Pome2584C29ED70FTheorem 19.6.13.2584C29ED70F The Legendre transform and inverse Legendre transform are inverse to each other, defining a bijection between $\mathrm{TC}(\Delta )$ and the set of test functions on $\Delta $.
Under this bijection, $\mathrm{TC}^1(\Delta )$ corresponds to test functions on $\Delta $ with finite energy and $\mathrm{TC}^{\infty }(\Delta )$ corresponds to bounded test functions on $\Delta $.
Proof. Thanks to 19.6.9, in order to prove the first assertion, it only remains to see that the Legendre transform and the inverse Legendre transform are inverse to each other, which is immediate by definition.
It is obvious that $\mathrm{TC}^{\infty }(\Delta )$ corresponds to bounded test functions. Moreover, a direct computation shows that if $\Delta _{\bullet }\in \mathrm{TC}(\Delta )$, then
concluding the $\mathrm{TC}^{1}(\Delta )$ case. $\square$
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