Question

A function : (0,0) - 0.00) is called a Young function, if 1. w is not identically 0. 2. lm, (u) = (0) = 0, 3. is convex on (0

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Answer #1

SOLUTION:

Given data from above question \varphi is not identically '0'

Then

The second and third condition states that the function \varphi is convex and is lower semi-continuous. By the Fenchel-Moreau theorem,

we know that for any extended real valued function f in a Hausdorff locally convex space,

  f^{**}=f

if and only if f is convex and is lower semi-continuous (or identically -\infty or \infty ). Since \varphi satisfies the conditions.

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