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<p>bounded in .</p>

<p><big> TVS case </big></p>
<p>Let  be <a href="page.php?w=topological_vector_space">topological vector space</a>s and  a linear operator. </p>

<p>The following statements are equivalent, and different authors may pick any one of these as the principal definition for "<b>' is a compact operator":</b></p>

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* there exists a <a href="page.php?w=neighbourhood_%28mathematics%29">neighborhood</a>  of the origin in ' and  is a relatively compact subset of <b>';<br/>
* there exists a <a href="page.php?w=neighbourhood_%28mathematics%29">neighborhood</a></b></p><p>
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