Why the inverse of $ L\in Pic(X) $ is the dual bundle?

Question

Given $ L\in Pic(X) $, the inverse of $ L $, denoted by $ L^{*} $ is the dual bundle $ Hom(L, \mathbb{C}) $, the transition functions of $ L^{*} $ are $ \{ g_{\alpha\beta}^{-1}, U_{\alpha} \} $. And we know that the unit element of $ Pic(X) $ is the trivial line bundle $ X\times\mathbb{C} $. But how to prove that $ L\otimes L^{*}\cong X\times\mathbb{C} $ ? And I am confused about the definition of the dual line bundle $ L^{*} $, for example how to define the topology of $ L^{*} $ ?