A normed linear space $(X, \lVert \cdot \lVert)$ can form a metric space induced by the norm and thereby a topological vector space. Thus the analytical concepts can be taken along with $X$. The axiomatic properties in the defintion of $\lVert \cdot \lVert$ is enough to say the topology induced by the same will be compatible (continuity of 'vector addition' map and 'scalar multiplication' map) with the linear structure.
My Doubt: What is the actual role of sub-multiplicative property ($\lVert AB \lVert\leq \lVert A \lVert\lVert B \lVert$) and $\lVert I \lVert \geq 1$ in the definition of 'matrix norm' ? Are they again compatibility conditions?