In Boyd's Convex optimization, he states that
If $f(u, v) = f_{1}(u) + f_{2}(v)$, where $f_{1}$ and $f_{2}$ are convex with convex conjugates $f_{1}^{\ast}$, $f_{2}^{\ast}$, respectively, then $f^{\ast}(w,z) = f_{1}^{\ast}(w) + f_{2}^{\ast}(z)$
I wonder why we need $f_{1}$, $f_{2}$ should be convex.
My proof is followings:
$f^{*}(w,z) = \sup_{u,v} (\langle (w,z), (u,v)\rangle - f(u,v)) = \sup_{u,v} ( w^{T}u-f_{1}(u) + z^{T}v - f_{2}(v) ) = \sup_{u} (w^{T}u - f_{1}(u)) + \sup_{v} (z^{T}v - f_{2}(v)) = f_{1}^{*}(w) + f_{2}^{*}(z)$
I think convexity does not affect to my proof.