Proving something is an inner product

Question

I am faced with the following question:

Prove that this is an inner product on V.

I understand to show something is an inner product I must verify the following conditions:

1. $\langle\,f,g\rangle = \overline{\langle\,g,f\rangle}$ (Conjugate symmetry)

2. $\langle\,\overline\alpha (f + g),h\rangle = \overline\alpha(\langle\,f,h\rangle + \langle\,g,h\rangle)$ (Linearity in the first argument)

3. $\langle\,f,f\rangle \ge 0$ with equality iff $f=0$

(Note: conditions (1) and (2) are slightly different to usual as we are dealing with complex numbers)

I have seen other examples of this being done, however, I am not sure how to do it in this particular case. Any help with this would be greatly appreciated.

Answer 1

If you have $A$ and $B$, compute $C_{ij} := \sum_k A^*_{ki}B_{kj}$. That's what $A^\dagger B$ is all about, and with it you can compute its trace, giving \begin{align} \langle A, B\rangle &= \sum_{i} A^*_{ki}B_{ki}. \end{align}

Straightforward this way to work through each of the conditions you list for an inner product.