Dirac ket-bra simplification over here.

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I usually know how to manipulate bras and kets. But I probably found difficulty because of the double summation. How did Sakurai manage to go from the second line to the ird line in the attachment?

SECTION 4.4 in Sakurai.
 
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  • Handwritten-style derivation of bra-ket manipulations: |tilde-beta> = sum over a' of <a'|beta>* U |a'> equals DC arrow sum over a' of <a'|beta> <a'| U-dagger; <tilde-beta|tilde-alpha> = double sum over a'' and a' of <a''|beta> <a''|U-dagger U|a'> <alpha|a'>; = sum over a' of <alpha|a'> <a'|beta> = <alpha|beta>; = <beta|alpha>*.
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Using the orthogonality condition
 
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I was thinking about taking duality but that didn't seem to work.
 
M. next said:
How come? If so I will end up with a zero, no?
No. What is [itex]\langle a''|a' \rangle[/itex] equal to and what does this imply for [itex]\Sigma_{a''} \langle a''|a' \rangle[/itex]?
 
It is equal to 1 if a=a'' and 0 otherwise if we take orthogonality into consideration.. It implies that the summation = 1 when a'' = a' and 0 for all the other summations. Right?
 
Yes. If you put Bill's comment and this together, do you still miss something to go from line two to line three?