(a) Let $T: X \rightarrow Y$ be a linear map between normed spaces. What does it mean to say that $T$ is bounded? Show that $T$ is bounded if and only if $T$ is continuous. Define the operator norm of $T$ and show that the set $\mathcal{B}(X, Y)$ of all bounded, linear maps from $X$ to $Y$ is a normed space in the operator norm.

(b) For each of the following linear maps $T$, determine if $T$ is bounded. When $T$ is bounded, compute its operator norm and establish whether $T$ is compact. Justify your answers. Here $\|f\|_{\infty}=\sup _{t \in[0,1]}|f(t)|$ for $f \in C[0,1]$ and $\|f\|=\|f\|_{\infty}+\left\|f^{\prime}\right\|_{\infty}$ for $f \in C^{1}[0,1]$.

(i) $T:\left(C^{1}[0,1],\|\cdot\|_{\infty}\right) \rightarrow\left(C^{1}[0,1],\|\cdot\|\right), T(f)=f$.

(ii) $T:\left(C^{1}[0,1],\|\cdot\|\right) \rightarrow\left(C[0,1],\|\cdot\|_{\infty}\right), T(f)=f$.

(iii) $T:\left(C^{1}[0,1],\|\cdot\|\right) \rightarrow\left(C[0,1],\|\cdot\|_{\infty}\right), T(f)=f^{\prime}$.

(iv) $T:\left(C[0,1],\|\cdot\|_{\infty}\right) \rightarrow \mathbb{R}, T(f)=\int_{0}^{1} f(t) h(t) d t$, where $h$ is a given element of $C[0,1]$. [Hint: Consider first the case that $h(x) \neq 0$ for every $x \in[0,1]$, and apply $T$ to a suitable function. In the general case apply $T$ to a suitable sequence of functions.]