Let $V$, $V^{'}$ be vector spaces over a field $K$ and let $V \times V^{'} \longrightarrow K$ be a bilinear map with left and right kernel $W$, $W^{'}$ respectively. If $V^{'}/W^{'}$ is finite dimensional, then according to theorem 6.4 p. 145 in Lang's Algebra, $V^{'}/W^{'}$ is isomorphic with $\left(V/W\right)^{\vee}$. I understand the arguments based on the two induced injective homomorphisms $V/W \longrightarrow (V^{'}/W^{'})^{\vee}$ and $V^{'}/W^{'} \longrightarrow (V/W)^{\vee}$. What i can't see is why these are inverses of each other, as is stated in the last sentence of the proof. Any insights?
Duality theorem of vector spaces
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abstract-algebra
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0I don't understand either how these homomorphisms could be inverses of each other since the range of one is not the domain of the other. Anyway I think this is not needed to prove the theorem: if you have an injective homomorphism between two vector spaces of equal finite dimension, then it's an isomorphism. – 2011-09-11