Wikipedia defines a matrix norm as follows:
Let $\alpha$ be a scalar in $\Bbb K$ and $A,B$ be $m\times n$ matrices over the field $\Bbb K$. Then a matrix norm $\|\cdot\|$ must satisfy the following properties, if $|\cdot|$ is an absolute value on $\Bbb{K}$:$$\begin{align}&(1)\quad\|A\|\ge0\\&(2)\quad \|A\|=0\iff A=0_{m,n}\\&(3)\quad\|\alpha A\|=|\alpha|\cdot\|A\|\\&(4)\quad\|A+B\|\le\|A\|+\|B\|\end{align}$$
And as almost an afterthought, they mention that some authors require the following property, and that only some matrix norms satisfy the following property:
$$(5)\quad\|AB\|\le\|A\|\cdot\|B\|$$
Which seems to be optional.
Now it has already been proved on this site that any eigenvalue of $A$ is lesser than $A$'s matrix norm, if you take the optional requirement $(5)$ on matrix norms. This is used to justify the Cauchy Integral Formula for matrix functions, which is an interest of mine at the moment, and as I am taking notes I would love for those notes to be correct. I try to keep things as abstract as possible, so I wonder if the statement:
$$\lambda\le\|A\|$$
Holds true in general, if you drop requirement $(5)$ and rely only on requirements $1-4$ and standard linear algebra - abstractly! A particular proof for a particular vector space is not what I am looking for. I have not been able to make any headway at all, and do not even know if the end goal is possible.
No, it doesn't hold true, because without $(5)$, one could always consider a new norm $\| \cdot\|_* := \alpha \| \cdot \|$ for $\alpha > 0$. By making $\alpha$ sufficiently small, we can make $\|A\|_*$ smaller than $\lambda$ (given eigenvalues don't depend on the choice of norm).