Example of a non-closed subspace such that the quotient is not a Banach space

Question

As I've learnt recently in my Functional Analysis course, it is well known that if $X$ is a normed Banach space and $Y$ is a closed subspace, then the quotient $X/Y$ is a Banach space (e.g. How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$?)

However, I've been trying to find an explicit example of a normed Banach space $X$ and a non-closed subspace $Y$ such that $X/Y$ is not a Banach space, but I haven't come to something yet.

Can you help me to find such spaces? It would be great to read your answers, there may be some interesting examples out there.

Answer 1

Let $X$ be a Banach space and let $\alpha\colon X\longrightarrow\mathbb R$ be a discontinuous linear form. Then $\ker\alpha$ is a dense subspace of $X$. And $X/\ker\alpha$ is not a Banach space simply because the norm$$\|x+\ker\alpha\|=\inf\{\|x+y\|\,|\,y\in\ker\alpha\}$$is not a norm. In fact, it follows from the density of $\ker\alpha$ that$$(\forall x\in X):\|x+\ker\alpha\|=0.$$

Answer 2

You don't need to "find" examples, any non-closed subspace gives an example.

Say $Y$ is a subspace of $X$ and $Y$ is not closed. Say $y_n\in Y$, $y_n\to x$ and $x\notin Y$. Then $$||x+Y||=0$$although $x+Y\ne0$.