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I've been reading about Hilbert spaces and I noticed something that bugs me. All the math books state that the scalar product is linear in its first argument and antilinear in the second, while all the physics books say it's linear in the second argument and antilinear in the first.

I'm confused. Am I right to assume that the conversion between mathematical and bra-ket notation is:

$$\langle a,b\rangle=\langle b|a\rangle$$

or is there something else going on here?

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Yes, the standard convention differs between mathematics and physics, and yes, $$ \langle a,b\rangle = \langle b\vert a\rangle$$ would be one way to deal with that when translating from one field to the other. There's nothing going on here except that the physicist conceives of the ket as the "normal" vector and the bra as its conjugate, so multiplying the ket with a scalar must be linear and the multiplying the bra must be anti-linear.

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