Corollary 11.12.63DE300A9B2AD

Proof. By proposition 11.12.2, the polar square is a positive continuous plurisubharmonic complex homogeneous function of degree two, and by lemma 11.12.3 it has the strong convexity in the fibers required in proposition 11.10.5. Applying proposition 11.11.2 to $F=E^*$, whose dual $F^*=E$ is ample, gives the positive smooth strictly plurisubharmonic homogeneous function $H$ on $E^*$. Hence proposition 11.10.5 applies. Its proof uses theorem 11.10.4, which preserves strong convexity in the fibers, and the square root of the resulting function is the asserted smooth norm. $\square$