In order to check wheter a transformation is linear and bounded i need to show that
$T(ax +by) = aT(x) + bT(y)$
But when the examples get harder, I have trouble doing so. For instance the following three questions stumpled me. I am only used to proving linearity to mappings on the form $x \mapsto f(x)$.
b) $T: \ \mathbb{R} \to \mathbb{R}, \ \ t^3 \mapsto 3t^2$
This is just derivation right? And derivation is a linear and bounded transformation..
d) $T: \ P_n(\mathbb{R}) \to P_n(\mathbb{R}), \ \ \sum_{j=0}^n a_j x^j \mapsto \sum_{j=1}^n j a_j x^{j-1}$
with $\| \sum a_j x^j\|_{P_{n}(\mathbb{R})} = \|(a_0, \ldots, a_n)\|_{\mathbb{R}^{n+1}}$.
f) $C^1([0,1],\mathbb{R}) \cap BC([0,1],\mathbb{R})\to BC([0,1,\mathbb{R}]), \qquad f \mapsto f'$ where the $BC([0,1],\mathbb{R})$-norm on the dense subspace $C^1([0,1],\mathbb{R})$ of $BC([0,1],\mathbb{R})$