Let $X$ be a Banach space and $T: X \rightarrow X$ a bounded linear map. Define the spectrum $\sigma(T)$, point spectrum $\sigma_{p}(T)$, resolvent $R_{T}(\lambda)$, and resolvent set $\rho(T)$. Show that the spectrum is a closed and bounded subset of $\mathbb{C}$. Is the point spectrum always closed? Justify your answer.

Now suppose $H$ is a Hilbert space, and $T: H \rightarrow H$ is self-adjoint. Show that the point spectrum $\sigma_{p}(T)$ is real.