* Scalar product (20 min)

a) Use the bra and ket notation to show that for an orthonormal basis $\{ \vert\psi_n \rangle\}$ and two Hilbert space vectors $\vert\psi\rangle$ and $\vert\chi\rangle$, one has

$$\langle\psi\vert\chi\rangle=\sum_{n=0}^{\infty}\langle\psi\vert\psi_n\rangle\langle\psi_n\vert\chi\rangle.$$ (28)

b) Show that in the case of vectors ${\bf x}$, ${\bf y} \in R^d$, this reduces to the standard formula for the scalar product in $R^d$,

$$\langle {\bf x} \vert{\bf y} \rangle =\sum_{i=1}^{d}x_i^*y_i.$$

c) Use Eq.(3.28) and Eq.(3.27) to prove

$$\frac{\pi^6}{960}= \sum_{k=0}^{\infty} \frac{1}{(2k+1)^6}$$