I have this subspace of $C[-1,1]$ with inner product $\langle f,g\rangle = \int_{-1}^1f(x)\cdot \bar g(x)\,dx$: $$ E=\left\{f : \int_{-1}^0f=\int_{0}^1f\right\} $$
need to prove that $E^\bot=\{0\}$
2026-04-26 06:11:55.1777183915
Show that orthogonal complement is trivial
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Show that any element in $E^\perp$ must satisfy $$ f(x) = \cases{c & if $x>0$ \cr -c & if $x<0$} $$ for some constant $c$.