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A non-reflexive Grothendieck space that does not contain $$l_{\infty }$$. (English) Zbl 1358.46007
Summary: A compact space $$S$$ is constructed such that, in the dual Banach space $$C(S)^*$$, every weak$$^*$$ convergent sequence is weakly convergent, while $$C(S)$$ does not have a subspace isomorphic to $$l_{\infty }$$. The construction introduces a weak version of completeness for Boolean algebras, here called the Subsequential Completeness Property. A related construction leads to a counterexample to a conjecture about holomorphic functions on Banach spaces. A compact space $$T$$ is constructed such that $$C(T)$$ does not contain $$l_{\infty }$$ but does have a “bounding” subset that is not relatively compact. The first of the examples was presented at the International Conference on Banach spaces, Kent, Ohio, 1979.

##### MSC:
 46B03 Isomorphic theory (including renorming) of Banach spaces 46G20 Infinite-dimensional holomorphy 32A10 Holomorphic functions of several complex variables
##### Keywords:
Grothendieck space; bounding subset
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##### References:
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