I am considering a Hilbert space $X$, endowed with its weak topology. I need to work with a subset (but not a subspace) $S$ of $X$.
However I need to endow $S$ with the subspace topology (so $U$ is weakly open in $S$ if and only if there is a weakly open set $V$ in $X$ such that $U=S \cap V$).
I have quite a lot to write about this, and I am worried that the reader will misunderstand when I speak about $S$ with the subspace topology, thinking that I have misunderstood $S$ to be a vector subpspace. I am also worried that when I say subspace topology the reader will not realise it's the one induced from $X$ with its weak topology.
Is there a better way to phrase myself? Perhaps another way to refer to the subspace topology as to not confuse $S$ with a vector subspace of $X$? I am struggling to find a concise and clear way to phrase it.