Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be a countably infinite set where the $x_i$ are pairwise distinct elements. We denote by $\mathscr{P}(E)$ the set of subsets of $E$. Let $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ be the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$. If $f \in \mathscr{B}(\mathscr{P}(E), \mathbb{R})$, we set $$\|f\| = \sup\{|f(A)|, \quad A \in \mathscr{P}(E)\}.$$ Show that $\|\cdot\|$ defines a norm on the vector space $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$.
Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be a countably infinite set where the $x_i$ are pairwise distinct elements. We denote by $\mathscr{P}(E)$ the set of subsets of $E$. Let $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ be the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$. If $f \in \mathscr{B}(\mathscr{P}(E), \mathbb{R})$, we set
$$\|f\| = \sup\{|f(A)|, \quad A \in \mathscr{P}(E)\}.$$
Show that $\|\cdot\|$ defines a norm on the vector space $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$.