Let A $\in R^{m\times n}$. What dose it mean when the book then write A : $R^{n} \rightarrow R^{m}$?
I try to understand this inequality: $\frac{1}{\sqrt m} \|\mathbf{A}\|_1 \le \|\mathbf{A}\|_2 \le \sqrt n \|\mathbf{A}\|_1$
I do not understand why is should be $\sqrt n$ and $\sqrt m$ on the other side... so I try to understand the calculation here: Inequality between 2 norm and 1 norm of a matrix
They say,
We have this, $\mathbf{A} : R^{n} \rightarrow R^{m}$
I understand why we get square root of a constant, the length of x vector in the definition of the norm, ut when should the constant be $\sqrt n$ and $\sqrt m$?
In the linked proof, the constants arise from these two specific steps: $$\Vert A\Vert_1\Vert x\Vert_1 \le \sqrt n \Vert A\Vert_1\Vert x\Vert_2$$ and $$\Vert Ax\Vert_1\le \sqrt m \Vert Ax\Vert_2.$$
In the first one, $x$ lives in $\Bbb R^n$, which is why $\Vert x\Vert_1 \le \sqrt n\Vert x\Vert_2$ with an $n$ in the constant.
In the second, $Ax$ is now the vector we consider, and it lives in $\Bbb R^m$, which is why the constant uses $m$ this time.