Notation question: $T[X]=\bigcup\limits_{n=1}^{\infty} nT[B]$.

Question

I'm reading the following text: https://www.ucl.ac.uk/~ucahad0/3103_handout_7.pdf

I am reading the proof of Lemma 7.17 on page 16.

$B$ is the open unit ball in X. What I am wondering is what is $nT[B]$? A follow up question would be given $T \in B(X,Y)$ for Banach spaces $X$ and $Y$, how would $T[X]=\bigcup\limits_{n=1}^{\infty} nT[B]$.

My guess is that we're multiplying the radius of B by n.

Answer 1

This notation means that we multiply each element of the set $T[B]$ by $n$: $nT[B]=\{nT(x)\,|\,x\in B\}$. From this you should be able to deduce why $T[X]=\bigcup_{n=1}^{\infty} n T[B]$.