I have recently studied about dual and double dual of a vector space $V$ in my Linear Algebra course. But, though I am able to feel what about $V^*$, I don't think I understand $V^{**}$ properly.
So, $V^*$ is a set of all linear functionals from $V$ to $\mathbb{F}$, wherein each functional (which is a vector in $V^*$) is like a linear transformation and hence can be represented using a $1\times n$ matrix if $V$ is $n$ dimensional.
$V^{**}$ is dual of $V^*$ and hence collection of all maps from $V^*$ to $\mathbb{F}$ and this collection forms a vector space, which I think is easily proved. Each vector in this double dual vector space is a linear transformation from $V^*$ to $\mathbb{F}$. So, in a sense I am associating a number with a function. This sounds really weird because this is nothing like I have ever seen I guess. Am I right in interpreting $V^{**}$ as a collection of linear transformations which associate a function (more precisely a functional) with a scalar?
Now, if I am not wrong until here, then in essence, members or vectors in double dual space are linear transformations from a $n$ dimensional vector space ($\because V^*$ is $n$ dimensional) to a one dimensional vector space ( which is the field $\mathbb{F}$ itself). Hence, I should be able to represent vectors in $V^{**}$ as $1\times n$ matrices. But I am unable to think how will I be constructing these matrices and what will they be operating on, since I am unable to wrap my head around this scenario wherein a matrix is operating on a function. I don't know whether this makes any sense...
Hence, I seek your help.
Thanks!
For finite dimensional cases a space one can build intuition for why a space and its duel are actually the same space and the double duel is the original space using matrix multiplication. To see why this is lets examine the product $a^Tx$ with $a$ and $x$ both real column vectors and $a^T$ the transpose of $a$, and so $a^Tx$ is just the usual dot product. By holding $a$ constant and allowing $x$ to vary we can interpret this as a linear functional since it maps vectors to elements of the base field and is easily shown to be linear. So $a$ defines a specific linear functional.
Now to consider all the linear functionals we instead just allow $a$ to vary and hold $x$ constant. The equation $a^Tx$ being the usual dot product satisfies $a^Tx=x^Ta$ so we can simply reverse the roles of $a$ and $x$. But what have we really done here? The names $a$ and $x$ were arbitrary but they're both vectors in vector spaces of the same dimension. This suggests that these vector spaces are isomorphic. If we do this process twice we're simply back to the form $a^Tx$ so the double duel must be the original space.