Gram-Schmidt process for polynomial space

Question

I'm looking for a orthonormal basis to 2 polynomials $f,g ∈ \Bbb R_2[x]$
we will define inner product space:
$<f,g>=f\left(0\right)\cdot g\left(0\right)+f\left(1\right)\cdot \:g\left(1\right)+f\left(2\right)\cdot \:\:g\left(2\right)$
I know how to do Gram-Schmidt process to vectors
But how do you make it for polynomial?
** For example in vectors it is easy to see vectors like $\vec v=(0,1),\vec u=(1,0)$
And see that all vectors in $\Bbb R_2$ can appear as linear combination of $\vec v,\vec u$
**
But how does it work with polynomials?
Thank you.

Answer 1

Definition: A vectorial space $V$ over some field $F$ is a vectorial space with inner product if there exists and function $<.>: V \times V \longrightarrow F$ su h that for any $u,v,w\in V$ and $\alpha \in F$:

$<v,v> \, >0$

$<v,v>=0$ iff $v=0$

$<u,v>=\overline{<u,v>}$

$<u,v+w>=<u,v>+<u,w>$

$<u+w,v>=<u,v>+<w,v>$

$<\alpha u,v>=\alpha <u,v>$

Definition: Let $V$ a vector space with inner product over some field $F$,given $v\in V$ we define the norm of $v$ by $||v||=\sqrt{<v,v>}$

${\displaystyle \mathrm {proj} _{\mathbf {u} }\,(\mathbf {v} )={\langle \mathbf {u} ,\mathbf {v} \rangle \over \langle \mathbf {u} ,\mathbf {u} \rangle }{\mathbf {u} },}$

Finally given a finite vector space $V$ over $F$, with inner product, then $V$ have a ortonormal basis.

Construction

Let $S= \lbrace v_1,v_2, \cdots v_n \rbrace $ basis of $V$.Pick $v_1\in S$. $u_1=v_1$

$u_2=v_2-Proj_{u_1}(v_2)$

$u_3=v_3-Proj_{u_1}(v_3)-Proy_{u_2}(v_3)$

and notice

$u_n=v_n-\sum_{j=1}^{n}{Proy_{u_j}(v_n)}$

And consider $e_i=\frac{u_i}{||U_i||}$

Then the set $U=\lbrace e_1,e_2, \cdots, e_n \rbrace$ is a ortonormal basis of $V$.

In your case you had that the polinomials are a vector space with inner product given by $<p,q>=\sum_{i=0}^{n}{a_ib_i}$ where $a_i,b_i$ are the coeficients of your polinomial(you can comprove it).

Now you only had to follow the process.

I hope that my comment helps you