Let $X, Y$ be Banach spaces and let $\mathcal{B}(X, Y)$ denote the space of bounded linear operators $T: X \rightarrow Y$.

(a) Define what it means for a bounded linear operator $T: X \rightarrow Y$ to be compact. Let $T_{i}: X \rightarrow Y$ be linear operators with finite rank, i.e., $T_{i}(X)$ is finite-dimensional. Assume that the sequence $T_{i}$ converges to $T$ in $\mathcal{B}(X, Y)$. Show that $T$ is compact.

(b) Let $T: X \rightarrow Y$ be compact. Show that the dual map $T^{*}: Y^{*} \rightarrow X^{*}$ is compact. [Hint: You may use the Arzelà-Ascoli theorem.]

(c) Let $X$ be a Hilbert space and let $T: X \rightarrow X$ be a compact operator. Let $\left(\lambda_{j}\right)$ be an infinite sequence of eigenvalues of $T$ with eigenvectors $x_{j}$. Assume that the eigenvectors are orthogonal to each other. Show that $\lambda_{j} \rightarrow 0$.