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Pseudodistances invariantes sur les domaines d’un espace localement convexe. (Invariant pseudodistances on domains of a locally convex space). (French) Zbl 0603.46052
The authors prove that the Kobayashi and Carathéodory invariant metrics (and infinitesimal metrics) coincide on a convex domain in a locally convex space. Applications include showing the existence of geodesics (in the sense of Vesentini) in convex domains of a reflexive Banach space.

MSC:
46G20 Infinite-dimensional holomorphy
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
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References:
[1] N. Bourbaki , Topologie générale, chapitre I , Hermann , Paris , 1965 . MR 244924
[2] H. Cartan , Sur les fonctions de plusieurs variables complexs. L’itération des transformations intérieures d’une domaine borné , Math. Z. , 35 ( 1932 ), pp. 760 - 773 . MR 1545327 | Zbl 0004.40602 | JFM 58.0349.02 · Zbl 0004.40602
[3] J.-F. Colombeau - D. Lazet , Sur les théorèmes de Vitali et de Montel en dimension infinie , C.R. Acad. Sci. , Paris , A . 274 ( 1972 ), pp. 185 - 187 . MR 295022 | Zbl 0229.46045 · Zbl 0229.46045
[4] S. Dineen , Complex analysis on locally convex spaces , North-Holland Math . Studies 57, Amsterdam , 1981 . MR 640093 | Zbl 0484.46044 · Zbl 0484.46044
[5] S. Dineen , The Cartan-Thullen theorem for Banach spaces , Ann. Scuola Norm. Sup. Pisa ( 3 ), 24 ( 1970 ), pp. 667 - 676 . Numdam | MR 277754 | Zbl 0235.46037 · Zbl 0235.46037
[6] T. Franzoni - E. Vesentini , Holomorphic maps and invariant distances , North-Holland Math . Studies 40, Amsterdam , 1980 . MR 563329 | Zbl 0447.46040 · Zbl 0447.46040
[7] L.A. Harris , Schwarz-Pick systems of pseudometrics for domains in normed linear spaces , in Advances in Holomorphy , North-Holland Math . Studies 34, Amsterdam , 1979 , pp. 345 - 406 . MR 520667 | Zbl 0409.46053 · Zbl 0409.46053
[8] L.A. Harris - J.-P. Vigué , A metric condition for equivalence of domains , Atti Accad. Naz. Lincei , 67 ( 1979 ), pp. 402 - 403 . MR 625903 | Zbl 0472.46033 · Zbl 0472.46033
[9] E. Hille - R. Phillips , Functional analysis and semi-groups , Amer. Math. Soc. Colloq. Publ. , 36 , A.M.S ., Providence , 1957 . MR 89373 | Zbl 0078.10004 · Zbl 0078.10004
[10] W. Kaup - H. Upmeier , Jordan algebras and symmetric Siegel domains in Banach spaces , Math. Z. , 147 ( 1977 ), pp. 179 - 200 . Article | MR 492414 | Zbl 0357.32018 · Zbl 0357.32018
[11] W. Kaup , A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces , Math. Z. , 183 ( 1983 ), pp. 503 - 529 . Article | MR 710768 | Zbl 0519.32024 · Zbl 0519.32024
[12] S. Kobayashi , Hyperbolic manifolds and holomorphic mappings , Marcel Dekker , New York , 1971 . MR 277770 | Zbl 0207.37902 · Zbl 0207.37902
[13] S. Kobayashi , Intrinsic distances, measures and geometric function theory , Bull. Amer. Math. Soc. , 82 ( 1976 ), pp. 357 - 416 . Article | MR 414940 | Zbl 0346.32031 · Zbl 0346.32031
[14] L. Lempert , La métrique de Kobayashi et la représentation des domaines sur la boule , Bull. Soc. Math. France , 109 ( 1981 ), pp. 427 - 474 . Numdam | MR 660145 | Zbl 0492.32025 · Zbl 0492.32025
[15] L. Lempert , Holomorphic retracts and intrinsic metrics in convex domains , Anal. math. , 8 ( 1982 ), pp. 257 - 261 . MR 690838 | Zbl 0509.32015 · Zbl 0509.32015
[16] H. Royden - P. Wong , Carathéodory and Kobayashi metric on convex domains , à paraitre.
[17] E. Vesentini , Complex geodesics , Compositio math. , 44 ( 1981 ), pp. 375 - 394 . Numdam | MR 662466 | Zbl 0488.30015 · Zbl 0488.30015
[18] E. Vesentini , Complex geodesics and holomorphic maps , Sympos. math. , 26 ( 1982 ), pp. 211 - 230 . MR 663034 | Zbl 0506.32008 · Zbl 0506.32008
[19] E. Vesentini , Invariant distances and invariant differential metrics in locally convex spaces , Spectral theory, Banach center publications , 8 ( 1982 ), pp. 493 - 512 . Article | MR 738313 | Zbl 0505.32020 · Zbl 0505.32020
[20] J.-P. Vigué , Le groupe des automorphismes analytiques d’un domaine borné d’un espace de Banach complexe. Application aux domaines bornés symétriques , Ann. Scient. Ec. Norm. Sup , ( 4 ), 9 ( 1976 ), pp. 203 - 282 . Numdam | MR 430335 · Zbl 0333.32027
[21] J.-P. Vigué , Automorphismes analytiques des produits continus de domaines bornés , Ann. scient. Ec. Norm. Sup ( 4 ), 11 ( 1978 ), pp. 229 - 246 . Numdam | MR 510550 | Zbl 0405.32007 · Zbl 0405.32007
[22] J.-P. Vigué , Sur la convexité des domaines bornés cerclés homogènes , Séminaire Lelong-Skoda , Lecture Notes 822 , 1980 , Springer , Berlin , pp. 317 - 331 . MR 599035 | Zbl 0444.32016 · Zbl 0444.32016
[23] J.-P. Vigué , Sur les applications holomorphes isométriques pour la distance de Carathéodory , Ann. Scuola Norm. Sup. Pisa ( 4 ), 9 ( 1982 ), pp. 255 - 261 . Numdam | MR 674974 | Zbl 0507.32017 · Zbl 0507.32017
[24] J.-P. Vigué , Géodésiques complexes et points fixes d’applications holomorphes , Adv. in Math. , 52 ( 1984 ), pp. 241 - 247 . MR 744858 | Zbl 0555.32015 · Zbl 0555.32015
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