We consider the two fields: $\mathbb{C}^m$ and $\mathbb{C}^n$ with the norm $||.||$.
From this, we can construct the induced norm for operators $T \in \mathcal{L}(\mathbb{C}^m,\mathbb{C}^n)$
$$||T|| = Sup(||T(x)||, ||x||=1)$$
This is explained in https://en.wikipedia.org/wiki/Matrix_norm#Matrix_norms_induced_by_vector_norms
My question:
What "disturbs" me in the fact that the same notation is used for the norm of the operator and the norm of the vectors in $\mathbb{C}^m$ or $\mathbb{C}^n$. For me, from this definition they are "in principle" two different objects.
I wondered if the reason of this notation is because $\mathcal{L}(\mathbb{C}^m,\mathbb{C}^n) \cong \mathbb{C}^{m*n}$ so that actually the induced norm coincides with the norm of $\mathbb{C}^{m*n}$ ?
Note: I am new in learning this, I would like simple explanation not involving too much extra knowledge :)
The induced norm is usually fairly different that the norm inducing it. If you use the Euclidean norm in $\mathbb C^n$, i.e. $\|x\|=\left(\sum_j|x_j|^2\right)^{1/2}$, then the operator norm for $T\in L(\mathbb C^n,\mathbb C^m)$ is given by the largest singular value of $T$, and this almost never agrees with the Euclidean norm of $T$ .
There is nothing disturbing about using the same notation. Most of the time you are using a single norm per context, so you'll have one norm for vectors and one for operators; there is little chance of confusing them.