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I am given a space $M:=\left\{\begin{pmatrix}x&y\\0&x\end{pmatrix}: x,y\in \mathbb{C}\right\}$ with the usual addition and multiplication of matrices. I want to show that $M$ is a Banach space under the norm $$\left\|\begin{pmatrix}x&y\\0&x\end{pmatrix}\right\|:=|x|+|y|.$$ I have already shown that $\|\cdot\|$ defines a norm on $M$. I want to prove that $M$ is complete. Usually, I pick a sequence $\{f_n\}_{n=1}^\infty$ in $M$ and then show that $\lim_{n\to \infty}\limits\|f_n-f\|=0$ for some $f\in M$. However, these being matrices, I failed to see how such sequences look like. I failed to prove that $M$ is complete.

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  • $\begingroup$ Your space $M$ is essentially (read: isometrically isomorphic to) $\mathbb R^2$ equipped with the norm $\|(x,y)\| = |x| + |y|$. $\endgroup$ – gerw Jun 5 '18 at 14:53
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It is a finite dimensional vector space, and every norm defined on a finite dimensional vector space is complete.

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