Let $B=-B$ be a nowhere dense bounded closed convex set in the Hilbert space $\ell\_2$ such that the linear hull of $B$ is dense in $\ell\_2$.



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> **Question.** Is there a non-compact linear bounded operator $T:\ell\_2\to \ell\_2$ such that $T(B)$ is compact?
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Compact images of nowhere dense closed convex sets in a Hilbert space
Your revised assumption is that the norm (rather than semi-norm after the revision) on $\ell\_2$ given by sup-ing against vectors in $B$ is not equivalent to the usual norm on any finite codimensional subspace. So there is an ON sequence $x\_n$ in $\ell\_2$ s.t. $\sup \{\langle x\_n,b\rangle : b\in B \} \to 0$ as $n\to \infty$. Take for $T$ the orthogonal projection onto the closed span of the $x\_n$.


