Let the states form a discrete ONB for the space of single particle, and let and be the wavefunctions for the state in the position and wavevector representations, respectively. Prove the so-called closure relations:

and

The second part for k should be the same as the first.

I am pretty sure that can be rewritten as which would give

Since form a discrete ONB we should get

I don't know how to justify

Or am I totally off in left field again?

Thank you for any help, clarification, or sanity check.