Exercise 11.7 (The Second Dual Space)

Question

Let $V$ be a vector space. For each $v \in V$, define a linear functional $\xi(v): V^{*}	o \mathbb{R}$ by
\begin{equation*}
  \xi(v)(\omega) = \omega(v) 	ext{ for }\omega \in V^{*}.
\end{equation*}

\begin{enumerate}[label=(\alph*)]
\item For any $v \in V$, show that $\xi(v)(\omega)$ depends linearly on $\omega$, so $\xi(v)\in V^{**}$.
  \item Show that the map $\xi:V 	o V^{**}$ is linear.
\end{enumerate}

Elu Thingol · Accepted Answer

Fix $v \in V$. Let $\omega, \omega^{\prime} \in V^{*}$ and $c \in \mathbb{R}$ be arbitrary. We then have
$$\xi(v)(c \omega + \omega^{\prime}) = (c\omega+\omega^{\prime})(v) = c \omega(v)+\omega^{\prime}(v) = c\xi(v)(\omega) + \xi(v)(\omega^{\prime}).$$
Now let $v,v^{\prime}$ and $c \in \mathbb{R}$ be arbitrary. We then have, for all inputs $\omega \in V^{*}$,
\begin{equation*}
  \xi(cv + v^{\prime})(\omega) = \omega(cv+v^{\prime}) = c\omega(v) + \omega(v^{\prime}) = c\xi(v)(\omega) + \xi(v^{\prime})(\omega).
\end{equation*}

Exercise 11.7 (The Second Dual Space)

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Exercise 11.7 (The Second Dual Space)

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