Q:

Norms with complex numbers over Hilbert Spaces

Let $H$ be a Hilbert space and $v,w \in H$ ans a be a scalar.
Prove that $\|v\| \leq \|v+aw\|$ for all scalar a iff (v,w)=0 for real and complex cases. 
I want to choose a such that $\bar{a}(v,w)$ will be real and negative but "close" to zero. 
Can i then say that $2Re(\bar{a}(v,w)) \leq |a|^2\|w\|^2$? How do i choose such an a? 

A:

If $(v,w) \ne 0$, then choose $\alpha$ so that $(v-\alpha w)\perp w$, which is to say that $\alpha=\frac{(v,w)}{(w,w)}$. Then $\alpha\ne 0$, which leads to 
$$
          \|v\|^2=\|(v-\alpha w)+\alpha w\|^2=\|v-\alpha w\|^2+\|\alpha w\|^2 > \|v-\alpha w\|^2.
$$
Conversely, if $(v,w)=0$, then the following holds for all $\alpha$:
$$
        \|v-\alpha w\|^2=\|v\|^2+\|\alpha w\|^2 \ge \|v\|^2
$$

