HW 03

Problem

If $⟨{\varphi }_n|{\varphi }_m⟩={\delta }_{mn}$ and ${\varphi }_n$ span the space of functions for which $x(t)$ and $y(t)$ are members and $x(t)=\sum _n{a}_n{\varphi }_n(t)$ and $y(t)=\sum _n{b}_n{\varphi }_n(t)$, then show

1. $⟨x|y⟩=\sum _n{a}_n{b}_n^{*}$
2. $⟨x|x⟩=\sum _n{\left|a_n\right|}^2$

Notes

$⟨x|y⟩=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(t)y(t{)}^{*}dt$