questions about advanced Linear algebra

Question

I have some questions about advanced linear algebra. Let $V$ be a vector space and $V^*$ be the dual space.

1. Why is $V=V^*$ called non-natural, and $V=V^{**}$ called natural?

2. $V$ is a vector space with dimension $\dim V=\infty$. Give an example where $V$ is not equal to $V*$ and $V$ is not equal to $V**$ ?

3. If $\langle .,.\rangle$ is a non-degenerate scalar product on $V$, and $$\varphi: V \to V^{*}: v \mapsto L_v(w) = \langle w,v \rangle$$ is not an isomorphism. Give an example.

Answer 1

First, $V\ne V^*$ and $V\ne V^{**}$ always, but in finite dimension $V\cong V^*$ and $V\cong V^{**}$ trivially because all have the same dimension. But in the second case there is a natural (definable independently of basis) isomorphism, namely: $$v\longmapsto e_v,\qquad e_v(f) = f(v),\qquad v\in V, f\in V^*.$$ Now, check what happens when $\dim V = \infty$. For example, $V = \Bbb R\oplus\Bbb R\oplus\cdots =$ space of eventually zero real sequences.