I have been thinking about the definition of a bounded operator and the following concern has arisen.
The definition of an linear bounded operator $A:X\to Y$ ($X, Y$ Banach spaces) is $$|Ax|_{Y}\leq C|x|_{X},\quad \forall x\in X$$
for some $C>0$.
Question 1. An linear operator $A:D(A)\subset X\to Y$ with $X, Y$ Banach spaces, is bounded if
$$|Ax|_{Y}\leq C|x|_{X},\quad \forall x\in D(A)\text{ Right?}$$
Question 2. If $D(A)$ is dense in $X$, then $$|Ax|_{Y}\leq C|x|_{X},\quad \forall x\in X?$$
That is, the inequality extends to the entire space $X$ by density.
Best Answer
Yes to both questions. The first one is just the definition of a bounded operator. As for the second one: let $x \in X $. Since the desired inequality is true if $x \in D(A)$ (by assumption), we can suppose without loss of generality that $x \notin D(A)$. Since $D(A)$ is dense in $X$, there exists a sequence $(x_n)_{n \in \mathbb{N}}$ such that $$\lim_{n \to \infty}x_n = x$$ And since linear bounded operators between normed spaces are continuous and limits preserve $\leq$ inequalities, we have that:
$$\|Ax \| = \lim_{n \to \infty} \| Ax_n \| \leq C \lim_{n \to \infty} \|x_n \| = C \|x \|$$
where we also used that $x \mapsto \|x\|$ is continuous.