Prove that a projector is continuous using Cauchy-Schwarz's inequality

Question

Given a vector $a$ in an Euclidean Space with $a\cdot a = 1$ ($\cdot$ = scalar product), then $P(b) = (a \cdot b)a$ defines the orthogonal projection $P$ on vector $a$.

How do you show that $P$ is continuous using Cauchy-Schwarz's inequality?

Thanks for your help.

Answer 1

This is really simple: $$ Px - Px_0 = (a\cdot x)a - (a\cdot x_0)a = \left(a\cdot x - a\cdot x_0\right)a = \big(a\cdot(x-x_0)\big)a. $$ Hence, $$ \|Px - Px_0\| = |a\cdot(x-x_0)|\|a\| = |a\cdot(x-x_0)|. $$ Now, use Cauchy-Schwarz.