Weak topologies on subspaces of 𝐶(𝑆)

Shapiro, Joel H.

doi:10.1090/S0002-9947-1971-0415285-1

Weak topologies on subspaces of $C(S)$
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by Joel H. Shapiro PDF

Trans. Amer. Math. Soc. 157 (1971), 471-479 Request permission

Abstract:

Let $S$ be a locally compact Hausdorff space, $E$ a linear subspace of $C(S)$. It is shown that the unit ball of $E$ is compact in the strict topology if and only if both of the following two conditions are satisfied: (1) $E$ is the Banach space dual of $M(S)/{E^0}$ in the integration pairing, and (2) the bounded weak star topology on $E$ coincides with the strict topology. This result is applied to several examples, among which are ${l^\infty }$ and the space of bounded analytic functions on a plane region.

References

R. Creighton Buck, Bounded continuous functions on a locally compact space, Michigan Math. J. 5 (1958), 95–104. MR 105611
Heron S. Collins, On the space $l^{\infty }\,(S)$, with the strict topology, Math. Z. 106 (1968), 361–373. MR 239406, DOI 10.1007/BF01115085
K. de Leeuw, Banach spaces of Lipschitz functions, Studia Math. 21 (1961/62), 55–66. MR 140927, DOI 10.4064/sm-21-1-55-66

Distributions and Fourier transforms

Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
Taqdir Husain, The open mapping and closed graph theorems in topological vector spaces, Clarendon Press, Oxford, 1965. MR 0178331
J. A. Johnson, Banach spaces of Lipschitz functions and vector-valued Lipschitz functions, Trans. Amer. Math. Soc. 148 (1970), 147–169. MR 415289, DOI 10.1090/S0002-9947-1970-0415289-8

Topologische lineare Räume

33

L. A. Rubel and J. V. Ryff, The bounded weak-star topology and the bounded analytic functions, J. Functional Analysis 5 (1970), 167–183. MR 0254580, DOI 10.1016/0022-1236(70)90023-6
L. A. Rubel and A. L. Shields, The space of bounded analytic functions on a region, Ann. Inst. Fourier (Grenoble) 16 (1966), no. fasc. 1, 235–277. MR 198281