The Dual Space of a Real Normed Space and the Dual Norm

In the setting of The Real Numbers: Standing Notation and Background, let $E$ with norm $\lVert\cdot\rVert$ be a real normed space.

1. (Bounded linear functionals)¶ A linear functional on $E$ is a map $\ell:E\to\mathbb{R}$ with $\ell(u+v)=\ell(u)+\ell(v)$ and $\ell(\lambda v)=\lambda\,\ell(v)$ for all $u,v\in E$ and $\lambda\in\mathbb{R}$. A real number $C$ is a bound for $\ell$ if $|\ell(v)|\le C\,\lVert v\rVert$ for every $v\in E$, and $\ell$ is bounded if it has a bound.

2. (Dual space)¶ The dual space $E^{*}$ of $E$ is the set of bounded linear functionals on $E$, with the operations

These maps belong to $E^{*}$. They are linear, addition and multiplication in $\mathbb{R}$ being associative, commutative and distributive. If $C$ and $C'$ are bounds for $\ell$ and $m$, then $C+C'$ is a bound for $\ell+m$, since $|\ell(v)+m(v)|\le|\ell(v)|+|m(v)|\le(C+C')\lVert v\rVert$ by the triangle inequality for $|\cdot|$; and $|\lambda|\,C$ is a bound for $\lambda\ell$, since $|\lambda\,\ell(v)|=|\lambda|\,|\ell(v)|\le|\lambda|\,C\,\lVert v\rVert$ because $0\le|\lambda|$.

3. (Dual norm)¶ For $\ell\in E^{*}$ let $B_{\ell}$ be the set of bounds $C$ for $\ell$ with $0\le C$. It is nonempty: if $C$ is a bound for $\ell$ and $0\le C$ then $C\in B_{\ell}$, and if $C<0$ then $|\ell(v)|\le C\lVert v\rVert\le0=0\cdot\lVert v\rVert$ for every $v\in E$, since $0\le\lVert v\rVert$ by Real Normed Space and Real Banach Space §norm, so $0\in B_{\ell}$. It is bounded below by $0$. The dual norm of $\ell$ is the greatest lower bound

which exists by The Real Numbers: Standing Notation and Background §bounds.