Relation between scalar product and distance in $L^2$ hilbert spaces

Question

I think that scalar product and distance between two vector is somehow related in spaces $\mathbb{C}^n$, but I cannot understand how:

• Eucledean distance: $ |u-v| = \sqrt{\sum_{i} (u_i - v_i)^2 } = \sqrt{ \sum_{i} u_i^2 + v_i^2 - 2v_iu_i} = \sqrt{ |u| + |v| -2\langle v, u \rangle} $
• Scalar product: $ \langle u,v \rangle = \sum_{i} u_i^*v_i$

Moreover, on some lecture notes, I have found that, for unit vectors, the real part of their inner product equals $1-\frac{1}{2}|v-u|^2$. Sorry, but I cannot understand the relationship between the two.

Answer 1

The distance in any inner product space is given by $$|u-v| = \sqrt{ \langle u-v,u-v \rangle }.$$

Answer 2

A scalar product $\langle \cdot\, , \cdot \rangle$ on a space $X$ allows you to define a norm by $\|x\| = \sqrt{\langle x\, , x\rangle} $. The norm in turns allows you to define a distance by $d(x,y) = \|x-y\|$.