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Bokuan Li 373bc8a223 Adjusted wording in Fenchel-Moreau.
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2026-06-24 23:21:52 -04:00
Bokuan Li e50cc6fb46 Updated gitignore. 2026-06-24 23:14:50 -04:00
2 changed files with 3 additions and 1 deletions
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@@ -14,3 +14,5 @@ codebook.toml
*.out *.out
*.pdf *.pdf
spec.db spec.db
spec.db-shm
spec.db-wal
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@@ -150,7 +150,7 @@
f^{**}(x) = \sup\bracs{\dpn{x, \phi}{\lambda} - \alpha|(\phi, \alpha) \in F \times \real, (\phi, \alpha) \le f} f^{**}(x) = \sup\bracs{\dpn{x, \phi}{\lambda} - \alpha|(\phi, \alpha) \in F \times \real, (\phi, \alpha) \le f}
\] \]
\item $\text{epi}(f^{**})$ is the $\sigma(E \times \real, F \times \real)$-closed convex hull of $\text{epi}(f)$. \item $\text{epi}(f^{**})$ is the $\sigma(E \times \real, F \times \real)$-closed convex hull of $\text{epi}(f)$.
\item The biconjugate $f^{**}$ is the greatest convex and $\sigma(E, F)$-lower semicontinuous function bounded above by $f$. \item $f^{**}$ is the greatest convex and $\sigma(E, F)$-lower semicontinuous function bounded above by $f$.
\item $f = f^{**}$ if and only if $f$ is convex and $\sigma(E, F)$-lower semicontinuous. \item $f = f^{**}$ if and only if $f$ is convex and $\sigma(E, F)$-lower semicontinuous.
\end{enumerate} \end{enumerate}
\end{theorem} \end{theorem}