Let $T:X\to Y$ be a bounded operator between Banach spaces $X$ and $Y$. Assume that for any $\epsilon >0$ there is a finite-dimensional subspace $Y_\epsilon\subset Y$ so that $\|Q_\epsilon T\|<\epsilon$, where $Q_\epsilon : Y\to Y/Y_\epsilon$ is the quotient map. Define $$K=\{y\in Y_\epsilon : dist(y, T(B(0,1))<\epsilon\}$$ (here $B(0,1)$ indicates the unit ball in $X$)
Then the text I am reading says "$K$ is bounded and thus, as $dim Y_\epsilon <\infty$, totally bounded".
I do not see why, does someone has any hint? Thanks
Paola
Since dim $Y_\varepsilon < \infty$ and $K$ is bounded, $\overline{K}$ in $Y_\varepsilon$ is compact and we can take a finite subcover of $\cup_{y\in K} B(y,\delta)$, for all $\delta>0$.