completeness of eigenfunctions
Suppose you have a set of paint colours and you want to know: can you mix every possible colour from these, or are some shades forever out of reach? A set of building blocks is complete if nothing is missing — if everything you might want can be built from them. Completeness of eigenfunctions asks exactly this of the modes of a Sturm-Liouville problem: is every reasonable function reachable as a combination of them, or do some functions slip through the gaps?
Formally, the eigenfunctions {y_n} of a regular Sturm-Liouville problem are complete in the space of square-integrable functions on [a, b] with weight w: for any such function f, its eigenfunction expansion sum of c_n y_n converges to f in the mean-square sense, meaning the weighted integral of (f minus the partial sum)^2 tends to zero as you include more terms. Equivalently, no nonzero function is orthogonal to every eigenfunction — there is no 'hidden direction' the eigenfunctions fail to span. Completeness is the partner of orthogonality: orthogonality lets you compute the coefficients cleanly, but only completeness guarantees the series actually adds back up to f and not to something less.
This is the deep theorem that makes the whole enterprise legitimate. Without completeness, you could expand a function, sum the series, and land somewhere other than where you started, and separation of variables for partial differential equations would be unjustified. The Sturm-Liouville framework guarantees completeness for the regular case, which is precisely why Fourier series, Bessel series, and Legendre series can be trusted to represent arbitrary data. Completeness is also subtle: convergence is in the averaged mean-square sense, so it tolerates disagreement at isolated points and at jumps.
The sines {sin(n pi x / L)} are complete on [0, L]: every square-integrable function with zero ends can be reconstructed from them. If they were missing even one mode, say sin(2 pi x / L), then that very function would be orthogonal to all the rest and could never be rebuilt — the set would have a gap and fail to be complete.
Completeness means no function is orthogonal to the whole set; remove a single eigenfunction and that gap reappears.
Completeness is convergence in the mean-square (averaged) sense, which is weaker than pointwise convergence. A complete series can still disagree with f at individual points or fail to converge everywhere; the energy of the error, not its value at every point, is what goes to zero.