Consider all polynomials $\mathbb R[x]$ and the subspace of polynomials of degree $0$, which we will refer to by the letter $U$. Is this subspace closed with respect to the inner product: $\langle f,g \rangle = \int_0^1 f(x)g(x)dx$
My teacher says that they must not be closed, since if they were, then $(U^{\perp})^{\perp} = U$, but $(U^\perp)^{\perp} = \mathbb R[x]$. I'm having a hard time believing this without seeing an actual example of a limit point of $U$ that is not in $U$.
So if/since it is not closed, could someone please provide an example of one of its limit points not in the space itself?