I'm having trouble while doing this exercise, it says: 

> 
> In the vector space of the continuous functions in [0,1] with the inner product :
>  $$\langle f,g \rangle = \int\_{0}^{1}f(x)g(x)dx$$
>  a) Orthonormalize the set of functions $\left\{1,x\right\}$
> 
> 
> 

So I assumed $B$ as a base of the vector space $ \prod^{1} $ (the canonical base) $B = \left\{(1,0),(0,1)\right\} $ and applied Gram-Schmidt method to orthonormalize the functions.

The first one it's already normalized for that inner product $\vec{u\_{1}} = \frac{\vec{v\_{1}}}{\|\vec{v\_{1}}\|} = (1,0)$

So, for the second one $\vec{u\_{2}} = \frac{\vec{v\_{2}}-\langle \vec{v\_{2}},\vec{u\_{1}}\rangle \vec{u\_{1}}}{\|\vec{v\_{2}}-\langle \vec{v\_{2}},\vec{u\_{1}}\rangle \vec{u\_{1}}\|} = \frac{(0,1)-\left(\frac{1}{2},0\right)}{\|\left(-\frac{1}{2},1\right)\|}=(-6,12)$

But then $\|\vec{u\_{2}}\|=\sqrt{\langle \vec{u\_{2}},\vec{u\_{2}}\rangle} = \sqrt{12} \neq 1 $ ; so $\vec{u\_{2}}$ is not normal.

I don't know whether I'm all wrong assuming the base or doing it wrong with the Gram-Schmidt method, but I've been stuck for more than an hour with this :$.