# Quotient Banach space whose dual map sends the ball onto a given convex subset

Let $$X$$ be a Banach space and let $$A$$ be a closed, convex and balanced subset of $$B_{X^{*}}$$ (where $$B_{X^{*}}$$ denotes the closed unit ball of the dual $$X^{*}$$). Is there a closed subspace $$M$$ of $$X$$ such that $$Q^{*}_{M}$$ maps $$B_{(X/M)^{*}}$$ onto $$A$$, where $$Q_{M}:X\rightarrow X/M$$ is the quotient map?

• What happens when $X=\mathbb{R}^n$? Suppose $A$ is any closed convex balanced set with nonempty interior, other than the ball itself. If $M \ne 0$ then $Q_M^*$ has rank less than $n$ and so its image cannot cover $A$, and if $M=0$ then $Q_M^*$ is the identity map and it maps the ball to itself. – Nate Eldredge Feb 15 at 13:33
• Thanks, Nate. What happens if $X$ is infinite-dimensional? – Dongyang Chen Feb 15 at 14:36
• I was just thinking about that. More generally, the image of $Q_M^*$ will always equal the annihilator of $M$, right? If $M \ne 0$ this is a proper closed subspace. So take any $A$ which is not contained in a proper closed subspace (e.g. any $A$ with nonempty interior) and is not the ball, and I think that is a counterexample. – Nate Eldredge Feb 15 at 14:39
• Indeed, $Q^{*}_{M}B_{(X/M)^{*}}$ is equal to the closed unit ball of the annilator of $M$. If we take $A$ with nonempty interior, then $M=0$. – Dongyang Chen Feb 15 at 14:57