Banach spaces and quotient space

Question

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?

Answer 1

Though solution is very simple I'll give a refernece to the book where it is solved. See page 35 exircise 1.27 of the book Banach space theory. The basis for linear and non-linear analysis. M. Fabian, P. Habala, P. Hajek, V. Montesinos, V. Zizler. This is a must read book for Banach space theory learners.

If you drop requirement for $M$ and $X/M$ both to be Banach, then result is not true. Consider $X=c_{00}(\mathbb{N})\oplus \ell_2(\mathbb{N})$, and set $M=c_{00}(\mathbb{N})$. $X$ is not complete though $X/M\cong\ell_2(\mathbb{N})$ is complete.

Answer 2

Let $(x_n)$ be a Cauchy sequence in $X$.

• Show that $(x_n+M)$ is a Cauchy sequence in $X/M$. Therefore, $x_n+M \to x+M$ in $X/M$ for some $x \in X$.
• Let $m_n \in M$ such that $||x_n-x+M|| \leq ||x_n-x+m_n|| \leq ||x_n-x+M||+ \epsilon$. Show that $(m_n)$ is a Cauchy sequence in $M$. Therefore, there exists $m \in M$ such that $m_n \to m$.
• Show that $x_n \to x+m$.

Remark: The norm on $X/M$ I used is $|| \cdot || : x \mapsto \inf\limits_{m \in M} ||x+m||$.