Question: Let a Banach space $B=M \oplus N$ where $M$ and $N$ are closed linear subspaces of $B$.

If $z=x+y$ is the unique representation of $z$ in $B$ where $x$ and $y$ are from $M$ and $N$ respectively then define a new norm by $\|z\|'=\|x\|+\|y\|$.

**If $B'$ symbolises the linear space B equipped with this norm $\|\cdot\|'$ then show that $B'$ is a Banach space.**

I have got that $M$,$N$ are complete subspaces of $B$.How do I show that a Cauchy sequence in $B'$ converges?

Answer: 
It isn't too much work to show that $D$ is closed (it is, for instance, the inverse image of the closed set $\{0\}\subseteq \Bbb R$ for some suitable continuous function $\Bbb R^3\to\Bbb R$), and that it is bounded (none of the coordinates can exceed $3$ in absolute value). And for subsets of $\Bbb R^n$, [closed and bounded means compact](https://en.wikipedia.org/wiki/Heine%E2%80%93Borel_theorem).
Alt. hint: $\;\sqrt[3]{|x| \cdot |y| \cdot 2 |z|} \le \sqrt{\dfrac{x^2+y^2+4z^2}{3}}=\sqrt{2}\;$ by the [root-mean square vs. GM inequality](https://artofproblemsolving.com/wiki/index.php?title=Root-Mean_Square-Arithmetic_Mean-Geometric_Mean-Harmonic_mean_Inequality), and the equality holds iff $\,|x|=|y|=2|z|\,$.