Previous |  Up |  Next

Article

Keywords:
weakly compact operator on $C_0(T)$; representing measure; lcHs-valued $\sigma $-additive Baire (or regular Borel; or regular $\sigma $-Borel) measures
Summary:
Let $T$ be a locally compact Hausdorff space and let $C_0(T)$ be the Banach space of all complex valued continuous functions vanishing at infinity in $T$, provided with the supremum norm. Let $X$ be a quasicomplete locally convex Hausdorff space. A simple proof of the theorem on regular Borel extension of $X$-valued $\sigma $-additive Baire measures on $T$ is given, which is more natural and direct than the existing ones. Using this result the integral representation and weak compactness of a continuous linear map $u\: C_0(T) \rightarrow X$ when $c_0 \lnot \subset X$ are obtained. The proof of the latter result is independent of the use of powerful results such as Theorem 6 of [6] or Theorem 3 (vii) of [13].
References:
[1] S. K. Berberian: Measure and Integration. Chelsea, New York, 1965. MR 0183839 | Zbl 0126.08001
[2] J. Diestel and J. J.  Uhl: Vector measures. In Survey, No. 15, Amer. Math. Soc., Providence, 1977. MR 0453964
[3] N.  Dinculeanu and I.  Kluvánek: On vector measures. Proc. London Math. Soc. 17 (1967), 505–512. MR 0214722
[4] N. Dunford and J. T.  Schwartz: Linear Operators, General Theory. Part I. Interscience, New York, 1958.
[5] R. E.  Edwards: Functional Analysis, Theory and Applications. Holt, Rinehart and Winston, New York, 1965. MR 0221256 | Zbl 0182.16101
[6] A. Grothendieck: Sur les applications linéares faiblement compactes d’espaces du type  $C(K)$. Canad. J.  Math. 5 (1953), 129–173. DOI 10.4153/CJM-1953-017-4 | MR 0058866
[7] P. R.  Halmos: Measure Theory. Van Nostrand, New York, 1950. MR 0033869 | Zbl 0040.16802
[8] I.  Kluvánek: Characterizations of Fourier-Stieltjes transform of vector and operator valued measures. Czechoslovak Math.  J. 17(92) (1967), 261–277. MR 0230872
[9] C. W.  McArthur: On a theorem of Orlicz and Pettis. Pacific J. Math. 22 (1967), 297–302. MR 0213848 | Zbl 0161.33104
[10] T. V.  Panchapagesan: On complex Radon measures I. Czechoslovak Math.  J. 42(117) (1992), 599–612. MR 1182191 | Zbl 0795.28009
[11] T. V.  Panchapagesan: On complex Radon measures II. Czechoslovak Math. J. 43(118) (1993), 65–82. MR 1205231 | Zbl 0804.28007
[12] T. V.  Panchapagesan: Applications of a theorem of Grothendieck to vector measures. J.  Math. Anal. Appl. 214 (1997), 89–101. DOI 10.1006/jmaa.1997.5589 | MR 1645515
[13] T. V.  Panchapagesan: Characterizations of weakly compact operators on $C_0(T)$. Trans. Amer. Math. Soc. 350 (1998), 4849–4867. DOI 10.1090/S0002-9947-98-02358-7 | MR 1615942 | Zbl 0906.47021
[14] A.  Pelczyński: Projections in certain Banach spaces. Studia Math. 19 (1960), 209–228. MR 0126145
[15] W.  Rudin: Functional Analysis. McGraw-Hill, New York, 1973. MR 0365062 | Zbl 0253.46001
[16] M.  Sion: Outer measures with values in topological groups. Proc. London Math. Soc. 19 (1969), 89–106. MR 0239039
[17] E.  Thomas: L’integration par rapport a une mesure de Radon vectorielle. Ann. Inst. Fourier (Grenoble) 20 (1970), 55–191. DOI 10.5802/aif.352 | MR 0463396 | Zbl 0195.06101
[18] Ju. B. Tumarkin: On locally convex spaces with basis. Dokl. Akad. Nauk. SSSR 11 (1970), 1672–1675. MR 0271694 | Zbl 0216.40701
[19] H.  Weber: Fortsetzung von Massen mit Werten in uniformen Halbgruppen. Arch. Math. XXVII (1976), 412–423. DOI 10.1007/BF01224694 | MR 0425070 | Zbl 0331.28005
Partner of
EuDML logo