find the orthogonal projection of 1

Question

Let $V = C[0,π/2]$ and consider the inner product $\langle ·, · \rangle : V × V → R$ defined by

${\langle f,g\rangle} =\int_0^{\pi/2}{f(x)g(x)} dx$

for all $f, g ∈ V$. Find the orthogonal projection of $1$ onto $\text{span}({\cos(x),\sin(x)})$.

$\langle\cos(x), \sin(x)\rangle$

$\langle\cos(x), \cos(x)\rangle$

$\langle\sin(x), \sin(x)\rangle$

I'm pretty sure we need to use those to find the orthogonal projection of $1$. However, I am not quite sure how to do it

Can anyone help?

Answer 1

Applying the Gram-Schmidt process to $\{\sin,\cos\}$, you will get that an orthonormal basis of $\operatorname{span}\bigl(\{\sin,\cos\}\bigr)$ is $\{f_1,f_2\}$, with$$f_1(x)=\frac2{\sqrt\pi}\sin(x)\text{ and }f_2(x)=\frac2{\sqrt{\pi(\pi^2-4)}}\bigl(\pi\cos(x)-2\sin(x)\bigr).$$So, the orthogonal projection $p_1$ that you're after is $\langle1,f_1\rangle f_1+\langle1,f_2\rangle f_2$; in other words, it is the function$$p_1(x)=\frac{4\bigl(\sin (x)+\cos (x)\bigr)}{2+\pi}.$$