The Exercise

Take vectors $|w_1\rangle, \dots, |w_n\rangle$ and $|v\rangle$ of an inner product space, and complex amplitudes $\lambda_1, \dots, \lambda_n$ . Write $(|a\rangle, |b\rangle)$ for the inner product of $|a\rangle$ with $|b\rangle$ . Exercise 2.6 asks to show that any inner product is conjugate linear in its first argument: when the superposition $\sum_i \lambda_i |w_i\rangle$ stands in the first slot, against $|v\rangle$ in the second, each amplitude comes out as its complex conjugate $\lambda_i^*$ .^✦

The Claim

Theorem 1. Let $V$ be an inner product space over $\mathbb{K}$ , the real or the complex numbers, and write $\lambda^*$ for the complex conjugate of a scalar $\lambda$ . Let $S$ be a finite set of indices, with a scalar $\lambda_i$ and a vector $|w_i\rangle$ of $V$ for each $i \in S$ , and let $|v\rangle$ be a vector of $V$ . Then $\Bigl(\sum_{i \in S} \lambda_i |w_i\rangle,\, |v\rangle\Bigr) = \sum_{i \in S} \lambda_i^* \, \bigl(|w_i\rangle, |v\rangle\bigr).$ (1)^✦

With $\mathbb{K}$ the complex numbers this is the exercise.

What Is Checked

The $|w_i\rangle$ need not be normalized or orthogonal. For every such choice the library proves the identity (1).