Let $X$ and $Y$ be Banach spaces. Define what it means for a linear operator $T: X \rightarrow Y$ to be compact. For a linear operator $T: X \rightarrow X$, define the spectrum, point spectrum, and resolvent set of $T$.

Now let $H$ be a complex Hilbert space. Define what it means for a linear operator $T: H \rightarrow H$ to be self-adjoint. Suppose $e_{1}, e_{2}, \ldots$ is an orthonormal basis for $H$. Define a linear operator $T: H \rightarrow H$ by setting $T e_{i}=\frac{1}{i} e_{i}$. Is $T$ compact? Is $T$ self-adjoint? Justify your answers. Describe, with proof, the spectrum, point spectrum, and resolvent set of $T$.