Inner product of vectors

Question

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?

Answer 1

You missed part, it says,

but for the complex conjugation of the components of the first vector, ⟨V|V⟩ would not even be real, not to mention positive. [emphasis added].

Because the first term is conjugated ($v_i^\star$), we know that each term $v_i^\star v_i$ is real and nonnegative so the sum $\left<V|V\right>$ will also real and positive (assuming V is not the zero vector).

The phrase tells me that the inner product above is not real.

In general, if there's no other restriction on two vectors V and W which have complex elements, then you are correct, it's possible their inner product is not real.

Edit: you edited your question to say,

The phrase tells me that the inner product ⟨V|V⟩ is not real.

No, it says if you didn't take the conjugate of the first term in each product, it might not be real.

Since we do conjugate the first term, the inner product of a complex-valued vector with itself will be real.