orthogonal vectors in pre-Hilbert space

[waytogo]

Hi,

I must proove the following statement: in any pre-Hilbert space \(\displaystyle x\perp y \Leftrightarrow \forall k \in K\) \(\displaystyle \mid\mid x \mid\mid \leq \mid\mid x +ky \mid\mid\), where \(\displaystyle K\) is set of real or complex numbers.
Any ideas?

 

[MATHNEM]

Hi,
In any pre-Hilbert space \(\displaystyle x\perp y \Leftrightarrow \forall k \in K\) \(\displaystyle \mid\mid x \mid\mid \leq \mid\mid x +ky \mid\mid\), where \(\displaystyle K\) is set of real or complex numbers.

For the first implication it may be useful the fact that \(\displaystyle x\perp y\implies \mid\mid x+y \mid\mid ^2=\mid\mid x \mid\mid ^2+\mid\mid y \mid\mid ^2\)

 

[waytogo]

For the first implication it may be useful the fact that \(\displaystyle x\perp y\implies \mid\mid x+y \mid\mid ^2=\mid\mid x \mid\mid ^2+\mid\mid y \mid\mid ^2\)

yes, thanks, this implication is easy to proove. but it seems a bit tricky for me to proove the other one.