While reading Shankar Ramamurti's book on Principles of Quantum Mechanics, p. 10, I came across the following lines under the concept of inner product of two vectors (in terms of orthonormal basis):
$$\langle V|W \rangle= \sum_{i} {v_i}^{*} w_i \tag{1.2.5}$$ [...] You can now appreciate the first axiom; but for the complex conjugation of the components of the first vector, $\langle V|V \rangle$ would not even be real, not to mention positive.
The first axiom under discussion is the skew-symmetry of vectors: $\langle V|W \rangle={\langle W|V \rangle}^{*}$. The trouble I am facing is understanding the phrase "$\langle V|V \rangle$ would not even be real"
The phrase tells me that the inner product $\langle V|V \rangle$ is not real. The axiom—positive semidefiniteness—tells me otherwise. Am I misinterpreting the language here or is my understanding of the axiom askew?