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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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    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.2012-07-24
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    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 $V\to V^{**}:v\mapsto \phi_v$.2012-07-24
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    @ 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.2012-07-24
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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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