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<p>of . That is,</p>

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:</p>

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:</p>

<p><big>Mathematical theorem</big></p>
<p>Let X and Y be <a href="page.php?w=Banach_spaces">Banach spaces</a>,  and  be convex functions and  be a <a href="page.php?w=bounded_operator">bounded</a> <a href="page.php?w=linear_map">linear map</a>.  Then the Fenchel problems:<br/>
:<br/>
:satisfy <a href="page.php?w=weak_duality">weak duality</a>, i.e. .  Note that  and  are the convex conjugates of  and , respectively, and  is the <a href="page.php?w=adjoint_operator">adjoint operator</a>.  The <a href="page.php?w=perturbation_function">perturbation function</a></p><p>
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