Completeness, Orthonormality and Orthogonality in QM

What are the properties of orthogonal functions?  

Purpose of it in power series expanstion?

https://physics.stackexchange.com/questions/295064/orthonormality-and-ket-expansion-for-uncountable-basis-kets-such-as-the-positio  

http://web.mst.edu/~parris/QuantumTwo/Class_Notes/GeneralFormulation.pdf   

https://ocw.mit.edu/courses/physics/8-05-quantum-physics-ii-fall-2013/lecture-notes/MIT8_05F13_Chap_04.pdf   

https://www.tcm.phy.cam.ac.uk/~pdh1001/thesis/node12.html  

https://www2.ph.ed.ac.uk/~ldeldebb/docs/QM/lect4.pdf  

https://en.wikipedia.org/wiki/Orthogonality  

https://arxiv.org/pdf/1709.07652.pdf

https://arxiv.org/pdf/quant-ph/9911046.pdf

http://www.physics.sdsu.edu/~johnson/phys410/lecture9of410.pdf

https://www.quora.com/How-do-you-explain-the-completeness-relation-in-Quantum-Mechanics 

https://physics.stackexchange.com/questions/241673/on-the-completeness-relation-in-quantum-mechanics/241677 

https://physics.stackexchange.com/questions/251803/what-is-meant-by-the-term-completeness-relation

http://iopscience.iop.org/article/10.1088/2399-6528/aa9fc0

https://www.youtube.com/watch?v=Wq0gYWJjtMc 

https://www.youtube.com/watch?v=LyyZJT5FXIY