Proof of the Cauchy-Schwarz Inequality

$${\large \text{Cauchy-Schwarz Inequality}}$$ $${\lvert \langle f, g \rangle \rvert}^2 \leq \langle f, f \rangle \langle g, g \rangle$$

For two vectors $f$ and $g$ in an inner product space, pick an arbitrary real scalar $\lambda$ and write:

$$\lvert \langle \lambda f + g, \lambda f + g \rangle \rvert \geq 0$$

which is true because it’s a length squared, so

$$\lvert \langle \lambda f + g, \lambda f + g \rangle \rvert = {\lambda}^2 \langle f, f \rangle + 2\lambda\lvert\langle f, g \rangle\rvert + \langle g, g \rangle \geq 0$$

The expression on the left is quadratic in $\lambda$, and since it’s always greater than or equal to zero, it has zero real roots (i.e. entirely above the x axis) or one real root (i.e. just touching the x axis).

It can’t have two real roots, because that would require that the expression be negative for some $\lambda$ (i.e. go underneath the x axis), and that can’t be the case since it’s a length squared.

Altogether, this means that the quadratic’s discriminant is less than or equal to zero:

$$b^2 - 4ac \leq 0$$ $$4 {\lambda}^2 {\lvert\langle f, g \rangle\rvert}^2 - 4 {\lambda}^2 \langle f, f \rangle \langle g, g \rangle \leq 0$$ $$4 {\lambda}^2 {\lvert\langle f, g \rangle\rvert}^2 \leq 4 {\lambda}^2 \langle f, f \rangle \langle g, g \rangle$$ $${\lvert\langle f, g \rangle\rvert}^2 \leq \langle f, f \rangle \langle g, g \rangle$$

Equality occurs when $f$ and $g$ are linearly dependent. This means $f = \lambda g$ for some scalar $\lambda$, and we can show this directly by writing:

$${\lvert \langle f, g \rangle \rvert}^2 ={\lvert\langle \lambda g, g \rangle \rvert}^2 = {\lvert \lambda \rvert}^2 {\langle g, g \rangle}^2 = {\lvert \lambda \rvert}^2 \langle g, g \rangle \langle g, g \rangle = \langle f, f \rangle \langle g, g \rangle$$ $${\lvert \langle f, g \rangle \rvert}^2 = \langle f, f \rangle \langle g, g \rangle$$

See some similar/alternate proofs and applications of the Cauchy-Schwarz Inequality here: http://cnx.org/content/m10757/latest/

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