Hi!

The map $\displaystyle \langle ; \rangle : X \times X \to \mathbb{K}$ (it is a dot product) is continuous.

Proof:

$\displaystyle |\langle x,y \rangle - \langle x' , y' \rangle|$

$\displaystyle = |\langle x-x' , y \rangle - \langle x' , y'-y \rangle |$

$\displaystyle \le |\langle x-x' , y \rangle| - |\langle x' , y'-y \rangle |$

$\displaystyle \le ||y||\cdot ||x-x'|| + ||x'||\cdot ||y'-y|| $

So what is the argument to imply that the dot product continuous?

Any help would be much appreciated. Thank you!

Rapha