Relation between boundedness of a function and an operator

Question

I've a little confusion about a bounded function and a bounded operator.

I know that , "Every operator is a function but not conversely". For an operator the co-domain set should must be a linear space but for a function co-domain is any arbitrary set.

Now two definitions :

" A function $f:A\to B$ is said to be bounded if there exists a positive constant $M$ such that $|f(x)|\le M$ , for all $x\in A$."

"An operator $T:X\to Y$ is said to be bounded if there exists a positive constant $M$ such that $\lVert T(x)\rVert\le M\lVert x\rVert$ , for all $x\in X$."

My Question : As an operator is a function so how I relate the above two definitions ? That means how I can deduce the 1st definition from the 2nd one ?

Answer 1

The ONLY linear operator which is a bounded function is the identically zero one, for if $Tx=y\ne 0$, for some $x$, then $$ \|T(nx)\|=n\|y\|\to \infty, \quad\text{as}\,\,\,n\to\infty. $$ Hence, if $T\not\equiv 0$, then $T$ is not a bounded function.

A bounded linear operator is one which is bounded in any bounded set.

This is equivalent to being continuous. $T$ is bounded iff $T$ is continuous iff $T$ is continuous at $0$.

Answer 2

You can't, unless you restrict the operator to a bounded domain like a ball of finite radius. The term bounded has, for historical reasons, a special meaning for (linear) operators, which is de facto different from what you use for general functions. In other words, a linear operator which is bounded will not be bounded in the other sense, unless it is the zero operator.