Can anyone explain this isometry to me? $T: (\mathbb R^n,\|\cdot\|_1)^*\to (\mathbb R^n,\|\cdot\|_{\infty})$, $ T(x)(y) = \sum_{i=1}^n x_i y_i$ Can anyone explain this isometry to me?
$$T: (\mathbb R^n,\|\cdot\|_1)^*\to (\mathbb R^n,\|\cdot\|_{\infty}),\qquad  T(x)(y) = \sum_{i=1}^n x_i y_i$$
I don't get what the domain and image of $T$ are.
Any help would be appreciated!
 A: Complementing the very well detailed answer of Zev Chonoles, I think you've got it in the opposite direction. $T$ as is defined in the title is the mapping $x\mapsto T(x)=x^T$, such that $T(x)(y)=x^Ty$ is a linear transformation of the vector $y$. This means that the domain of $T$ is $(\mathbb R^n, \Vert \cdot \Vert_\infty)$ and its codomain is the dual space of linear transformations in $(\mathbb R^n, \Vert \cdot \Vert_1)$.
If you are wondering where the $1$ and $\infty$ came from or why the map is an isometry, you have to take a closer look at the norm of the space $(\mathbb R^n, \Vert \cdot \Vert_1)^*$. As Zev said, the norm of a linear transformation $ \ell$ of a vector $y\in (\mathbb R^n, \Vert\cdot \Vert_1)$ is
$$\Vert \ell \Vert= \sup_{\Vert y\Vert_1=1}|\ell(y)|.$$
This means that if $\ell$ is the image under $T$ of some vector $x$, then 
$$\Vert \ell \Vert=\Vert T(x) \Vert= \sup_{\Vert y\Vert_1=1}|T(x)(y)|= \sup_{\Vert y\Vert_1=1}|x^Ty|.$$
It turns out that the expression $\sup_{\Vert y\Vert_1=1}|x^Ty|$ is exactly equal to $\Vert x \Vert_\infty$! (You can try to prove this yourself, it's not so hard.) That is why the mapping is an isometry only when the domain is taken to be euclidean space with the $\Vert \cdot \Vert_\infty$ norm.
One speaks of this phenomenon as $\Vert\cdot\Vert_1$ and $\Vert\cdot\Vert_\infty$ norms being dual of each other. 
You can also try to prove the duality the other way round, that is $\sup_{\Vert y\Vert_\infty=1}|x^Ty|=\Vert x \Vert_1.$
A: A pair $(V,\|\cdot\|)$ denotes  a vector space $V$ over  $\mathbb{R}$, together with a norm function $\|\cdot\|:V\to\mathbb{R}$. 
Thus, $(\mathbb{R}^n,\|\cdot\|_1)$ and $(\mathbb{R}^n,\|\cdot\|_\infty)$ mean "the vector space $\mathbb{R}^n$ equipped with the $L^1$ norm", and "the vector space $\mathbb{R}^n$ equipped with the $L^\infty$ norm", respectively. You can take a look at this Wikipedia article for a discussion of $L^p$ norms.
When we have a normed vector space $(V,\|\cdot\|)$, its dual is defined to be the vector space$$V^*=\{\text{continuous linear functions }f:V\to\mathbb{R}\}$$
together with the norm function $\|f\|=\sup\limits_{\|x\|=1}|f(x)|$.
The function $T$ is a continuous linear function from (the dual of ($\mathbb{R}^n$ under the $L^1$ norm)) to      ($\mathbb{R}^n$ under the $L^\infty$ norm).
