HW 03

Problem

If $\left\langle \phi _{n}|\phi _{m}\right\rangle =\delta _{mn}$ and $\phi _{n}\,\!$ span the space of functions for which $x(t)\,\!$ and $y(t)\,\!$ are members and $x(t)=\sum _{n}a_{n}\phi _{n}(t)\,\!$ and $y(t)=\sum _{n}b_{n}\phi _{n}(t)\,\!$, then show

1. $\left\langle x|y\right\rangle =\sum _{n}a_{n}b_{n}^{*}$
2. $\left\langle x|x\right\rangle =\sum _{n}\left|a_{n}\right|^{2}$

Notes

$\left\langle x|y\right\rangle =\int _{-\infty }^{\infty }x(t)y(t)^{*}\,dt$