Gram Schmidt Process Using Orthonormal Vectors

Question

Given $v_1,v_2,...,v_n$ vector the process is:

$u_1=v_1\Rightarrow e_1=\frac{u_1}{\|u_1\|}$

$u_2=v_2-\frac{<v_2,u_1>}{<u_1,u_1>}u_1\Rightarrow e_2=\frac{u_2}{\|u_2\|}$

And so on, I am trying to derive how can one use only the orthonormal vectors for example for $v_2$:

$$u_2=v_2-\frac{<v_2,u_1>}{<u_1,u_1>}u_1=v_2-\frac{<v_2,u_1>}{\|u_1\|^2}u_1=v_2-\frac{<v_2,u_1>}{\|u_1\|}\frac{u_1}{\|u_1\|}=v_2-\frac{<v_2,u_1>}{\|u_1\|}e_1$$

How can we go from $$\frac{<v_2,u_1>}{\|u_1\|}\Rightarrow <v_2,e_1>$$ Which properties of the inner product can we use?

Answer 1

$\frac {\langle v_2, u_1 \rangle } {\|u_1\|}=\langle v_2, \frac {u_1} {\|u_1\|} \rangle$ because $ \langle a, cb \rangle =c\langle a, b \rangle $ for $c$ real. [Here $c=\frac 1{\|u_1\|}$]. Now just use the defintion of $e_1$.