\[ \newcommand{\C}{\mathbb{C}} \newcommand{\cont}{\operatorname{\mathsf{C}}} \newcommand{\haar}{\mathsf{m}} \newcommand{\B}{\mathscr{B}} \newcommand{\D}{\mathscr{D}} \newcommand{\P}{\mathcal{P}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\sph}{\mathsf{S}^\mathsf{1}} \newcommand{\g}{>} \newcommand{\l}{<} \newcommand{\intd}{\,\mathsf{d}} \newcommand{\Re}{\mathsf{Re}} \newcommand{\area}{\mathop{\mathsf{Area}}} \newcommand{\met}{\mathop{\mathsf{d}}} \newcommand{\emptyset}{\varnothing} \DeclareMathOperator{\borel}{\mathsf{Bor}} \DeclareMathOperator{\baire}{\mathsf{Baire}} \newcommand{\symdiff}{\mathop\triangle} \newcommand{\lp}[1][2]{\operatorname{\mathscr{L}^{\mathsf{#1}}}} \newcommand{\Lp}[1][2]{\operatorname{\mathsf{L}^{\!\mathsf{#1}}}} \]

1. Let $p,q > 1$ satisfy \[ \tfrac{1}{p} + \tfrac{1}{q} = 1 \] and fix a measure space $(X,\B,\mu)$. Fix $f$ in $\Lp[p](X,\B,\mu)$ and define \[ \xi_f : \Lp[q](X,\B,\mu) \to \R \] by \[ \xi_f(g) = \int f \cdot g \intd \mu \] for all $g \in \Lp[q](X,\B,\mu)$. Verify that $\xi_f$ is well-defined and linear.
2. Let $\mu$ be Lebesgue measure on $\R$.
   1. If $f \in \Lp[1](\R,\borel(\R),\mu)$ must \[ g(x) = x f(x) \] belong to $\Lp[1](\R,\borel(\R),\mu)$?
   2. If $f : \R \to \R$ is differentiable must $f'$ belong to $\Lp[1](\R,\borel(\R),\mu)$?