I am reading the book "The Index Formula for Dirac Operators: an Introduction” (via this link http://www.impa.br/opencms/pt/biblioteca/pm/PM_10.pdf)
I am having trouble understanding the middle of pp.53 (I have the relevant screenshot below):
Notice in the above image how the author says, If the inner product ⟨ , ⟩ on V is bilinearly extended to V ⊗ C we see easily that
However, since the extension of J is unitary with respect to this extended inner-product, the direct-sum decomposition into W and its conjugate is orthogonal. Therefore, we also have
Therefore, this extended bilinear-product is identically zero, and thus degenerate. I am wondering, how then can the statement in italics be true? Perhaps it has to do with the fact that I am implicitly assuming that the extended inner-product is defined as
If so, what is the correct definition of the extended inner-product?