Let $\phi: S^{k-1}\to E\setminus Q$ be a continuous map, here

- $E$ is an infinite dimensional Hilbert space
- $Q$ is a compact set in $E$

I need extend $\phi$ to $B^{k}$ such that $\phi: B^{k}\to E\setminus Q$ is still continuous.

If $Q := \{u\in E:\|u\|_E=1\}\setminus E_{k+1}~ \text{where}~ E_{k+1} ~\text{is any $k+1$-dimension subspace of} ~E$ (not compact), that's Ok, how about other $Q$?