This induces a Hermitian inner product on the tangent space to each point of Teichm�ller space, and hence a Riemannian metric.
2.
Since the Riemann surface has a natural hyperbolic metric, at least if it has negative Euler characteristic, one can define a Hermitian inner product on the space of quadratic differentials by integrating over the Riemann surface.
3.
In the case of complex Hilbert spaces, we use a Hermitian inner product; it will be conjugate-linear ( complex anti-linear ) in " u " ( math convention ) or " v " ( physics convention ), and complex-linear in the other variable.
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