completeness of eigenfunctions
When you build a vector out of coordinate axes, you trust that the axes are enough — that nothing is left over that the axes cannot reach. Completeness is that same trust for eigenfunctions: the modes u_1, u_2, u_3, ... are rich enough to represent every reasonable function, with nothing left out. It is the half of the Sturm-Liouville theorem that says the basis is not missing any directions.
Formally, the eigenfunctions {u_n} are complete in the weighted square-integrable space (L^2 with weight w) when every such function f can be approximated arbitrarily well, in mean square, by partial sums of its eigenfunction expansion: the error integral of (f - sum from 1 to N of c_n u_n)^2 w dx goes to 0 as N goes to infinity. Equivalently, Parseval's identity holds — the total weighted 'energy' of f equals the sum of the squared coefficients, integral of f^2 w = sum of c_n^2 times the norms — so no energy is lost in passing to the series.
Why it matters: orthogonality alone is not enough. You could have an orthogonal family that omits some directions, and then certain functions could never be expanded. Completeness is the guarantee that the eigenfunction expansion really equals f, that the method of separation of variables reconstructs the true solution and not just part of it. The Sturm-Liouville theorem delivers completeness automatically for regular problems; the related Mercer theorem expresses the same fact through the kernel of the operator.
The sines {sin(n*pi*x/L)} are complete on [0, L]: every L^2 function — even a discontinuous step or a jagged sawtooth — equals its Fourier sine series in the mean-square sense. Parseval says integral from 0 to L of f^2 dx = (L/2) sum of b_n^2: the function's energy is fully accounted for by its coefficients, none lost.
Completeness plus Parseval: the series captures all of f, energy and all.
Completeness is in the mean-square (L^2) sense by default, which does not imply the series converges pointwise everywhere — at jumps it overshoots (Gibbs) and converges to the midpoint. Stronger (uniform, pointwise) convergence needs extra smoothness of f, not just completeness.