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Eh, I don't quite understand the first question. Can someone explain it?

And for the second question, can I say that they have the same dimension. And since the kernel is ${0}$, it is injective. Thus I've solved the second question. What's more, what is the relationship between the dual space of the dual space, which is $V^{**}$ and $V^*$?

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    $\begingroup$ Do you understand what $\lambda_v$ is? If so, what's unclear about the first part of the question. If not, what are you saying has a kernel of $\{0\}$? $\endgroup$ – Milo Brandt Apr 16 '16 at 1:21

The isomorphism is simply this: \begin{align*} V&\longrightarrow V^{**}\\ v&\longmapsto (\lambda_v\colon f\in V^*\mapsto f(v)) \end{align*} In other words, it associates to each vector $v\in V$, the evaluation morphism at $v$.

It's an isomorphism because both spaces have the same dimension, and $\lambda$ is injective.


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