So this is the last question of the problem, it asks: Let $C^0$ be the vector space of continuous functions on the interval $[−2, 2]$. Consider this an inner product space with the inner product $<f, g> = $$f_{X,Y}(x,y) 1 = \int_{ -2 }^{2} f \, g \, dx $
Find any non-zero element in $V^⊥$ where $V$ is the subspace of $C^0$ spanned by ${\cos x, \cos^2 x, \cos^3 x, \cos^4 x, \cdots }.$
So I am thinking maybe to do Gram-Schmidt to kind of find $V^⊥$? But I think it is going to be hard though I have not thought it through ... Do I have any easier way to find any non-zero element in $V^⊥$ where $V$ is the subspace of $C^0$ spanned by ${\cos x, \cos^2 x, \cos^3 x, \cos^4 x, \cdots}.$?
Notice that for every $n\geq 1$, the function $\cos^n(x)$ is even. So if you take an odd function $f(x)$, the function $f(x)\cdot\cos^n(x)$ will be odd for every $n\geq 1$, and then, you'll have $\int_{-2}^2f(x)\cos^n(x)dx=0$ for every $n\geq 1$. In particular, you can take $\sin(x)$, which is an odd function (and is non-zero).