# $V^⊥$ where V is spanned by ${\cos x, \cos^2 x, \cos^3 x, \cos^4 x, . . . }$

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 $$ = f_{X,Y}(x,y) 1 = \int_{ -2 }^{2} f \, g \, dx$$

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 $<\cos x , \cos ^2 x , \cdots, \cos^n x> = <\cos x , \cos 2 x , \cdots, \cos n x>$. Mar 23, 2016 at 1:42

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).