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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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    $\begingroup$ Can you show that there is an injective linear transformation between a space $V$ and its dual $V^*$? If you can, then it's just one more line to answer your question. $\endgroup$ – Raskolnikov Jul 24 '12 at 6:53
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    $\begingroup$ Define the evaluation maps $\phi_v: V^*\to F:f\mapsto f(v)$ for each $v\in V$. Note that $\phi_v\in (V^*)^*$. Then consider $V\to V^{**}:v\mapsto \phi_v$. $\endgroup$ – anon Jul 24 '12 at 6:54
  • $\begingroup$ @ Raskolinikov..Is there any way you can help me get started with showing that there is an injective linear transformation between a space $V$ and its dual $V^{*}$, I am having a hard time understanding these concepts. Thanks. $\endgroup$ – Melky Jul 24 '12 at 8:13
  • $\begingroup$ Just try to construct some nontrivial linear map $V\to V^*$...there is a very obvious one! Then check injectivity. $\endgroup$ – wildildildlife Jul 24 '12 at 10:25
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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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  • $\begingroup$ The mathematicians do too :) $\endgroup$ – wildildildlife Jul 24 '12 at 10:23
  • $\begingroup$ Some do, for sure. Hopefully not many get to talk about bras and kets ;) $\endgroup$ – Martin Argerami Jul 24 '12 at 10:47
  • $\begingroup$ 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. $\endgroup$ – Melky Aug 3 '12 at 23:48

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