# Math Help - Show that in an inner product space

1. ## Show that in an inner product space

Dear Colleagues,

Show that in an inner product space, $x\bot y$ if and only if $||x+\alpha y||\geq ||x||$ for all scalars $\alpha$.

Remark: I have already proved that $x\bot y$ implies that $||x+\alpha y||\geq ||x||$ for all scalars $\alpha$, it remains the converse.

Regards,

Raed.

2. Take the squares and expand the inner product.

3. Thank you very much for your reply, I have already done that and the result is
$2Re(\alpha {\bar} \langle x,y \rangle) + |\alpha|^{2}\langle y,y \rangle \geq 0$, where $Re$ denotes the real part of a complex number.

But what then?

4. You can divide by $|\alpha|^2$.