User talk:Thielum

Welcome! Kiefer.Wolfowitz (talk) 09:35, 23 December 2009 (UTC)

Linear transformations and convex conjugacy
Hello Thielum: I believe your recent edits replace valid formulas for $$A\in{L(X,Y)}$$ by less understandable formulas restricted to automorphisms. The original formula is stated on page 76 of Zalinescu's Convex Analysis in General Vector Spaces, for example. Thank you for your consideration. Kiefer.Wolfowitz (talk) 09:39, 23 December 2009 (UTC)


 * Hello Kiefer.Wolfowitz:
 * I had and still have a hard time making sense of $$Af$$ in the LHS of formula $$ \left(A f\right)^\star = f^\star A^\star $$. With $$A$$ defined from Rn to Rm how could it be a function defined on Rm?
 * Thanks for your help!
 * Thielum (talk) 23:34, 25 December 2009 (UTC)


 * Hello Thielum:
 * This formula follows from the definition of conjugacy. An accessible proof is contained in Theorem E.2.1.1 (page 222) of Fundamentals of Convex Analysis by Hiart-Urruty and Lemaréchal; this result makes some hypotheses about the domain of the functions, so that the WP entry may need to be updated. (Zalinescu's book is very abstract, unfortunately, so my first reference was less accessible.)
 * Remark: One should have some care with the adjoint, because the Banach-space adjoint differs somewhat from the Hilbert-space adjoint, as discussed in Kreyszig's friendly textbook on Functional Analysis.
 * Best regards, Kiefer.Wolfowitz (talk) 20:55, 26 December 2009 (UTC)