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If $(X,||\cdot||)$ is a Banach space and $K$ is a closed subspace, then $X/K$ becomes a Banach space when endowed with the quotient norm. Is it also true that if $(X,||\cdot||)$ is a normed space and $K$ is a complete subspace with $X/K$ complete, then $X$ is complete?

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    This is a good question for beginners to contemplate! In fact, lots of interesting questions have the form: "If $K$ and $X/K$ both have property $P$, then must $X$ itself have property $P$?" – GEdgar Jan 26 '22 at 06:09
  • See also https://math.stackexchange.com/questions/1937284/the-space-is-complete-iff-the-subspace-and-the-quotient-space-is-complete – F_M_ Jan 26 '22 at 06:16

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I propose the following proof:

Let $(x_n)_{n \geq 1}$ be a Cauchy sequence in $X$. Then $x_n \to x$ for some $x \in \overline{X}$. Notice that the sequence $(x_n + K)_{n \geq 1}$ in $X/K$ is Cauchy, and therefore converges to $x' + K \in X/K$. Since $K$ is complete in $X$, $K$ is also complete in $\overline{X}$, and is therefore closed in $\overline{X}$, allowing us to consider the quotient space $\overline{X}/K$ equipped with the suitable norm. Since $x_n \to x$, we also have that $x_n + K \to x+K$, and so $x + K = x' + K$. Thus, $x-x' \in K$, allowing us to conclude that $x \in X$.