[Journal of mathematics at home University] 053rd legensid Transformation

$ \ BF question $. set $ \ calx $ to A $ B $ space, $ F: \ calx \ To \ overline {\ BBR} \ sex {\ equiv \ BBR \ cap \ sed {\ infty} $ is a continuous convex function and $ f (x) \ not \ equiv \ infty $. if $ f ^ *: \ calx ^ * \ To \ overline {\ BBR} $ is defined as $ \ Bex f ^ * (x ^ *) =\ sup _ {x \ In \ calx} \ sed {\ SEF {x ^ *, x}-f (x )} \ quad \ sex {\ forall \ x ^ * \ In \ calx ^ *}. \ EEx $ verification: $ f ^ * (x ^ *) \ not \ equiv \ infty $.

Proof: Set $ x_0 \ In \ calx $ to $ F (x_0) <\ infty $. it is convex by $ F $ and consecutively known at $ x_0 $ \ p f (x_0) \ NEQ \ emptyset $. make $ x_0 ^ * \ In \ p f (x_0) $, then $ \ Bex f (x) \ geq F (x_0) + \ SEF {x_0 ^ *, x-x_0} \ quad \ sex {\ forall \ x \ In \ calx}, \ EEx $ and $ \ Bex \ SEF {x_0 ^ *, x}-f (x) \ Leq \ SEF {x_0 ^ *, x_0}-F (x_0) <\ infty, \ EEx $ \ Bex f ^ * (x_0 ^ *) \ Leq \ SEF {x_0 ^ *, x_0}-F (x_0) <\ infty. \ EEx $