I am faced with the following question:

enter image description here

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.

  • 3
    $\begingroup$ Could you be more specific about where you're running into trouble? If you tried copying what you've seen for similar examples, then where exactly did you get stuck? $\endgroup$ Jan 26, 2021 at 20:38
  • 2
    $\begingroup$ @James, this is one of those grind-em-out problems, but straightforward. Write out the products and compute the traces. If it's any comfort for you, all of this holds for $n \times n$ matrices too. $\endgroup$ Jan 26, 2021 at 20:40
  • $\begingroup$ Shouldn't 2. be $\langle\bar{\alpha}f+\bar{\beta}g,\,h\rangle=\bar{\alpha}\langle f,\,h\rangle+\bar{\beta}\langle g,\,h\rangle$? $\endgroup$
    – J.G.
    Jan 26, 2021 at 20:51
  • $\begingroup$ @Aruralreader sorry I don't fully understand what you mean by this? $\endgroup$
    – James
    Jan 26, 2021 at 20:57
  • $\begingroup$ @BenGrossmann I got stuck at knowing where to begin. The other examples didn't involve complex numbers and were more straightforward. $\endgroup$
    – James
    Jan 26, 2021 at 21:00

1 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.


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