How do we prove this:

Btw, here are the properties of inner products:

**<u,v>=<v,u>** (

*symmetry property*)

**<u+v,w>=<u,w>+<v,w>** (a

*dditivity property*)

**<ku,v>=k<u,v>** (

*homogeneity property*)

**<v,v> ≥ ****0 and <v,v>=0 if and only if v=0** (

*positivity property*)

So, I

*think* to show that a mapping is linear we must show that it satisfies the additivity and homogeneity conditions. But I don't know how to prove these conditions here but here's what I think so far

So, for vectors $\displaystyle x_1,x_2$ and scalars a and b,

**T(x)= <x,u> v** becomes

$\displaystyle T(ax_1+bx_2)= \left\langle ax_1+bx_2,u \right\rangle v$

By the additivity property we get

$\displaystyle = (\left\langle ax_1,u \right\rangle + \left\langle bx_2,u \right\rangle )v$

I don't know what to do after this point. Can anyone help?