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Complete set and Klein-Gordon equation

In http://www.physics.ucdavis.edu/~cheng/teaching/230A-s07/rqm2_rev.pdf, it says that when there is some external potential, the Klein-Gordon equation is altered, and it says the following:

The solution $\Psi$ can always be expressed as a superposition of free particle solutions, provided that the latter form a complete set. They from a complete set only if the negative energy components are retained, so they cannot be simply discarded.

Can anyone explain what this is talking about? First of all, what is a complete set? And why does forming a complete set require negative energy components being retained? Why is the equation (solutions, I guess) need to be a complete set?