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

  • 2
    $\begingroup$ 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. $\endgroup$ Jul 24, 2012 at 6:53
  • 5
    $\begingroup$ 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, 2012 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, 2012 at 8:13
  • $\begingroup$ Just try to construct some nontrivial linear map $V\to V^*$...there is a very obvious one! Then check injectivity. $\endgroup$ Jul 24, 2012 at 10:25

1 Answer 1


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.

  • $\begingroup$ The mathematicians do too :) $\endgroup$ Jul 24, 2012 at 10:23
  • $\begingroup$ Some do, for sure. Hopefully not many get to talk about bras and kets ;) $\endgroup$ Jul 24, 2012 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, 2012 at 23:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.