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I am reading through a short text on theta functions and am having trouble with a certain concept. The idea is that if $V$ is a finite dimensional vector space with a lattice $\Gamma$, then there is a dual space $V^*$ and a dual lattice $\Gamma'$ which is defined by: $x\in V^*$ such that $<x,y>\in \mathbb{Z}$ for all $y\in \Gamma$.

I don't really have a problem with this, however there is this passage from the text that I can't quite wrap my head around.

Suppose $V$ has a symmetric, bilinear form, $B(x,y) = x\cdot y$ which is positive and non-degenerate. This simplifies the situation above as $V^*$ can be identified with $V$ using the form, and $\Gamma'$ is a lattice in $V$.

Why do these conditions on $V$ make $\Gamma'$ a lattice on $V$?

Since $V$ is finite dimensional then then $B$ is a isomorphism between $V$ and $V^*$. I am guessing it has to do with this fact but I don't know where to go next. Thanks for your help!

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    $\begingroup$ Are you happy that $\Gamma'$ is a lattice in $V^*$? As $B$ defines an isomorphism between $V$ and $V^*$, surely you should be happy that the image of $\Gamma'$ is a lattice in $V$? $\endgroup$ – Lord Shark the Unknown Aug 28 '17 at 15:15

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