A new characterization of the dual space of Henstock–Kurzweil integrable functions over $$\mathbb {R}$$

Abstract Recently we have shown in a previous article a full characterization of the dual space of the Henstock–Kurzweil integrable function over a bounded interval. In this manuscript we construct the Henstock–Kurzweil integral over $$\mathbb {R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> as an extension of a linear form initially defined on the classical space $$L^1(\mathbb {R})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . In particular, we give a full characterization of the dual space of Henstock–Kurzweil integrable functions over $$\mathbb {R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> in terms of a quotient space. Furthermore, the non-reflexivity of these spaces is also shown.