Let $(X,||.||_X)$ be a normed space, M,N two subspaces with norms $||.||_M,||.||_N$
The identity maps are cont. Now I can define the norm $||x||_{M+N}=inf\{||m||_M+||n||_N:m\in M, n\in N, x=m+n\}$
1)I already showed that $||x||_{M+N}=0$ iff $x=0$ and homogenity. But what about the traingle inequality ? I am not sure how to write $||x+y||_{M+N}$.Maybe $||x+y||_{M+N}=inf\{||m||_M+||n||_N:m\in M, n\in N, x+y=m+n\}$ ?
2) If $M,||.||_M$ and $N,||.||_N$ are complete (i.e $||x_n-x_m||<\epsilon$ and $||y_n-y_m||<\epsilon$) how can I follow that $(M+N,||.||_{M+N})$ is complete ?