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.gitignore
vendored
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.gitignore
vendored
@@ -14,3 +14,5 @@ codebook.toml
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*.out
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*.out
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*.pdf
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*.pdf
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spec.db
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spec.db
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spec.db-shm
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spec.db-wal
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@@ -150,7 +150,7 @@
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f^{**}(x) = \sup\bracs{\dpn{x, \phi}{\lambda} - \alpha|(\phi, \alpha) \in F \times \real, (\phi, \alpha) \le f}
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f^{**}(x) = \sup\bracs{\dpn{x, \phi}{\lambda} - \alpha|(\phi, \alpha) \in F \times \real, (\phi, \alpha) \le f}
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\]
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\]
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\item $\text{epi}(f^{**})$ is the $\sigma(E \times \real, F \times \real)$-closed convex hull of $\text{epi}(f)$.
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\item $\text{epi}(f^{**})$ is the $\sigma(E \times \real, F \times \real)$-closed convex hull of $\text{epi}(f)$.
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\item The biconjugate $f^{**}$ is the greatest convex and $\sigma(E, F)$-lower semicontinuous function bounded above by $f$.
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\item $f^{**}$ is the greatest convex and $\sigma(E, F)$-lower semicontinuous function bounded above by $f$.
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\item $f = f^{**}$ if and only if $f$ is convex and $\sigma(E, F)$-lower semicontinuous.
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\item $f = f^{**}$ if and only if $f$ is convex and $\sigma(E, F)$-lower semicontinuous.
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\end{enumerate}
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\end{enumerate}
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\end{theorem}
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\end{theorem}
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