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There are two things which I'm unsure about with this question which I believe are stopping me from proceeding to find a solution.

1) Since we're told that the map given is a valid norm in $\mathbb{R}^m$ and that it is invertible we know that $W$ must be a $m \times m$ matrix right?

2) What does the norm with a matrix as a subscript mean? Does it mean the induced norm given by the dimensions of a matrix (E.g. A is $2 \times 3$ so $\|x\|_A = \|x\|_{2,3}$) or something else?

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    $\| x \|_W$ is being defined by the right side: by definition $\| x \|_W = \| Wx \|_2$ (where the subscript $2$ just means you're taking the Euclidean norm on $\mathbb{R}^m$). And yes $W$ must be $m \times m$ in this context. – 2017-02-24
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    Yes, $W$ is an $m$ by $m$ matrix. Otherwise, $Wx$ could not be considered an element of $\mathbb{R}^m$. $||x||_W$ is literally defined to be $||Wx||_2$. – 2017-02-24
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    Thanks, I missed that initially. I see now that $\|x\|_W$ has literally just been defined to be $\|Wx\|_2$. – 2017-02-24
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    I like to read matrix subscripts as "the norm of $x$ with respect to the matrix $A$," for example. Here it appears that the measurement taken on $x$ is simply the length of the transformed vector from the calculation $W\cdot x.$ – 2017-02-24

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If $W$ is injective then $\|x\|_W = \|Wx\|_2$ is a norm.

$\|tx\|_W = \|W (tx)\|_2 = |t| \|Wx\|_2 = |t| \|x\|_W$.

$\|x+y\|_W = \|W(x+y)\|_2 \le \|Wx\|_2 + \|Wy\|_2 = \|x\|_W + \|y\|_W$.

Suppose $\|x\|_W = 0$, then $\|Wx\|_2 = 0$ and hence $Wx = 0$. Since $W$ is presume injective we have $x=0$.

Note, it does not have to be the Euclidean norm $\| \cdot \|_2$, any norm will do.

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    You are welcome! Glad to be able to help. – 2017-02-24