Let $V=\mathbb{R}[x]$ be the vector space of all polynomials with coefficients in $\mathbb{R}$ with the inner product defined by $\langle f,g\rangle=\int_{0}^{1}f(t)g(t)dt$.
Let $W$ the subspace of $\mathcal{L}(V)$ generated by those linear operators which have no adjoint. What is the dimension of $W$?
I know that the differentiation operator belongs to $W$, which means that $W$ is a nonempty set.
I would appreciate you help.