Math. J. 14 (1967), 107-116; MR 34 #6562.
[121] B. Grünbaum, Projection constants, Trans. Amer. Math. Soc. 95 (1960), 451-465; MR 22 #4237.
[122] B. Grünbaum, Projections onto some function spaces, Proc. Amer. Math. Soc. 13 (1962), 316-324; MR 36 #6109.
[123] H. Hahn, Über lineare Gleichungssysteme in linearen Räumen, J. Reine Angew. Math. 157 (1927), 214-229.
[124] I. Halperin, The supremum of a family of additive functions, Canad. J. Math. 4 (1952), 463-479.
[125] J. D. Halpern, The independence of the axiom of choice from the Boolean prime ideal theorem, Fund. Math. 55 (1964), 57-66; MR 29 #2182.
[126] M. Hamana, Injective envelopes of operator systems, Publ. Res. Inst. Math. Sci. 15 (1979), 773-785.
[127] M. Hamana, Injective envelopes of C*-algebras, J. Math. Soc. Japan 31 (1979), 181-197.
[128] R. E. Harte, A generalization of the Hahn-Banach theorem, J. London Math. Soc. 40 (1965), 283-287; MR 30 #3379.
[129] M. Hasumi and F. Seever, The extension and lifting properties of Banach spaces, Proc. Amer. Math. Soc. 15 (1964), 773-775; MR 29 #6277.
[130] M. Hasumi, The extension property of complex Banach spaces, Tôhoku Math. J. 10 (1958), 135-142; MR 20 #7209.
[131] T. L. Hayden, The extension of bilinear functionals, Pacific J. Math. 22 (1967), 99-108; MR 37 #3320.
[132] R. Haydon, Injective Banach lattices, Math. Z. 156 (1977), 19-47; MR 58 #12293 (see also MR 57 #13438).
[133] A. Hayes, Additive functionals on groups, Math. Proc. Cambridge Philos. Soc. 58 (1962), 196-205.
[134] S. Heinrich and P. Mankiewicz, Applications of ultrapowers to the uniform and Lipschitz classification of Banach spaces, Studia Math. 73 (1982), 225-251; MR 84h: 46026.
[135] Y. Hellegouarch, Une généralisation du théorème de Hahn-Banach, C. R. Acad. Sci. Paris 274 (1972), A1615-A1618.
[136] E. Helly, Über lineare Funktionaloperationen, Sitzungsber. Akad. Wiss. Wien 121 (1912), 265-297.
[137] E. Helly, Über Mengen konvexer Körper mit gemeinschaftlichen Punkten, Jahresber. Deutsch. Math.-Verein. 32 (1923), 175-176.
[138] J. Hennefeld, A decomposition for B(X)* and unique Hahn-Banach extensions, Pacific J. Math. 46 (1973), 197-199.
[139] N. Hiramos, H. Komiya and W. Takahashi, A generalization of the Hahn-Banach Theorem, J. Math. Anal. Appl. 88 (1982), 330-340.
[140] H. Hochstadt, Eduard Helly, father of the Hahn-Banach theorem, Math. Intelligencer 2 (1980), 123-125.
[141] W. Hodges, Krull implies Zorn, J. London Math. Soc. (2) 19 (1979), 285-287; MR 80f: 04004.
[142] J. A. R. Holbrook, Concerning the Hahn-Banach Theorem, Proc. Amer. Math. Soc. 50 (1975), 322-327; MR 51 #6368.
[143] O. Hustad, Extension of positive linear functionals, Math. Scand. 11 (1962), 63-68; MR 26 #5399.
[144] O. Hustad, Linear inequalities and positive extension of linear functionals, ibid. 8 (1960), 333-338; MR 23A #A2026.
[145] O. Hustad, On positive and continuous extension of positive functionals defined over dense subspaces, ibid. 7 (1959), 392-404; MR 22 #6991.
[146] A. W. Ingleton, The Hahn-Banach theorem for non-Archimedean-valued fields, Proc. Cambridge Philos. Soc. 48 (1952), 41-45; MR 13-659.
[147] A. D. Ioffe, A new proof of the equivalence of the Hahn-Banach extension and the least upper bound properties, Proc. Amer. Math. Soc. 82 (1981), 385-389; MR 82j: 46003.
[148] J. R. Isbell, Three remarks on injective envelopes of Banach spaces, J. Math. Anal. Appl. 27 (1969), 516-518; MR 40 #4739.
[149] H. Ishihara, On the constructive Hahn-Banach Theorem, Bull. London Math. Soc. 21 (1989), 79-81; MR 90a: 46197.
[150] R. C. James, Projections in the space (m), Proc. Amer. Math. Soc. 6 (1955), 899-902; MR 17-877.
[151] T. Jech, The Axiom of Choice, North-Holland, Amsterdam 1973.
[152] L. Jekl, An application of the Hahn-Banach Theorem in convex analysis, Comment. Math. Univ. Carolin. 22 (1981), 799-807.
[153] P. Jiménez Guerra, Sur une généralisation du théorème de Hahn-Banach, C. R. Acad. Sci. Paris 278 (1974), A1087-A1089.
[154] P. Jiménez Guerra, Generalizaciones del teorema de Hahn-Banach para semimodulos preordenados, Mem. Real Acad. Cienc. Exact. Fís. Natur. Madrid 12 (1979), 1-119.
[155] R. V. Kadison and I. M. Singer, Extension of pure states, Amer. J. Math. 81 (1959), 547-564.
[156] S. Kakutani, Some characterizations of Euclidean space, Japan. J. Math. 16 (1939), 93-97; MR 1-146.
[157] S. Kakutani, Concrete representation of abstract (M)-spaces, Ann. of Math. 42 (1941), 994-1024; MR 3-205.
[158] S. Kakutani, Simultaneous extension of continuous functions considered as a positive linear operation, Japan. J. Math. 17 (1940), 1-4; MR 2-204.
[159] N. J. Kalton, N. T. Peck and J. W. Roberts, An F-space Sampler, London Math. Soc. Lecture Note Ser. 89, Cambridge Univ. Press, 1985.
[160] N. J. Kalton, Basic sequences in F-spaces and their applications, Proc. Edinburgh Math. Soc. (2) 19 (1974), 151-167.
[161] N. J. Kalton, The atomic space problem and related questions for F-spaces, in: Proc. Orlicz Memorial Conf., Univ. of Mississippi, March 21-23, 1991.
[162] L. V. Kantorovich, Concerning the problem of moments for a finite interval, Dokl. Akad. Nauk SSSR 14 (1937), 531-536.
[163] R. Kaufman, Extension of functionals and inequalities on an Abelian semigroup, Proc. Amer. Math. Soc. 17 (1966), 83-85; MR 31 #2498.
[164] R. Kaufman, Remark on invariant means, ibid. 18 (1967), 120-122; MR 34 #3344.
[165] R. Kaufman, A type of extension of Banach spaces, Acta Sci. Math. (Szeged) 27 (1966), 163-166; MR 34 #4872.
[166] R. Kaufman, Interpolation of additive functionals, Studia Math. 27 (1966), 269-272; MR 34 #587.
[167] R. Kaufman, Semigroups and de Leeuw's convexity theorem, Proc. Amer. Math. Soc. 17 (1966), 1317-1319; MR 34 #3371.
[168] R. Kaufman, Positive semicharacters, J. London Math. Soc. 42 (1967), 264-266; MR 34 #7685.
[169] R. Kaufman, Semicharacters on subsemigroups of an Abelian topological group, Proc. Amer. Math. Soc. 17 (1966), 1097-1098; MR 34 #1440.
[170] J. L. Kelley, Averaging operators on $C^∞(X)$, Illinois J. Math. 2 (1958), 214-223; MR 21 #7286.
[171] J. L. Kelley, Measures on Boolean algebras, Pacific J. Math. 9 (1959), 1165-1177; MR 21 #2179.
[172] J. L. Kelley, Banach spaces with the extension property, Trans. Amer. Math. Soc. 72 (1952), 323-326; MR 13-659.
[173] J. Kindler, A Mazur-Orlicz type theorem for submodular set functions, J. Math. Anal. Appl. 120 (1986), 533-546; MR 88a: 28014.
[174] K. Iséki and S. Kasahara, On Hahn-Banach type extension theorem, Proc. Japan Acad. 41 (1965), 29-30.
[175] V. L. Klee, Invariant extensions of linear functionals, Pacific J. Math. 4 (1954), 37-46; MR 15-631.
[176] Y. Kobayashi, Conditions for commuttive semigroups to have nontrivial homomorphisms into nonnegative (positive ) reals, Proc. Amer. Math. Soc. 65 (1977), 199-203.
[177] I. Kolumban, On the uniqueness of the extension of linear functionals, Mathematica (Cluj) 4 (27) (1962), 267-270.
[178] H. König, On certain applications of the Hahn-Banach and minimax theorems, Arch. Math. (Basel) 21 (1970), 583-591.
[179] H. König, Neue Methoden und Resultate aus Funktionalanalysis und Konvexer Analysis, Oper. Res. Verf. 28 (1978), 6-16.
[180] H. König, Der Hahn-Banach-Satz von Rodé für unendlichstellige Operationen, Arch. Math. (Basel) 35 (1980), 292-304; MR 81j: 46012.
[181] H. König, On the abstract Hahn-Banach theorem due to Rodé, Aequationes Math. 34 (1987), 89-95; MR 89f: 46010.
[182] H. König, Sublineare Funktionale, Arch. Math. (Basel) 23 (1972), 500-508; MR 49 #3511.
[183] P. Kranz, Additive functionals on abelian semigroups, Comment. Math. Prace Mat. 16 (1972), 239-246; MR 49 #3016.
[184] P. Kranz, Extension of additive functionals and semicharacters on Abelian semigroups, Res. Report N15, Nanyang Univ., Singapore 1976.
[185] P. Kranz, Sandwich and extension theorems on semigroups and lattices, Comment. Math. Prace Mat. 18 (1975), 193-200; MR 52 #14119.
[186] S. K. Kranzler and T. S. McDermott, Extending continuous linear functionals in convergence inductive limitet spaces, Proc. Amer. Math. Soc. 43 (1974), 357-360; MR 48 #11964.
[187] M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspekhi Mat. Nauk. 3 (1) (1948), 3-95 (in Russian); Amer. Math. Soc. Transl. 26 (1950).
[188] W. Krull, Algemeine Bewertungstheorie, J. Reine Angew. Math. 167 (1932), 160-196.
[189] R. G. Laatsch, Extension of subadditive functions, Pacific J. Math. 14 (1964), 209-215.
[190] M. Landsberg and W. Schirotzek, General extension theorems for linear functionals, Math. Nachr. 61 (1974), 111-122; MR 50 #10744.
[191] M. Landsberg and W. Schirotzek, Extremal and invariant extensions of linear functionals, ibid. 71 (1976), 191-202; MR 54 #3359.
[192] E. P. Lane, Insertion of a continuous function, Pacific J. Math. 66 (1976), 181-190.
[193] R. Larsen, Functional Analysis, Marcel Dekker, New York 1973.
[194] A. Lau and M. To, Extension of invariant linear functionals : a sequel to Fan's paper, Proc. Amer. Math. Soc. 63 (1977), 259-262.
[195] Å. Lima, Uniqueness of Hahn-Banach extensions and liftings of linear dependences, Math. Scand. 53 (1983), 97-113.
[196] J. Lindenstrauss, On the extension property for compact operators, Bull. Amer. Math. Soc. 68 (1962), 484-487.
[197] J. Lindenstrauss, On the extension of operators with range in a C(K) space, Proc. Amer. Math. Soc. 15 (1964), 218-225; MR 29 #5089.
[198] J. Lindenstrauss, On a problem of Nachbin concerning extension of operators, Israel J. Math. 1 (1963), 75-84; MR 28 #478.
[199] J. Lindenstrauss, Some results on the extension of operators, Bull. Amer. Math. Soc. 69 (1963), 582-586; MR 27 #579.
[200] J. Lindenstrauss, On nonseparable reflexive Banach spaces, ibid. 72 (1966), 967-970; MR 34 #4875.
[201] J. Lindenstrauss, On the extension of operators with a finite-dimensional range, Illinois J. Math. 8 (1964), 488-499; MR 29 #6317.
[202] J. Lindenstrauss, Extension of compact operators, Mem. Amer. Math. Soc. 48 (1964); MR 31 #3828.
[203] J. Lindenstrauss and L. Tzafriri, On the isomorphic classification of injective Banach lattices, Adv. in Math. 7B (1981), 489-498; MR 82k: 47055.
[204] Z. Lipecki and W. Thomsen, Extensions of positive operators and extreme points. IV, Colloq. Math. 46 (1982), 269-273; MR 84b: 47046b.
[205] Z. Lipecki, Extension of vector lattice homomorphisms, Proc. Amer. Math. Soc. 79 (1980), 247-248; MR 81b: 46009.
[206] Z. Lipecki, D. Plachky and W. Thomsen, Extensions of positive operators and extreme points. I, Colloq. Math. 42 (1979), 279-284; MR 82k: 46033.
[207] Z. Lipecki, Extensions of positive operators and extreme points. II, ibid., 285-289; MR 82k: 47056.
[208] Z. Lipecki, Extension of vector-lattice homomorphisms revisited, Indag. Math. 47 (1985), 229-233; MR 86i: 46008.
[209] Z. Lipecki, Extensions of positive operators and extreme points. III, Colloq. Math. 46 (1982), 263-268; MR 84b: 47046a.
[210] J. Łoś and C. Ryll-Nardzewski, On the applications of Tychonoff's theorem in mathematical proofs, Fund. Math. 38 (1951), 233-237.
[211] H. P. Lotz, Extensions and liftings of positive linear mappings in Banach lattices, Trans. Amer. Math. Soc. 211 (1975), 85-100.
[212] W. A. J. Luxemburg and A. R. Schep, An extension theorem for Riesz homomorphisms, Indag. Math. 41 (1979), 145-154; MR 80i: 47051.
[213] W. A. J. Luxemburg, Beweis des Satzes von Hahn-Banach, Arch. Math. (Basel) 18 (1967), 271-272.
[214] W. A. J. Luxemburg, Reduced powers of the real number system and equivalents of the Hahn-Banach Theorem, Technical Report 2, Cal. Inst. of Techn., 1967; MR 38 #5616.
[215] W. A. J. Luxemburg, A remark on Sikorski's extension theorem for homomorphisms in the theory of Boolean algebras, Fund. Math. 55 (1964), 239-247; MR 31 #2182.
[216] W. A. J. Luxemburg, Two applications of the method of construction by ultrapowers to analysis, Bull. Amer. Math. Soc. 68 (1962), 416-419; MR 25 #3837.
[217] W. A. J. Luxemburg, Nonstandard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers, Caltech, 1962.
[218] W. A. J. Luxemburg and A. C. Zaanen, Riesz Spaces I, North-Holland, Amsterdam 1971.
[219] G. W. Mackey, Note on a theorem of Murray, Bull. Amer. Math. Soc. 52 (1946), 322-325; MR 7-455.
[220] C. Malivert, J.-P. Penot and M. Thera, Un prolongement du théorème de Hahn- Banach, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), A165-A168; MR 56 #7115.
[221] P. J. Mangheni, The classification of injective Banach lattices, Israel J. Math. 84 (1984), 341-347; MR 86d: 46020.
[222] P. Mankiewicz, An example of a not well-located subspace of a nuclear space, Bull. Acad. Polon. Sci. 17 (1969), 285-288.
[223] M. Marcus and V. J. Mizel, Extension theorems of Hahn-Banach type for nonlinear disjointly additive functionals and operators in Lebesgue spaces, J. Funct. Anal. 24 (1977), 303-335; MR 57 #1102.
[224] J. Martínez-Maurica and C. Pérez-García, The Hahn-Banach extension property in a class of normed spaces, Quaestiones Math. 8 (1986), 335-341.
[225] B. Maurey, Un théorème de prolongement, C. R. Acad. Sci. Paris Sér. A 279 (1974), 329-332; MR 50 #8013.
[226] S. Maury, Un substitut du théorème de Hahn-Banach, Travaux du Séminaire d'Analyse Unilatérale, Vol. II, Exp. 2, 4pp, Univ. Montpellier, Montpellier 1969; MR 41 #7403.
[227] S. Mazur, Über konvexe Mengen in lineare normierte Räumen, Studia Math. 4 (1933), 70-84.
[228] S. Mazur et W. Orlicz, Sur les espaces métriques linéaires II, ibid. 13 (1953), 137-179; MR 16-932.
[229] E. K. McLachlan, Extremal elements of a convex cone of subadditive functions, Proc. Amer. Math. Soc. 12 (1961), 77-83.
[230] T. Mitchell, Constant functions and left invariant means on semigroups, Trans. Amer. Math. Soc. 119 (1965), 244-261.
[231] E. Michael and A. Pełczyński, A linear extension theorem, Illinois J. Math. 11 (1967), 563-579; MR 36 #671.
[232] E. Michael, Selected selection theorems, Amer. Math. Monthly 63 (1956), 233-238.
[233] D. P. Milman, On the set of minimal extensions of a subadditive functional, in: Topics in Functional Analysis. Essays Dedicated to M. G. Krein on the Occasion of His 70th Birthday, Adv. in Math. Suppl. Stud. 3, Academic Press, New York 1978, 227-240.
[234] D. P. Milman, On sublinear extensions of functionals, Dokl. Akad. Nauk SSSR 186 (2) (1969), 257-260 (in Russian).
[235] A. F. Monna, Functional Analysis in Historical Perspective, Wiley, New York 1973.
[236] A. Monteiro, Généralisation d'un théorème de R. Sikorski sur les algèbres de Boole, Bull. Sci. Math. (2) 89 (1965), 65-74; MR 32 #4054.
[237] V. B. Moscatelli, Note on the Hahn-Banach problem in bornology, J. London Math. Soc. (2) 6 (1973), 593-597; MR 48 #828.
[238] C. J. Mulvey and J. Wick Pelletier, A globalization of the Hahn-Banach theorem, Adv. in Math. 89 (1991), 1-59.
[239] F. J. Murray, Linear transformations in $L_p$ (p>1), Trans. Amer. Math. Soc. 39 (1936), 83-100.
[240] F. J. Murray, Quasi-complements and closed projections, ibid. 58 (1945), 77-95; MR 7-124.
[241] F. J. Murray, On complementary manifolds and projections in spaces $L_p$ and $l_p$, ibid. 41 (1937), 138-152.
[242] F. J. Murray, The analysis of linear transformations, Bull. Amer. Math. Soc. 48 (1942), 76-93; MR 3-209.
[243] J. Musielak and W. Orlicz, A generalization of certain extension theorems, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 8 (1960), 531-534; MR 24A #A2225.
[244] L. Nachbin, A theorem of the Hahn-Banach type for linear transformations, Trans. Amer. Math. Soc. 68 (1950), 28-46; MR 11-369.
[245] L. Nachbin, On the Hahn-Banach theorem, An. Acad. Brasil Ciênc. 21 (1949), 151-154; MR 14-114.
[246] L. Nachbin, Some problems in extending and lifting continuous linear transformations, in: Proc. Internat. Sympos. on Linear Spaces, Jerusalem, July 5-12, 1960, 340-350; MR 24 #A2826.
[247] H. Nakano, On the Hahn-Banach theorem, Bull. Acad. Polon. Sci. 19 (1971), 743-745.
[248] M. M. Neumann, Generalized convexity and the Mazur-Orlicz theorem, in: Proc. Orlicz Memorial Conf., Univ. of Mississippi, March 21-23, 1991.
[249] M. M. Neumann, On the Mazur-Orlicz theorem, Czechoslovak Math. J. 41 (1991), 104-109.
[250] M. M. Neumann, Some unexpected applications of the sandwich theorem, in: Proc. Conf. on Optimization and Convex Anal., Univ. of Mississippi, 1989, 22-38.
[251] W. Oettli, On a general formulation of the Hahn-Banach principle with application to optimization theory, in: Optimization - Theory and Algorithms, J. B. Hiriart-Urruty et al. (eds.), Marcel Dekker, New York 1983, 91-101.
[252] M. Ohron, On the Hahn-Banach theorem for modules over C(S), J. London Math. Soc. (2) 1 (1969), 363-368; MR 40 #3290.
[253] T. Ono, On the extension property of normed spaces over fields with non-Archimedean valuations, J. Math. Soc. Japan 5 (1953), 1-5; MR 15 #717.
[254] A. T. Paterson, Amenability, Math. Surveys Monographs 29, Amer. Math. Soc., 1988.
[255] J. Pawlikowski, The Hahn-Banach Theorem implies the Banach-Tarski paradox, Fund. Math. 138 (1991), 20-22.
[256] A. Pełczyński, Projections in certain Banach spaces, Studia Math. 19 (1960), 209-228; MR 23 #A3441.
[257] A. Pełczyński, Linear extensions, linear averagings and their applications to linear topological classification of continuous functions, Dissertationes Math. (Rozprawy Mat.) 58 (1968); MR 37 #3335.
[258] C. Pérez-García, Compactoid sets in p-adic locally convex spaces with the Hahn- Banach extension property, Ann. Soc. Sci. Bruxelles Sér. I 102 (1988), 97-104; MR 90k: 46168.
[259] C. Pérez-García, The Hahn-Banach extension property in p-adic analysis, in: p-adic Functional Analysis, Lecture Notes in Pure and Appl. Math. 137, Marcel Dekker, New York 1991, 127-140.
[260] R. R. Phelps, Uniqueness of Hahn-Banach extension and unique best approximations, Trans. Amer. Math. Soc. 95 (1960), 238-255; MR 22 #3964.
[261] R. R. Phelps, Extreme positive operators and homomorphisms, ibid. 108 (1963), 265-274; MR 27 #6153.
[262] R. S. Phillips, On linear transformations, ibid. 48 (1940), 516-541; MR 2-318.
[263] R. S. Phillips, A characterization of Euclidean spaces, Bull. Amer. Math. Soc. 46 (1940), 930-933; MR 2-220.
[264] D. Pincus, The strength of the Hahn-Banach theorem, in: Proc. Victoria Sympos. on Nonstandard Anal., Springer, New York 1974, 203-208.
[265] D. Pincus, Independence of the prime ideal theorem from the Hahn Banach theorem, Bull. Amer. Math. Soc. 78 (1972), 766-770; MR 45 #6619.
[266] K. Pothoven, Projective and injective objects in the category of Banach spaces, Proc. Amer. Math. Soc. 22 (1969), 437-438; MR 39 #3291.
[267] E. T. Poulsen, Eindeutige Hahn-Banach Erweiterungen, Math. Ann. 162 (1966), 225-227; MR 33 #6340.
[268] V. Pták, On a theorem of Mazur and Orlicz, Studia Math. 15 (1956), 365-366; MR 18 #320.
[269] M. S. Putcha and T. Tamura, Homomorphisms of commutative cancellative semigroups into nonnegative real numbers, Trans. Amer. Math. Soc. 221 (1976), 147-157.
[270] M. Reghis et T. Ho-van, Une démonstration nouvelle du théorème de Hahn-Banach, Bull. Soc. Roy. Sci. Liège 38 (1969), 155-157; MR 40 #4745.
[271] M. Ribe, Examples for the nonlocally convex three space problem, Proc. Amer. Math. Soc. 237 (1979), 351-355; MR 81a: 46010.
[272] M. Ribe, Necessary convexity conditions for the Hahn-Banach theorem in metrizable spaces, Pacific J. Math. 44 (1973), 715-732; MR 48 #2711.
[273] B. Riečan, Continuous extension of monotone functionals of a certain type, Mat.-Fyz. Časopis Sloven. Akad. Vied 15 (1965), 116-125 (in Russian; English summary); MR 33 #247.
[274] J. Riedl, Partially ordered locally convex spaces and extensions of positive linear mappings, Math. Ann. 157 (1964), 95-124.
[275] F. Riesz, Sur les systèmes orthogonaux de fonctions, C. R. Acad. Sci. Paris 144 (1907), 615-619.
[276] F. Riesz, Sur certains systèmes singuliers d'équations intégrales, Ann. École Norm. Sup. (3) 28 (1911), 33-62.
[277] J. W. Roberts, A nonlocally convex F-space with the Hahn-Banach appproximation property, in: Lecture Notes in Math. 604, Springer, 1977, 76-81; MR 58 #30008.
[278] G. Rodé, Eine abstrakte Version des Satzes von Hahn-Banach, Arch. Math. (Basel) 31 (1978), 474-481; MR 80g: 46009.
[279] B. Rodríguez-Salinas, Algunas problemas y teoremas de extensión de applicationes lineales, Rev. Real Acad. Cienc. Exact. Fís. Natur. Madrid 65 (1971), 677-704.
[280] B. Rodríguez-Salinas, El problema de la extensión, Ann. Mat. Pura Appl. (4) 64 (1964), 133-189.
[281] B. Rodríguez-Salinas and L. Bou, A Hahn-Banach theorem for arbitrary vector spaces, Boll. Un. Mat. Ital. (4) 10 (1974), 390-393; MR 51 #1332.
[282] B. Rodríguez-Salinas, Sobre la prolongación de funcionales lineales, Collect. Math. 16 (1964), 67-78.
[283] B. Rodríguez-Salinas, Generalización sobre modulos del teorema de Hahn-Banach y sus aplicaciones, ibid. 14 (1962), 105-151.
[284] S. Rolewicz, Functional Analysis and Control Theory, PWN, Warszawa 1974 (in Polish); German edition by Springer, 1976; English edition by PWN and Reidel, 1987.
[285] H. Rosenthal, On injective Banach spaces and the spaces $L^∞(μ)$ for finite measures μ, Acta Math. 124 (1970), 205-248; MR 41 #2370.
[286] H. Rosenthal, On injective Banach spaces and the spaces C(S), Bull. Amer. Math. Soc. 75 (1969), 824-828; MR 39 #6071.
[287] K. A. Ross, A note on extending semicharacters to semigroups, Proc. Amer. Math. Soc. 10 (1959), 573-583; MR 22 #73.
[288] K. A. Ross, Extending characters on semigroups, ibid. 12 (1961), 988-990; MR 24A #A2629.
[289] Z.-J. Ruan, Injectivity of operator spaces, Trans. Amer. Math. Soc. 315 (1989), 89-104; MR 91d: 46078.
[290] H. Rubin and J. E. Rubin, Equivalents of the Axiom of Choice II, North-Holland, 1985; MR 27 #3553.
[291] Ju. A. Šaškin
[Yu. A. Shashkin], Remarks on the extension of linear functionals, Mathematica (Cluj) 11 (34) (1969), 147-154; MR 42 #819.
[292] I. Sawashima, Locally convex spaces with the extension property, Natur. Sci. Rep. Ochanomizu Univ. 11 (1960), 19-27.
[293] G. C. Schmidt, Extension of lattice homomorphisms, J. London Math. Soc. (2) 8 (1974), 707-710.
[294] L. M. Schmitt, An equivariant version of the Hahn-Banach Theorem, Houston J. Math. 18 (1992), 429-447.
[295] A. J. Schwartz, On certain conditions for the existence of invariant linear functionals, Proc. Amer. Math. Soc. 12 (1961), 857-861; MR 24A #A3522.
[296] G. L. Seever, A Hahn-Banach theorem for commutative semigroups, dissertation, Pasadena 1967.
[297] Z. Semadeni, Banach Spaces of Continuous Functions, Vol. I, Monograf. Mat. 55, PWN, Warszawa 1971; MR 45 #5730.
[298] Z. Semadeni, Extension of linear functionals in two-norm spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 8 (1960), 427-432; MR 24A #A987.
[299] Z. Semadeni, Embedding of two-norm spaces into the space of bounded continuous functions on a half-straight line, ibid., 421-426; MR 24A #A425.
[300] D. A. Gregory and J. H. Shapiro, Nonconvex linear topologies with the Hahn Banach extension property, Proc. Amer. Math. Soc. 25 (1970), 902-905; MR 41 #8957.
[301] J. H. Shapiro, Extension of linear functionals on F-spaces with bases, Duke Math. J. 37 (1970), 639-645; MR 42 #5004.
[302] R. Shukla, On unique extension of linear functionals, Proc. Nat. Inst. Sci. India Part A 32 (1966), 113-115; MR 40 #4743.
[303] W. Sierpiński, Fonctions additives non complètement additives et fonctions non mesurables, Fund. Math. 30 (1938), 96-99.
[304] R. Sikorski, On an analogy between measures and homomorphisms, Ann. Soc. Math. Polon. 23 (1950), 1-20; MR 12-583.
[305] R. Sikorski, On the extension of homomorphisms, ibid. 21 (1948), 332-335; MR 11-76.
[306] R. Sikorski, On a theorem of Mazur and Orlicz, Studia Math. 13 (1953), 180-182; MR 16-932.
[307] R. Sikorski, Boolean Algebras, third ed., Springer, 1969; MR 39 #4053.
[308] R. J. Silverman and T. Yen, The Hahn-Banach theorem and the least upper bound property, Trans. Amer. Math. Soc. 90 (1959), 523-526; MR 21 #1511.
[309] R. J. Silverman, Invariant means and cones with vector interiors, ibid. 88 (1958), 75-79, 327-330.
[310] R. J. Silverman, Means on semigroups and the Hahn-Banach extension property, ibid. 83 (1956), 222-237; MR 18-910.
[311] R. J. Silverman, Invariant linear functionals, ibid. 81 (1956), 411-424; MR 20 #1917a.
[312] S. Simons, Extended and sandwich versions of the Hahn-Banach theorem, J. Math. Anal. Appl. 21 (1968), 112-122.
[313] S. Simons, Minimal sublinear functionals, Studia Math. 37 (1970), 37-56; MR 42 #8238.
[314] S. Simons, A theorem on lattice ordered groups, results of Pták, Namioka, Banach and a front- ended proof of Lebesgue's Theorem, Pacific J. Math. 20 (1967), 149-154.
[315] S. Simons, Banach limits, infinite matrices and sublinear functionals, J. Math. Anal. Appl. 26 (1969), 640-655; MR 39 #3293a.
[316] A. Sobczyk, Projections of the space m on its subspaces $c_0$, Bull. Amer. Math. Soc. 47 (1941), 938-947; MR 3-205.
[317] A. Sobczyk, On the extension of linear transformations, Trans. Amer. Math. Soc. 55 (1944), 153-169; MR 5-272.
[318] A. Sobczyk, Projections in Minkowski and Banach spaces, Duke Math. J. 8 (1941), 78-106; MR 2-220.
[319] A. Sobczyk, Extension properties of Banach spaces, Bull. Amer. Math. Soc. 68 (1962), 217-224; MR 26 #1739.
[320] G. A. Soukhomlinoff
[G. A. Sukhomlinov], Über Fortsetzung von linearen Funktionalen in linearen komplexen Räumen und linearen Quaternionräumen, Mat. Sbornik (N.S.) 3 (1938), 353-358 (in Russian; German summary).
[321] A. V. Štraus
[A. V. Shtraus], Extensions and characteristic function of a symmetric operator, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 186-207 (in Russian); MR 37 #788.
[322] K. Strohle, Varianten des Satzes von Hahn-Banach mit Anwendungen, doctoral dissertation, Ludwigs-Maximilians Univ., 1973.
[323] A. Tarski, Algebraische Fassung des Massproblems, Fund. Math. 31 (1938), 47-66.
[324] A. Tarski, Une contribution à la théorie de la mesure, ibid. 15 (1930), 42-50.
[325] A. E. Taylor, The extension of linear functionals, Duke Math. J. 5 (1939), 538-547; MR 1-50.
[326] T. O. To, A note of correction to a theorem of W. E. Bonnice and R. J. Silverman, Trans. Amer. Math. Soc. 139 (1969), 163-166.
[327] T. O. To, The equivalence of the least upper bound property and the Hahn-Banach extension property in ordered linear spaces, Proc. Amer. Math. Soc. 25 (1970), 902-905; MR 43 #875.
[328] F. Topsοe, The naive approach to the Hahn-Banch theorem, Comment. Math. Prace Mat., tomus specialis in honorem Ladislai Orlicz (1978), part I, 315-330; MR 80b: 46006.
[329] H. Tuy, Convex inequalities and the Hahn-Banach Theorem, Dissertationes Math. (Rozprawy Mat.) 97 (1972).
[330] J. Vangeldère, Frank separation of two convex sets and Hahn-Banach Theorem, J. Math. Anal. Appl. 60 (1977), 36-46; MR 57 #10404.
[331] A. C. M. van Rooij, Non-Archimedean Functional Analysis, Monographs Textbooks Pure Appl. Math. 51, Marcel Dekker, New York 1978; MR 81a: 46084.
[332] A. C. M. van Rooij, The Axiom of Choice in p-adic functional analysis, in: p-adic Functional Analysis, Lecture Notes in Pure and Appl. Math. 137, Marcel Dekker, New York 1991, 151-156.
[333] G. Vincent-Smith, The Hahn-Banach theorem for modules, Proc. London Math. Soc. (3) 17 (1967), 72-90; MR 35 #766.
[334] D. Vuza, The Hahn-Banach extension theorem for modules over ordered rings, Rev. Roumaine Math. Pures Appl. 27 (1982), 989-995; MR 85d: 46008.
[335] S. Wagon, The Banach-Tarski Paradox, Cambridge Univ. Press, 1986; MR 87e: 04007.
[336] J. H. Wells and L. R. Williams, Embeddings and Extensions in Analysis, Ergeb. Math. Grenzgeb. 84, Springer, 1975.
[337] J. Weston, A note on the extension of linear functionals, Amer. Math. Monthly 67 (1960), 444-445; MR 22 #5867.
[338] A. W. Wickstead, Extensions of orthomorphisms, J. Austral. Math. Soc. Ser. A 29 (1980), 87-98.
[339] M. de Wilde, Quelques théorèmes des fonctionnelles linéaires, Bull. Soc. Roy. Sci. Liège 36 (1966), 552-557.
[340] M. Wilhelm, Existence of additive functionals on semigroups and the von Neumann minimax theorem, Colloq. Math. 35 (1976), 267-274.
[341] G. Wittstock, Extensions of completely bounded C*-module homomorphisms, in: Proc. Conf. on Operator Algebras and Group Representations (Neptun 1980), Monographs Stud. Math. 18, Pitman, Boston 1984, 238-250; MR 85i: 46080.
[342] G. Wittstock, Ein Operatorwertiger Hahn-Banach Satz, J. Funct. Anal. 40 (1981), 127-150.
[343] J. Wolfe, Injective Banach spaces of type C(T), Israel J. Math. 18 (1974), 133-140; MR 50 #5445.
[344] M. A. Woodbury, Invariant functionals and measures, Bull. Amer. Math. Soc. 56 (1950), 172.
[345] J. D. Maitland Wright, An extension theorem and a dual proof of a theorem of Gleason, J. London Math. Soc. 43 (1968), 699-702; MR 37 #4632.
[346] B. Yood, On fixed points for semigroups of linear operations, Proc. Amer. Math. Soc. 2 (1951), 225-233.
[347] D. Yost and B. Sims, Linear Hahn-Banach extension operators, Proc. Edinburgh Math. Soc. (2) 32 (1989), 53-57; MR 90b: 46042.
[348] L. Young, Mathematicians and Their Times, North-Holland Math. Stud. 48 Amsterdam, 1981.
[349] W. Żelazko, Concerning extension of multiplicative linear functionals in Banach algebras, Studia Math. 31 (1968), 495-499; MR 39 #1974.
[350] M. Zippin, The separable extension problem, Israel J. Math. 26 (1977), 372-387; MR 56 #1030.
[351] J. Zowe, Sandwich theorems for convex operators with values in an ordered vector space, J. Math. Anal. Appl. 66 (1978), 282-296; MR 54 #13527.

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1991 Mathematics Subject Classification: Primary 46A22, 46-02; Secondary 04A25, 46M10, 46P05, 47B55.

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