It is proved here that an isometry on the subset of all positive functions of <mml:math alttext="$L^1  \bigcap L^p \left( \mathbb{R} \right)$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msup><mml:mo>&#x22C2;</mml:mo><mml:msup><mml:mi>L</mml:mi><mml:mi>p</mml:mi></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>&#x211D;</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> can be characterized by means of a function <mml:math alttext="$h$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>h</mml:mi></mml:math> together with a Borel measurable mapping <mml:math alttext="$\phi $" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x3D5;</mml:mi></mml:math> of <mml:math alttext="$\mathbb{R}$" id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x211D;</mml:mi></mml:math>, thus generalizing the Banach-Lamparti theorem of <mml:math alttext="$L^p $" id="E5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi><mml:mi>p</mml:mi></mml:msup></mml:math> spaces.

International Journal of Mathematics and Mathematical Sciences

Isometries of a function space

Abstract

Copyright