How to show that the set of square matrices $R^{n \times n}$ is complete under the operator norm $\|A\| = \sup\limits_{\|x\|\leq 1} \|Ax\|$

Question

I want to show that the set of square matrices $R^{n \times n}$ is a Banach algebra with property $\|AB\| \leq \|A\|\|B\|$.

I have already showed that $R^{n \times n}$ is a linear space and it is a normed space with the operator norm $\|A\| = \sup\limits_{\|x\|\leq 1} \|Ax\|$, all that is left is to show that the space is complete.

I have never seen a cauchy series of matrices...can someone please show how I should proceed from here?

Answer 1

The idea is that on finite dimensional vector spaces, all norms are equivalent. In particular you can find $\alpha >0$ with $$\Vert A \Vert_\infty \le \alpha \Vert A \Vert$$ for all $A \in \mathbb R^{n \times n}$. Where $$\Vert A \Vert_\infty = \sup\limits_{1 \le i,j \le n} \vert a_{i,j} \vert.$$ And it is easy to prove that $\mathbb R^{n \times n}$ is complete for the norm $\Vert \cdot \Vert_\infty$.