Cauchy-Schwarz inequality

The single most-used inequality in linear algebra says the inner product of two vectors can never exceed the product of their lengths: |<u,v>| <= ||u|| ||v||. Geometrically it is just |cos(angle)| <= 1, but it holds in any inner product space — including infinite-dimensional spaces of functions where no picture is available.

The clean proof needs no coordinates. If v = 0 both sides are 0. Otherwise let w = u - (<u,v>/<v,v>) v, the part of u orthogonal to v. Expanding 0 <= <w,w> and simplifying gives exactly ||u||^2 ||v||^2 >= |<u,v>|^2. Taking square roots finishes it. The trick is that subtracting off the projection leaves something with nonnegative self-product.

Equality holds if and only if u and v are parallel (linearly dependent). That is the residual w being zero — u is entirely its own projection onto the line through v, with nothing left over.

Its reach is enormous: it proves the triangle inequality (so ||.|| really is a norm), it underlies the definition of angle in abstract spaces, and in probability it becomes the bound between covariance and the product of standard deviations. Master this proof and a dozen others fall out.