Description:
<p>We consider quasi-isometries in real continuous functions spaces and show that such a quasi-isometry can be well approximated by an affine surjective isometry.</p>
<p>On the other hand, we give an example of quasi-isometries of the unit ball <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Subscript upper H">
<mml:semantics>
<mml:msub>
<mml:mi>B</mml:mi>
<mml:mi>H</mml:mi>
</mml:msub>
<mml:annotation encoding="application/x-tex">B_H</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> in a Hilbert space <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H">
<mml:semantics>
<mml:mi>H</mml:mi>
<mml:annotation encoding="application/x-tex">H</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> that are far from any affine map of <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H">
<mml:semantics>
<mml:mi>H</mml:mi>
<mml:annotation encoding="application/x-tex">H</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> and from any isometry of <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Subscript upper H">
<mml:semantics>
<mml:msub>
<mml:mi>B</mml:mi>
<mml:mi>H</mml:mi>
</mml:msub>
<mml:annotation encoding="application/x-tex">B_H</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>.</p>