The definition I was given is that for a bounded linear transformation $T$, $||T||=\inf\{K\in\mathbb{R}:\forall x, ||T(x)||\leq K||x||\}$. After messing around with suprema and infima for a while, I've been able to show that $||T||=\sup\{\frac{||T(x)||}{||x||}:x\neq 0\}$ and thus $||T||=\min\{K\in\mathbb{R}:\forall x, ||T(x)||\leq K||x||\}$. I was curious, however, if it is also true that $||T||=\max\{\frac{||T(x)||}{||x||}:x\neq 0\}$. In other words, must there exist $x$ such that $||T(x)||=||T||\cdot||x||$? If not, what if assume the vector spaces in question are finite dimensional? What about the case $T:\mathbb{R}^n\to\mathbb{R}^m$ for some $n, m\in\mathbb{Z}^+$?
Edit: I am being told that it is true for finite dimensional spaces, but not necessarily for infinite dimensional spaces. If someone could give me an explanation or proof for why it's true for the finite dimensional case, I would greatly appreciate it. Also, a counterexample in the infinite dimensional case would be appreciated as well.