Let $X$ be a normed vector space, $M$ a closed subspace of $X$ such that $M$ and $X/M$ are Banach spaces.
Any hint to prove that $X$ must be a Banach space?
banach-spacesfunctional-analysisnormed-spaces
Let $X$ be a normed vector space, $M$ a closed subspace of $X$ such that $M$ and $X/M$ are Banach spaces.
Any hint to prove that $X$ must be a Banach space?
Best Answer
Let $(x_n)$ be a Cauchy sequence in $X$.
Remark: The norm on $X/M$ I used is $|| \cdot || : x \mapsto \inf\limits_{m \in M} ||x+m||$.