Differentiable functions and rough norms on Banach spaces

Abstract

The main result is that if <italic>X</italic> is a real Banach space, such that the density character of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X Superscript asterisk"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>X</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{X^\ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is greater than that of <italic>X</italic>, then there does not exist any real-valued Fréchet differentiable function on <italic>X</italic> with bounded nonempty support.