A reflexive space of holomorphic functions in infinitely many variables

Abstract

We show the existence of an infinite-dimensional Banach space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H left-parenthesis upper E right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H(E)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the space of holomorphic functions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, endowed with the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau Subscript omega"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>τ<!-- τ --></mml:mi> <mml:mi>ω<!-- ω --></mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\tau _\omega }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> topology is reflexive.