Operatornorm Definition

Question

Let $X$ and $Y$ be normed spaces, and $T\colon X\rightarrow Y$ be linear. Then the following is equivalent:

• $T$ is continuous
• There is an $M\ge 0$ such that $\| Tx\|\le M\|x\|\ \forall x\in X$

We then define $\|T\| :=\inf\{M\ge 0\colon \|Tx\|\le M\|x\|, \forall x\in X\}$

What would happen, if I use $\min$ instead of $\inf$? I suppose, they are not always equal. Which one is commonly used in the literature of functional analysis?

Answer 1

What would happen, if I use $\min$ instead of $\inf$?

Nothing, really. The given set is closed, and hence achieves its infimum.

Which one is commonly used in the literature of functional analysis?

Generally infimum, as one can define a (finite) infimum simply by proving a lower bound (in this case, $0$ is an obvious lower bound), whereas a minimum requires proof that the infimum also lies in the set. But again, logically, in this case, they are interchangeable.

Answer 2

The infimum here is same as minimum. The infimum is always attained.