Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality says the inner product of two vectors cannot exceed the product of their lengths. In familiar geometry this is the statement that the cosine of the angle between them lies between -1 and 1: the dot product equals |u| |v| cos(theta), and a cosine never has magnitude greater than one. Lifted to functions, the ‘dot product’ becomes the integral of f times g, and the ‘lengths’ become L^2 norms.
In the L^2 setting: for square-integrable f and g, the absolute value of the integral of f times the conjugate of g is at most ||f||_2 times ||g||_2. This is exactly Hölder's inequality at the self-conjugate exponent p = q = 2. More abstractly, in any inner-product space the inequality |<u, v>| <= ||u|| ||v|| holds, and it is what guarantees the inner-product norm satisfies the triangle inequality.
Equality holds if and only if f and g are linearly dependent — one is a scalar multiple of the other almost everywhere (allowing zero). The inequality is the workhorse behind Bessel's inequality, the definition of angle in Hilbert space, and countless estimates; whenever you must bound a single product integral with no slack to spare, Cauchy–Schwarz is usually the first thing to try.
For sequences (the l^2 case): |a1 b1 + a2 b2 + ...| <= sqrt(sum a_n^2) times sqrt(sum b_n^2). With a = (1, 1, 0, ...) and b = (1, -1, 0, ...): the left side is |1 - 1| = 0, the right side is sqrt(2) times sqrt(2) = 2. Here the slack is maximal because a and b are orthogonal.
Orthogonal vectors give a zero inner product — the extreme of the bound.