Note, $\langle \vec{a}\mid\vec{b}\rangle = \vec{a} \bullet \vec{b}$
What does the asterisk mean in the below properties?
I am learning vector math[1]. I have come across the following property:
The inner product has the following properties:
$$\ (i) \ \langle \vec{a}\mid \vec{b} \rangle = \langle\vec{b}\mid\vec{a}\rangle^{*}$$ $$\ (ii) \ \langle \vec{a}\mid \lambda\vec{b} + \mu\vec{c} \rangle= \lambda \langle\vec{a}\mid\vec{b} \rangle + \mu \langle\vec{a}\mid\vec{c} \rangle $$
Literature states that the asterisk means: "When used as a superscript, the asterisk is commonly voiced "a-star." A raised asterisk is used to denote the adjoint $a^*$, or sometimes the complex conjugate."[2]
We note that in general, for a complex vector space, (i) and (ii) imply that
$$\langle \lambda \vec{a} + \mu \vec{b}\mid\vec{c} \rangle = \lambda^{*} \mu \langle \vec{a}\mid \vec{c} \rangle + \mu^{*}\langle \vec{b}\mid\vec{c} \rangle$$ $$\langle \lambda \vec{a}\mid\mu \vec{b} \rangle = \lambda^{*} \mu \langle \vec{a}\mid\vec{b} \rangle$$
I am assuming it is the complex vector space, but I also know, from my math book[3]
$$ (a) \ \langle \vec{a}\mid\vec{b} \rangle = \langle \vec{b}\mid\vec{a} \rangle$$
without the asterisk.
What does the asterisk mean in the context of (i)?
References:
[1] Riley, K. F.. Mathematical Methods for Physics and Engineering: A Comprehensive Guide (Kindle Locations 6288-6293). Cambridge University Press. Kindle Edition.
[2] MathWorld. (2018) Asterisk. Retrieved (2018, May 13). Available from: http://mathworld.wolfram.com/Asterisk.html
[3] Anton, Howard. (1992). Calculus with analytic geometry-4th Edition. Anton Textbooks, Inc.