I consider a TVS locally convex and separated. I define on it the concave conjuguate of a concave and upper semi continuous function as
$$ f^{*}(x^{*}) =\inf_{y\in X}\left\{x^{*}(y) - f(y)\right\},\quad x^{*}\in X^* $$
Where $X^*$ is the topological dual.
I would like to show that the concave conjuguate is also upper semi continuous using the epigraph, . Here is my attempt :
$$ \left\{ (x^*,\lambda)\in X^{*}\times\mathbb{R} : f^{*}(x^{*})\geq\lambda\right\} =\cap_{y\in X} \left\{ (x^*,\lambda)\in X^{*}\times\mathbb{R} : x^{*}(y) - f(y)\geq\lambda\right\} $$
But for each $y\in Y$, we know that the map $x^{*}\to x^{*}(y)$ is continuous for the weak star topology, therefore is $x^{*}(y) - f(y)$ and we conclude that the set
$$ \left\{ (x^*,\lambda)\in X^{*}\times\mathbb{R} : x^{*}(y) - f(y)\geq\lambda\right\} $$
is closed w.r.t the induced product topology. This shows that $f^{*}$ is weak star upper semi continuous.
I would like to know if my proof is correct please since I am not used to the weak star topology. Thank you a lot.