# Math Help - Why does this proof imply 'continuous'?

1. ## Why does this proof imply 'continuous'?

Hi!

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

Proof:

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

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

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

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

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

Any help would be much appreciated. Thank you!

Rapha

2. Hello,

To prove that <,> is continuous, you just have to check that $\forall (x,y)\in X^2$, there exists M such that $\langle x,y\rangle\leq M \cdot \|x\|_X\cdot\|y\|_X$

And by the Cauchy-Schwarz inequality, this is obvious :
Spoiler:
$M=1$

3. Hi Moo.

Alright then

Thank you very much!

best regards
Rapha