I am having trouble completing the last step of my problem below:
Let $V= \beta [0,2] = \{ $All continuous function $f:[0,2] \rightarrow \mathbb{R} \}$. Let $W = span(1,x,x^2) \subset V$. Find orthonormal basis of W.
I defined the standard inner product to be $<f,g> = \int_{0}^{2} f(x)g(x) dx$. Let $w_1=1,w_2=x,w_3=x^2$. Then I used the graham-schmidt formula to determine the orthogonal basis, and I get the following:
$$v_1=1,v_2=x-1,v_3=x^2-2x+\frac{2}{3}$$ (I would appreciate it if someone could confirm that this is a valid orthogonal basis because I am not 100% sure that I applied graham-schmidt properly).
Now my question is how should I convert the following orthogonal basis $v_1,v_2,v_3$ into an orthonormal basis? I am not sure how to complete this final step because $v_1,v_2,v_3$ are not exactly vectors that I could just normalize. Thank you very much and any help will be greatly appreciated. Thank you.
If you normalise the orthogonal basis, you will get an orthonormal basis. See also here: Finding orthonormal basis using orthogonal basis.
Recall that, given any vector $x \neq 0$, its normalisation is: $$ \frac{x}{\| x \|} $$ Moreover, recall that the norm induced by an inner product is: $$ \| x \| := \, \sqrt{\langle x, x\rangle} $$