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What does it mean for a function to be "square-summable"? Has something to do with the scalar product in Hilbert space.

What is meant by a "complete set"? Something about multiplying a set of vectors in the Hilbert space by an orthonormal basis set and that being equal to a summation of scalars*basis vectors?

What is actually meant by eigenfunction/eigenvalues in the context of Hilbert spaces? All I know is if I apply some Hilbert space operator to a function, I get another function with a scalar out in front of it. That scalar is an eigenvalue and the whole function is an eigenfunction.

All I know about eigenvalues is that they're the roots of characteristic polynomials in linear algebra, 2nd order homogenous ODEs, the square of the frequencies in small oscillations problems... They're numbers that can make an equation go to zero maybe?