Hi I have a basic question regarding bra-ket notation. Given that $\{|e_n \rangle \}$ is a discrete orthonormal basis, $$\langle e_m | e_n \rangle = \delta_{mn}$$ then $$\sum_{n}|e_n \rangle \langle e_n | ={\bf 1}$$ (the identity operator). This follows since we can easily show (by expressing any vector $v$ in component form) that $$\sum_n |e_n \rangle \langle e_n | v \rangle = | v \rangle.$$
When dealing with a inner product which has a continuous orthonormalized basis, it is stated that $$ \langle e_z | e_z' \rangle = \delta (z-z')$$ then $$\int | e_z \rangle \langle e_z| dz = {\bf 1}.$$
How would you show the last two equations? I think the first equation holds because we simply define the inner-product in this way and can show that it is consistent with the properties of an inner-product.
