Variational methods - conjugate of function

  • #1
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Let F:H->R bar be a function and F*:H->R bar its conjugate. Fix aEH and show that the conjugate of the new function G(u)=F(u-a) is G*(u*)=F*(u*)+<a,u>
Verify the case where F:R^2->R, F(x)=1/2(x)^2 and a = (2,-1)

I don't really know how to show this. please help
 

Answers and Replies

  • #2
HallsofIvy
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First, define your terms. What kind of objects are H and R? Euclidean spaces? General vector spaces? Hilbert spaces? Is "R", at least, the set of real numbers? The fact that you then use R2 as a specific case implies that it is. Is <a, u> the inner product in H? Finally, what, precisely, is your definition of "conjugate"?
 

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