Example of unbounded operators defined on the entire complete space

Question

In functional analysis I have come across unbounded operators defined in the following way: $$A:\mathcal D(A)\subseteq X\to Y,$$ where $X,Y$ are Hilbert spaces and $A$ is defined only for $x\in\mathcal{D}(A)$. My question is does there exist an unbounded which is defined on the whole of $X$? If yes, can we construct such an example?

Answer 1

On every infinite dimensional normed linear space there is a linear functional which is not continuous. Such a functional is defined everywhere but it is unbounded.

Hints for construction of an unbounded operator: There exists a Hamel basis $A$ consisting of unit vectors and $A$ is an infinite set. Let $(x_n)$ be a sequence in $A$. Let $f(x_n)=n$ and $f(x)=0$ if $x \in A$ with $x \neq x_n$ for all $n$. We can now extend $f$ to the whole space by liearity to gegt an unbounded operator.

Answer 2

In the case where $A$ is a closed operator (which covers a large class in applications), if $\mathcal{D}(A)=X$ then by the closed graph theorem $A$ will be bounded.

Then you can try with non closed operators.