I have a problem when I should make a Legendre transform on a subspace ImG of a space Y both have an infinite dimension, I want to ask if there is any results about Legendre transform on a subspace?
I hope I understand your problem correctly. If f:I subset of R->R is a convex function and Legendre transformation f*: I*->R is defined, you are expecting I* as proper subset of R. This is my interpretation from your question. If not please ignore the remaining post. If function f(x) = cx is convex, for every x on I=R. Then, x*x − f(x) = (x* − c)x is never bounded from above as a function of x, unless x* − c = 0. Hence f* is defined on I* = {c} and f*(c) = 0.
https://en.wikipedia.org/wiki/Legendre_transformation
Please refer this page example 4
You can see this site
http://mathoverflow.net/questions/233629/definition-or-properties-of-legendre-transform-on-a-subspace-and-not-on-the-whol
and
https://books.google.com.sa/books?id=UG4Rh4OG-hUC&pg=PA357&lpg=PA357&dq=properties+of+Legendre+transform+on+a+subspace&source=bl&ots=jCB8HhN3zY&sig=sMWe4Ljyq_hNUsEzMIJmv0N62Zw&hl=en&sa=X&redir_esc=y#v=onepage&q=properties%20of%20Legendre%20transform%20on%20a%20subspace&f=false
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