I’m taking a look at Chapter 5 of A Course in Mathematics for Students of Physics (vol. 1) by Bamberg & Sternberg; on p. 181, they assert:
If $A$ is an invertible linear transformation from $V$ to $W$ (so, in particular, $V$ and $W$ have the same dimension), then we can find constants $k_1$ and $k_2 > 0$ so that $\|A\mathbf{v}\| \leq k_1 \|\mathbf{v}\|$ and $\|A^{-1}\mathbf{w}\| \leq k_2 \|\mathbf{w}\|$
Unfortunately, they provide no proof that such constants $k_1$ and $k_2$ exist. Any help in understanding what guarantees their existence would be much appreciated.
I further conjecture that the inequality $\|A\mathbf{v}\| \leq k \|\mathbf{v}\|$ holds for some $k > 0$ even if $A$ isn’t invertible; any help in proving this would also be appreciated.
If $A$ is any linear map between finite dimensional spaces and $(e_i)$ is a basis for the domain then $\|A \sum\limits_{k=1}^{n} a_ie_i\|\leq \sum\limits_{k=1}^{n} |a_i| \|Ae_i\|\leq C \sum\limits_{k=1}^{n} |a_i|$ where $C$ is the maximum of the numbers $\|Ae_i\|$. The proof of the fact that $\sum\limits_{k=1}^{n} |a_i| \leq \ D \|\sum\limits_{k=1}^{n} a_ie_i\|$ depends on what norm you are using. There is a theorem which says all norms are equivalent on finite dimensional spaces. For the Euclidean norm we can take $(e_i)$ to be an orthonormal basis in which case we have above inequality holds with $D=\sqrt n$.