(a) Let $X$ be a normed vector space and let $Y$ be a Banach space. Show that the space of bounded linear operators $\mathcal{B}(X, Y)$ is a Banach space.

(b) Let $X$ and $Y$ be Banach spaces, and let $D \subset X$ be a dense linear subspace. Prove that a bounded linear map $T: D \rightarrow Y$ can be extended uniquely to a bounded linear map $T: X \rightarrow Y$ with the same operator norm. Is the claim also true if one of $X$ and $Y$ is not complete?

(c) Let $X$ be a normed vector space. Let $\left(x_{n}\right)$ be a sequence in $X$ such that

$$\sum_{n=1}^{\infty}\left|f\left(x_{n}\right)\right|<\infty \quad \forall f \in X^{*}$$

Prove that there is a constant $C$ such that

$$\sum_{n=1}^{\infty}\left|f\left(x_{n}\right)\right| \leqslant C\|f\| \quad \forall f \in X^{*}$$