What kinds of structure are there of the elements in the double dual space $V^{**}$?

Question

Let $V=\mathbb{R}^3$ be a vector space over $\mathbb{R}$. The dual space of $V$ contains the elements of the form $ax+by+cz$. What kinds of structure are there of the elements in the double dual space $V^{**}$?

Answer 1

For finite-dimensional vector spaces, there exists a natural isomorphism between $V$ and its double dual. In other words, for all intents and purposes $V^{**}$ is the same as $V$, and its elements are simply the vectors of $V$.

More generally, $V^{**}$ by definition contains linear functions on $V^*$. So if $\phi:(x,y,z)\mapsto ax+by+cz$ is a member of $V^*$, then an element of $V^{**}$ maps $\phi$ to $\alpha a+\beta b+\gamma c$, where $\alpha$, $\beta$ and $\gamma$ parametrize the elements of $V^{**}$.

It is now obvious that you get a natural injection $V\to V^{**}$ by just setting $(\alpha,\beta,\gamma)$ to $(x,y,z)$. It's also obvious that for finite dimensions this gives an isomorphism.

To see that this doesn't hold for infinite-dimensional vector spaces, just note that vectors are linear combinations of finitely many basis vectors, but linear functions can be non-zero on infinitely many basis vectors. For example, for polynomials consider the linear function that maps $x^n$ to $1$ if $n$ is prime, and to $0$ otherwise. This is clearly a linear function, but $\sum_{n\text{ prime}} x^n$ is not a polynomial.