Wikipedia tells me a little about it. Following the wiki-link treasure hunt leads me to topics such as "p-norms on finite dimensional vector spaces".
Which makes me want to ask: what's a good textbook or reference where I can learn more about the p-norms on finite dimension vector spaces? I need to learn enough in order to answer the following questions given a proposed induced norm:
Why is it that in general $\|Ax\| \leq \|A\|\|x\|$ (where $\|A\|$ is the matrix norm, and $\|x\|$ is the vector norm)?
is the proposed induced norm a norm? (I think I know how to check for this now, as the Wikipedia page tells me what requirements a norm must satisfy to be considered one.)
given a proposed induced norm on a matrix, what is the corresponding vector norm?
Could you recommend a good resource to learn (notes, or textbook, or whatever)?
A textbook that is self-contained would be particularly valuable, but if I had to choose between an insightful text that was not self-contained, and a needlessly opaque text that was self-contained, I would pick the latter.
$\|Ax\|\leq \|A\|\|x\|$ because $\|A\|$ is essentially defined to be the smallest number $C$ for which $\|Ax\|\leq C\|x\|$ holds. For $x\neq0$ this directly follows from the usual definition and the definition of supremum. Just divide by $\|x\|$ and take the supremums on both sides.
The induced norm is always a matrix norm, the axioms follow from the corresponding properties for the vector norm. Induced norms have extra properties like $\|I\|=1$ and $\|AB\|\leq \|A\|\|B\|$, so the converse is not true.
Not every matrix norm is induced, so for some of them there is no corresponding vector norm. Vector norms that differ by a constant multiple induce the same matrix norm, as the definition implies by inspection. If you know in advance that a given norm is induced, the corresponding vector norm can be recovered up to a constant multiple by taking norms of rank $1$ matrices, i.e. the ones with the range spanned by a single vector.
Lancaster and Tismenetsky's Theory of Matrices has a chapter on matrix and vector norms that goes into the details of the relationship between them, and $p$-norms are used in examples.