Bessel's inequality

Suppose you project a vector onto some of the axes of a coordinate system. The shadow it casts can never be longer than the vector itself — you only ever capture part of its length. Bessel's inequality is this obvious geometric fact, stated for functions and any orthonormal family: the total energy stored in the coefficients cannot exceed the energy of the function.

Let (e_n) be an orthonormal system in an inner product space and let c_n = inner product of f with e_n be the generalized Fourier coefficients. Then sum of |c_n|^2 <= the squared norm of f. The proof is one line: the difference between f and its partial projection is orthogonal to that projection, so by the Pythagorean theorem the projection's energy plus the remainder's energy equals f's energy — dropping the non-negative remainder gives the inequality.

Two consequences are immediate and useful. First, the coefficient series sum |c_n|^2 converges (its partial sums are bounded by the norm), which forces c_n -> 0. Second, equality holds for every f exactly when the orthonormal system is complete — and then Bessel's inequality upgrades to Parseval's identity. So Bessel is the always-true half; completeness is what closes the gap.