Calculating a function using orthogonal projection of vector on a subspace

Question

Let $V$ be a closed subspace of $L^2[0,1]$ and $f,g \in L^2[0,1]$ be given by $f(x)=x,g(x)=x^2.$

If $V^{\perp}=span\{f\}$ and $Pg$ be the the orthogonal projection of $g$ on $V,$

then $(g-Pg)(x), x\in$ [0,1] is = ?

1. $\frac{3}{4}x$
2. $\frac{1}{4}x^2$
3. $\frac{3}{4}x^2$
4. $\frac{1}{4}x$

My thought:

First we have to find out $V$ and a orthonormal basis for it .Then using inner product on $L^2[0,1]$,I can calculate the orthogonal projection. But I don't know how to find V using the given conditions or does there exist any simple method to calculate orthogonal projection ? Please help.

Thanks.

Answer 1

By the Hilbert Projection Theorem, $g-Pg \in V^\perp$. Thus, $Pg=g-cf$ for some $c$.

$$0=\langle f,Pg \rangle=\langle f, g-cf \rangle = \int_0^1 x^3 \mathop{dx} + c \int_0^1 x^2 \mathop{dx} = \frac{1}{4} - \frac{c}{3}.$$

Thus, $c=3/4$, so $g-Pg=3f/4$.