Let $T:X\longrightarrow Y$ be a continuous linear operator , $X \;,\;Y$ normed spaces with
Give an example of a continuous linear operator such that the supremum not reached
$$\|T(x)\|<\|T\|\;\; ,\;\; \|x\|\le 1$$
If the space $X$ is finite dimensional the unit ball is compact then the supremum is reached
Any hints would be appreciated.