Below is an excerpt from the Quantum book by Shankar.  It’s a great help!  I am loving the explanations to supplement our Sakurai book, anyway — this was a particularly interesting paragraph, which I had to re-read a couple times.

which expresses the antilinearity of the inner product with respect to the first factor
in the inner product. In other words, the inner product of a linear superposition
with another vector is the corresponding superposition of inner products if the super-
position occurs in the second factor, while it is the superposition with all coefficients
conjugated if the superposition occurs in the first factor. This asymmetry, unfamiliar
in real vector spaces, is here to stay and you will get used to it as you go along.

Cheers!

–Justin