Define the spectrum $\sigma(T)$ and the approximate point spectrum $\sigma_{\mathrm{ap}}(T)$ of a bounded linear operator $T$ on a Banach space. Prove that $\sigma_{\mathrm{ap}}(T) \subset \sigma(T)$ and that $\sigma(T)$ is a closed and bounded subset of $\mathbb{C}$. [You may assume without proof that the set of invertible operators is open.]

Let $T$ be a hermitian operator on a non-zero Hilbert space. Prove that $\sigma(T)$ is not empty

Let $K$ be a non-empty, compact subset of $\mathbb{C}$. Show that there is a bounded linear operator $T: \ell_{2} \rightarrow \ell_{2}$ with $\sigma(T)=K .$ [You may assume without proof that a compact metric space is separable.]