TITLE: Injective linear transformation from a vector space to the dual space of its dual space
QUESTION [7 upvotes]: I am currently trying to understand some concepts from some Linear Algebra. I seem to be having quite some difficulty understanding dual spaces and their dual spaces. I found this problem and was wondering how to get started on it.
Let $V$ be a vector space over the field $F$. Let $V^{*}$ be the dual space of $V$ and let $V^{**}$ be the dual space of $V^{*}$. Show that there is an injective linear transformation 
$\phi : V \rightarrow V^{**}$.

REPLY [3 votes]: The canonical answer is the one given by anon above. The point is that when you write $f(v)$, for $v\in V$, $f\in V^*$, you can see it as "$f$ acting on $v$", or you can also see it as "$v$ acting on $f$"; this second point of view defines the injection you are looking for. The physicists write $f(v)$ as $\langle f,v\rangle$ to emphasize this duality, and that's where the word "dual" comes from.