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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 $\phi : V \rightarrow V^{**}$.

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    Just try to construct *some* nontrivial linear map $V\to V^*$...there is a very obvious one! Then check injectivity.2012-07-24

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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.

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    I thank everyone for there help. Something is wrong either with my computer or the page in my browser, where I cannot click on this answer was helpful or give feedback. I'll check in to it.2012-08-03