Mapping properties of the Hilbert and Fubini--Study maps in K\"ahler geometry

Suppose that we have a compact K"ahler manifold $X$ with a very ample line bundle $\mathcal{L}$. We prove that any positive definite hermitian form on the space $H^0(X,\mathcal{L})$ of holomorphic sections can be written as an $L^2$-inner product with respect to an appropriate hermitian metric on $\mathcal{L}$. We apply this result to show that the Fubini–Study map, which associates a hermitian metric on $\mathcal{L}$ to a hermitian form on $H^0 (X,$\mathcal{L}$)$, is injective.